Implementation of Triangulated Quadtree Sequencing for a Global Relief Data Structure  

By Jaime A. Lugo  

 

 

Submitted in partial fulfillment of the requirements for the degree of Master of Arts in Geography, Hunter College, The City University of New York July 18, 1994

ABSTRACT — This thesis reviews and assesses the implementation of a triangulated data structure for displaying global datasets onto three-dimensional global relief maps based on the eight individual facets of an octahedral projection. 

    

  

 

 

    

7 '1B/q1 Date

Implementation Of Triangulated Quadtree Sequencing For A Global Relief Data Structure

by

Jaime A. Lugo

Submitted in partial fulfillment of the requirements for the degree of

Master of Arts in Geography, Hunter College, The City University of New York

1994

Thesis Sponsor:

Signature

Keith C. Clarke Professor of Geography

LL ?~ \ Sean Ahearn Associate Professor of Geography

HUNTER COLLEGE OF THE CITY UNIVERSITY OF NE\·1 YORK

. THESIS APPROVAL FORM

MR. MS •. Jaime A. Lugo

Mankato State Uru.versity ·Department of Geography, Annstrong Hall Rm. #7, Mankato, MN 56001

(Address after graduation)

a candidate for the degree of

Master of hts ----------has satisfactorily completed a thesis* entitled:

"Implementatio;,_ of Triangulated Quadtree Sequencing for a Global Relief

Data Structure."

This work has been approved by the graduate program in

__ Geogr_....._a-"p __ h __ y _________ •

July 21, 1994 loatel ·

Graduate Advisor:

~~EDUJfi. I GEOGRAPH~ (.Signature)

Sara IYT..cLafferty, Ph.D. (Typed name)

Accepted in fulfillment of the thesis requirement for September 1994 (month/Year)

£ '\ \ /,-? .. ~/ >vi/I z'.'~SJ',,VZ.,-

(date) (Dean for Graduate Studies)

*If the work submitted is not a thesis, please cross out •thesis" and substitute. the name of the appropriate equivalent.

Implementation Of Triangulated Quadtree Sequencing For A Global Relief Data Structure

by

Mr. Jaime A. Lugo

Submitted in partial fulfillment of the requirements for the degree of

Master of Arts in Geography, Hunter College, The City University of New York

1994

ACKNOWLEDGEMENTS

Without encouragement from the following people, I may have ended my career in geography a long time ago:

To Professors Jeffrey Osleeb, Saul Cohen and DeWitt Davis, who discovered my potential and gave me the opportunity to continue on to graduate school.

To Professor Keith C. Clarke, who, as advisor and mentor, went beyond the call of duty to help with this thesis. He helped to project the World Database-I into a triangular projection he named the Interrupted Collignon Equal Area Projection, or "Clarke's Butterfly Projection." Thanks for helping to overcome my fear of C programming.

To my dear mother, Pilar Cintron, and Humberto Figueroa (my surrogate father), who always kept me in their prayers and encouraged me to go on. They know how much I love and missed being with them.

To the memory of Osnaldo Lugo, my father, who died before fulfilling his dream of being the first in my family to obtain a Ph.D. I promise to do my best and fulfill this dream for him.

Finally, to all the people I wished to include but have no room for (the list is too long-unless I wanted this thesis to become my dissertation).

"As a young man, my fondest dream was to become a geographer. However, while working in the Customs office, I thought deeply about the matter and concluded that it was far too difficult a subject. I then turned to physics as a substitute."

iv

- Albert Einstein -(unpublished letter)

TABLE OF CONTENTS

Acknowledgements iv Table of Contents v Table and Figures vi Abstract 1

Chapter 1 - Introduction 1.1 Justification of and need for multilateral structures 2 1.2 Problem statement 3 1.3 Limitations of the smdy 4 1.4 Thesis implementation 5

Chapter 2 - Literature Review 2.1 Platonic solids and the early literature 6 2.2 Square-to-hexagon transformation algorithms 10 2.3 Quadtrees for storage of global datasets 13

2.3.a The B+-Tree 14 2.3. b Morton sequencing / Two-dimensional run encoding 15 2.3.c The PM3' Quadtree 17

2.4 The Quaternary Triangular Mesh (QTM) 2.4.a: Introduction and significance 17 2.4.b: Triangulation and 2-bit addressing 18 2.4.c: Octahedral geometry and QlM resolution 19 2.4.d: Triangle Orientation after tessellation 20 2.4.e: QlM addressing methods 22 2.4.f: Concluding statements on the QlM 24

Chapter 3 - Triangular Tessellation and Quadtree Modifications 3.1 Triangular tessellation of the octahedron 25 3.2 The Triangulated Quadtree Structure (TQS) 26

Chapter 4 - Applied Map Projections 4.1 Geodesic concerns in choosing a map projection 29 4.2 The Collignon Projection 30 4.3 The Interrupted Collignon Equal Area Projection 32

Chapter 5 - Implementation and Tessellation of the Triangulated Quadtree Structure (TQS)

5.1 Introduction , 34 5.2 Recursive Tessellation for Determination of a TQS Address 34 5.3 Discussion of the TQS Addressing Method 39 5.4 Results 41

Chapter 6 - Future applications of the Triangulated Quadtree Structure 6.1 Conditions and data requirements 43 6.2 The application process 44 6.13 Future applications 46

V

Chapter 7 - Conclusion, Summary, and Importance 47

Selected References 49

Appendix A: C source code 53

Appendix B: Glossary of Geographic Terms 63

Appendix C: Information about the ETOP05 5-minute gridded elevation data 68

TABLES AND FIGURES

Table 2.1: Addresses and orders of QTM cells dominated by attractors, spiral form. 30

Table 5.1: Addresses and orders of QTM cells dominated by attractors, spiral form. 35

Figure 1.1: Octahedral skeleton showing a three-dimensional coordinate grid. 5

Figure 2.1: Characteristics of various decomposition structures. 6

Figure 2.2: Hexagonal vs. circular pixels. 8

Figure 2.3: Triangle orientations. 8

Figure 2.4: The square lattice A4 ,and the hexagonal lattice A6 , with origin (0). 10

Figure 2.5: Hexagons vs. squares ... which are better for drawing curves and circles? 11

Figure 2.6: Nearest Neighbor Transformation. 12

Figure 2.7: Results of a grid transformation (T1). 13

Figure 2.8: Monon sequence and number ordering. 15

Figure 2.9: A two-dimensional run encoded quadtree using Monon Sequencing. 16

Figure 2.10: Three-dimensional triangular tessellation levels. 18

Figure 2.11.a: A tessellated triangle produces more triangles. 19

Figure 2.11.b: Facet breakdown numbering. 19

vi

Figure 2.12.a: Octahedron face with four level-one triangles.

Figure 2.12.b: Triangular tessellation, tiling and locators.

Figure 2.13: Organizational suucture of triangles, showing the location of attractors and null-cells.

Figure 2.14: Spiral method for ordering QTM Attractors.

Figure 3.1: 2DRE/QTM/fQS Translation Diagrams.

Figure 4.1: Collignon projection.

Figure 4.2: The Interrupted Collignon projection.

Figure 5.1: Octant boundary for map coordinates.

Figure 5.2: Lattice transformations.

Figure 5.3: TQS addressing method.

Figure 5.4: Triangle rotations and translations.

Figure 5.5: Clockwise and counterclockwise rotation fonnulas.

Figure 5.6: The actual rotation and translation process.

Figure 6.1: Primary Intersect Nodes (PINs), midpoints, and primary angles.

vii

20

20

21

23

28

31

33

36

36

37

38

38

39

41

ABSTRACT

This thesis reviews and assesses the implementation of a three-dimensional, multilateral

data structure for displaying global datasets. First, an in-depth review of various geometric

figures, triangular tessellations and quadtree data structures is conducted. A global data

structure based on Morton sequencing, two-dimensional run encoding, and Dutton's

Quaternary Triangular Mesh is developed. This data structure, called a Triangulated

Quadtree Structure, or TQS, was developed for use along with a modified version of the

Collignon map projection, given a set of coordinates and the level of resolution desired. To

test the accuracy of the structure, the ETOP05 Global Relief Dataset was translated into a

triangulated coordinate matrix, georeferenced with TQS addresses, and projected into the

eight individual octants to serve as the facets of an octahedral projection. A future

application is also proposed by which the Triangulated Quadtree Structure can be

incorporated into a geographic information system for rendering global datasets onto a

volumetric, three-dimensional, graphically-oriented computer model. Both Morton, QTM

and TQS geocodes may be assigned to every point on the dataset, and linked to the

octahedral skeleton by use of the modified projection.

CHAPTER I Introduction

1.1 Justification and Need for Multilateral Structures

Why do pixels always have to be squares? Or, why tessellate with squares? Since

the introduction of computer graphics and its first applications to automated mapping,

cartographers have preferred to measure and model spatial data using square-based

algorithms and coordinate grids. Hence, most of today's research in hardware (and

software) design is applied using square-based structures such as quadtrees (Samet, 1990a).

A square-based structure, although excellent for planar geometry and two­

dimensional coordinate systems (such as the Cartesian Grid), is one of the least suitable

geometric models available to develop a data structure for the storage, linkage and

aggregation of spatial data in relation to a global three-dimensional database. A multilateral

(other than four-sided) data structure using a geometric object, such as a hexagon or a

triangle, is better suited to model a three-dimensional data (Wuthrich and Stucki, 1991). For

example, if a hexagon is subdivided into smaller and smaller hexagons, it will need a

relatively smaller number of subdivisions for it to resemble a circle. On the other hand, no

amount of subdivision will make a square resemble a circle, unless the subdivided squares

were piled in a circular pattern and the level of resolution was fine enough such that the

human eye could no longer discern squared edges along the circle's circumference.

Why spend the effort in building a planetary model using unconventional data

structures when the quality of resolution available for computer graphics today is so

outstanding (e.g., 64-bit color boards) that the difference in graphical imagery between

squares and hexagons becomes nil? To answer this argument, the proposed planetary relief

model is an exercise applied to a relatively new, complex and unexplored field in

geography, cartography or computer graphics, global data analysis and three-dimensional

rendering. The purpose of this research is to store a digital dataset in a triangular

2

hierarchical structure and link it directly onto a three-dimensional data structure. This data

structure, called a Triangulated Quadtree Structure (TQS), will be georeferenced to a

modified version of the Collignon Map Projection. A full glossary of geographic terms is

located in Appendix B.

1.2 Problem Statement

The main objective of this thesis is the implementation of a hierarchical indexing method

for assigning a unique geocode to a global-scale database - given a set of coordinates and

the level of resolution desired - and to propose a method by which it can be developed

into a powerful analytical tool for rendering global datasets onto a volumetric, three­

dimensional, graphically oriented computer model.

For the proposed model to succeed, the data structure must be tested using a global

database, which will be used to test the accuracy of the geocode addressing process by

geocoding, referencing and then linking its data to every coordinate point on a modified

map projection. To implement this concept, a unique geocode is assigned to every point on

the selected projection, which is then projected onto an octahedral skeleton. The

octahedron was the geometric solid selected because, once it is linked to the map

projection, it can be stored in a set of eight quadtrees, four for each hemisphere, with each

hemisphere divided into four quadrants. Each time a quadrant is subdivided (or tessellated),

the geocode address for each parent quadrant is modified and expanded to keep track of

every new point hierarchically created after every tessellation (points at higher resolutions

contain more complex geocodes).

Each section will help corroborate the use of a multilateral (rather than a square)

lattice structure to portray the Earth's geodesic nature. The choice of map projections, the

data structure used, the resolution available, and the graphical solution adopted are also

discussed in order to develop and implement a geocoding method specifically designed to

link a planetary relief dataset to the previously mentioned multilateral structure.

3

1.3 Limitations of the study

The ideas implemented in this thesis are only the beginning in an attempt to

implement a fully integrated three-dimensional model for the study of spatial distributions

at a global scale. A series of ideas, requirements and procedural steps for the completion of

the study is presented. This project was also limited by the small amount of previous

research available, especially in the field of nontraditional data structures for n-dimensional

spatial analysis (the oldest reference listed talks about the origins of FORTRAN in 1956

(Koffman and Friedman, 1990)).

Although multilateral algorithms had been developed in the late 1950s to early

1960s to improve the accuracy and precision of three-dimensional structures, early

computer technology allowed only square-based structures to develop above all other

multilateral based structures (Goodchild, 1988; Wuthrich and Stucki, 1991). This limitation

was imposed by outdated programming languages (such as FORTRAN) and the quality of

graphics technology available in the early computer era. The square grid will most probably

continue to be the norm in coordinate planetary models, since FORTRAN's programming

structure was originally designed to use a square rather than a multilateral grid1. This

structural weakness has led various geographers to consider FORTRAN as one of the

primary stumbling blocks in the development of multilateral data structures (Dutton,

1989b; Peuquet and Marble, 1990).

Finally, hardware limitations in producing hexagonal screens [hardware], and

applied algorithms [software] were two other primary barriers to multilateral modeling

(Wuthrich and Stucki, 1991). These limitations have been circumvented with the

appearance of programming languages such as C, in which bit manipulations are possible,

(e.g., shifting, masking, and bit addressing through structures (Lauzon, et al., 1985)).

1. FORTRAN (FORmula TRANslation): First used in 1957, Fortran was the first high-level programming language, primarily used for scientific computations. In the 1970s, it was modified to facilitate structured programs, which are needed for high-level graphics (Koffman and Friedman, 1990).

4

1.4 Implementation

To properly implement a three-dimensional planetary model georeferenced to a

map projection and linked to a global relief dataset, a specially designed hierarchical data

structure, called the Triangulated Quadtree Structure (TQS), was developed. This was

accomplished in three steps. First, the map projection was selected, modified and displayed

using a graphical interface. The TQS addressing scheme was then developed to fit the

chosen map projection. Third, the squared lattice structure of a global relief database named

ETOP05 was transformed into the map projection's triangular lattice. Finally, ETOP05

was geocoded and compressed (using the TQS scheme) to test the validity of the model.

This implementation has opened the door to a feasible application. Once every

ETOP05 data point was geocoded and linked to the selected projection, the Earth's surface

was ready to be graphically rendered using a display device in the shape of an octahedral

skeleton (Figure 1.1). To achieve this goal, a series of steps for the graphical visualization

of the three-dimensional model will be necessary, along with a satisfactory testing method

for the resulting data structure. For example, every vector throughout an octahedral

skeleton can be bisected, followed by the connection of all midpoints belonging to the

original facets. Elevation can be eventually portrayed by lowering or raising each midpoint,

based on the elevation for that TQS address. Such a process will result in the rendition of a

planetary model where relief (e.g. mountains, valleys, chasms) represents spatial trends

along the three-dimensional model's surface. Therefore, this process will allow the display

device to more closely resemble the Earth's surface with each increase in resolution.

Figure 1.1:

Octahedral skeleton showing a three­dimensional coordinate grid. Source: Dutton, 1989b.

5

CHAPTER2 Review of Pertinent Literature

2.1 Platonic Solids and the Early Literature

This section reviews the geometry of platonic solids available for developing a

discrete three-dimensional representation of the Earth. Of the geometric figures to be

described, the octahedron was selected because it retains its original shape after it has been

subdivided up to an infinite level of tessellations. In addition, each facet of the octahedron

forms a triangle where hexagons begin to form after only two subdivisions.

For a solid to serve as the basis for a space decomposition method, it should possess

two properties: 1) "The partition should be an infinitely repetitive pattern so that it can be

used for images of any size," and 2) the "partition should be infinitely decomposable into

increasingly finer patterns (i.e., higher resolution)" (Samet, 1990a, p.16). A multilateral

tessellation (decomposition) is regular if the component parts (or tiles) are composed of

regular polygons (i.e., squares), and isohedral if all the tiles forming the solid are

symmetrically equivalent. Also, a tessellation is limited when tiles are not similar, and

unlimited when the tessellation is infinitely decomposable (Samet, 1990a).

It is better to use a hexagonal rather than a square grid for the application of a global

coordinate system. "Rogers [1984] outlined that the best disposition of the points on a n-

Figure 2.1: Characteristlcs of various decomposition structures. Source: Samet, 1990a, pp. 16-19.

a) Square [4~: regular. unlimited tiliml. b) Triangle[4 ]: isohedr~ unlimited tiling. c) Isosceles 1riangle [4.8 ]: isohedral. unlimited tiling. d) 30-60 Rig~niangle [4.612): isohedral, unlimited tiling. e) Hexagon [6 ]: non isohedral, limited tiling.

6

dimensional plane could be reached if the points were [in the center of the hexagons within

a non-overlapping] hexagonal grid" (Wuthrich and Stucki, 1991). In addition, the distances

between hexagon centroids are identical, while the distances between square centroids vary

along the square diagonals. In other words, there are geometric figures - such as hexagons

and triangles - that provide a more accurate spatial centroid distribution, which in turn

allows better measurement of locational uncertainty.

Previous searches for alternative pixel configurations have typically begun with a

review of the five platonic solids - tetrahedron, hexahedron, octahedron, dodecahedron

and icosahedron- to determine their applicability to multilateral imaging (Peuquet, 1984;

Tobler and Chen, 1986; Laurini and Thompson, 1992). Although the icosahedron most

closely resembles a sphere over all the other platonic solids (the largest number of vertices,

edges and facets), the octahedron was regarded as a better selection because of its inherent

polar symmetry and its uniform triangular facets. A symmetrical octahedron is composed

of two tetrahedra, aligned such that they are hemispherically opposite to each other. Each

hemisphere is then composed of four triangular facets, or quadrants. As the octahedron is

sequentially subdivided (or tessellated, hexagons begin to form within its geometric

structure and the octahedron approaches the shape of a sphere (Laurini and Thompson,

1992). Although triangles are the base for the geometric structural matrix, these hexagons

can also serve as an alternate coordinate matrix used to compose the model onto a

hexagonal grid.

Rosenfeld (1974) had already proposed multilateral picture elements other than

squares for the graphical visualization of spherical objects and global coordinate systems.

Although he presented strong evidence of the advantages and feasibility of hexagonal­

based coordinate structures in computer graphics and spatial analysis, the current graphical

technologies were incapable of handling their algorithms and the huge amount of data

necessary to produce results. Thus, graphic developers continued to use square-based

picture elements (pixels) and compensated by improving the resolution of output devices.

7

The hexagon-based graphic revolution remained dormant for many years. Then,

when technology caught up with theory, researchers began to apply previously impossible

tasks - such as hexagon-based grids and other graphical applications - to geography.

These new technologies have many applications in multilateral coordinate research. If a

sphere is to be drawn, why not use circular pixels? Although it may seem that a true sphere

is more realistically represented by a circular pixel than by any other geometric shape, it

does not provide the best quality images because it leaves holes within its matrix (see

Figure 2.2a). Hexagons, on the other hand, prevent holes from corrupting the database by

forming hexagonal facets throughout the swface (Dutton, 1991) (see Figure 2.2b).

Figure 2.2: Hexagonal vs. Circular pixels.

In support of this idea, Wuthrich and Stucki (1991) claimed that only hexagons,

squares and triangles are fit to cover a coordinate plane. Of these, only squares and

hexagons are such that the centers of a regular polygon always lie directly in the center of

these geometric features, thus forming a simple lattice, while triangles need a more

complicated process - triangles, when tessellated, point in different directions (Figure

2.3).

Figure 2.3: Triangle Orientations.

apex

base

8

This statement is indirectly confinned by Ferenc Csillag (1993), although that was

not his intention. Csillag claimed that this difference in orientation was a strong point

against triangles in planar structuring methods. Csillag listed three advantages in using

squares for two-dimensional (planar) tessellations:

"The resulting decomposition [of a square] yields a partition (1) that is an infinitely repetitive pattern so that it can be used/or any size of space; (2) that is infinitely decomposable into increasingly finer patterns; and (3) whose tiles have similar orientation. Nonsquare decompositions of space do not meet all these criteria (e.g., hexagons do not meet property 2, equilateral triangles do not meet property 3)" (Csillag, 1993).

· Unlike in planar geometry, the fact that triangles do point in opposite directions

after decomposition is advantageous in solid, three-dimensional geometry by providing

orientation (Dutton, 1989b). In addition, when Csillag suggested that hexagons cannot be

decomposed into smaller hexagons he limited his scope to simple geometry instead of

applying a more complex vector geometry, such as the one suggested by Dutton in 1989b

and implemented in this project.

Other advantages hexagons have over squares include:

1. A tessellated octahedron is well suited for mapping a sphere on a two­dimensional surface, such as a map. This octahedron is tessellated into triangles, which then form hexagons (Dutton, 1984).

2. Algorithms exist that transform from square to hexagon lattices.

3. Unlike a square's two perpendicular axes, a hexagon possesses three directional axes, (Wuthrich & Stucki, 1991, Figures. 4-6).

4. Distance among hexagon centroids is identical.

The next section will expand upon this discussion by reviewing a set of algorithms

where, based on the work of Rosenfeld (1974), Li-De Wu (1982), S. Pham (1986, 1987),

Wuthrich and Stucki (1991) and others concluded that hexagons can produce straight lines

and curves as well as - and in many instances better than - squares.

9

2.2 Square-to-Hexagon Transformation Algorithms

A review of various terms and algorithms - ranging from geometry to computer

graphics and remote sensing - is needed to understand mathematical algorithms which

transform between square-based and hexagon-based matrices, such as those by Klette

(1985), Wuthrich & Stucki (1991), and Hedley (1992). Although hexagons have more

facets than squares, making them appear more complicated to store and portray, hexagons

are no more difficult to handle than squares "because the basic mathematical model for both

grids is the same" (Wuthrich and Stucki, 1991, p. 338). To show the process by which a

square grid can be transformed into a hexagonal grid, several terms must first be defined 1.

In coordinate projections, a lattice Aa is a discrete number of points within the set

of all real numbers in a two-dimensional space, such as a map projection, where CJ indicates

the geometric base of the lattice. Therefore, the lattice structures for both the square and

hexagonal grids are A4 and A6, respectively (CJ = 4 for the square lattice A4, and CJ = 6

for the hexagonal lattice A6).

~e 2.4: a) The Square lattice and b) "'the Hexagonal Lattice, with origin (0).

a) Y b) t I I I f I I I f f ---~- -~---~---·---·-' I I I I : : . : . : . : I f I I t ·---,- ·•---r--·r-··r· I I I I

: I I X ----~- -~---r---~--~--i ~ ! I I

•'

All map projections, by definition, must have a coordinate system for locating every

discrete point within a particular lattice. Every grid point in the lattice Aa is identified by

the ordered coordinate pairs (n1,n2), except that the coordinate pairs for the square and

hexagonal grids are different. The origin is the lattice point ( 1..0) , such that its coordinates

are (0,0) on both the square and hexagonal grids.

1. Since most of this section covers a series of specializ.ed mathematical concepts, a review of related terms is included in the glossary of terms found in Appendix B.

10

Note that the hexagonal lattice shows only two directional axes instead of three

(Figure 2.4, previous page). The reason is because only two axes are needed to locate points

within the hexagonal lattice - Wuthrich and Stucki (1991) have provided mathematical

proofs based on the mirror-image quality of the two diagonal axes. The First Intersection

Reduction Theorem, introduced by Wuthrich and Stucki (1991, p. 331), reduces the

intersections that must be computed to produce the transformation from three to two (See

Figure 2.4, or Wuthrich and Stucki (1991), p. 331, Figure 10).

This theorem also provides the coordinate pairs for the finite square and hexagonal

lattice dispositions in Figure 2.4, which are:

Coordinate pairs for the Square Grid Lattice:

Coordinate pairs for the hexagonal lattice:

From these and other algorithms, Wuthrich and Stucki (1991) concluded that a

circle - or a line drawn along the circumference of a sphere - is more realistically

represented when the picture elements, or pixels, are placed on a hexagonal rather than a

square lattice (Figure 2.5).

Figure 2.5: Hexagons vs. Squares ... Which are more accurate for drawing curves and circles?

To transform the real plane into a discrete set of hexagonal-based points, Wuthrich

and Stucki (1991) modified the square-based Nearest Neighbor Transformation from a

square into a hexagonal grid (a modified version of the Nearest Neighbor Transformation

11

was developed to determine the discrete value for each point on the original image, such as

in remotely sensed satellite data, by averaging the four square pixels surrounding each

particular location, such as in Figure 2.6 (Lillesand and Kieffer, 1987).

Figure 2.6: In the Nearest Neighbor Transformation. ilie values of the first four adjacent pixels are assigned to the center point, in order to average the values of the surrounding squares into one pixel.

Wuthrich and Stucki chose the Nearest Neighbor Transformation because it is a

simple, first-order algorithm. More accurate transformation techniques exist, such as the

bilinear interpolation and cubic convolution methods, along with a detailed description of

these resampling techniques and can be found in Lillesand and Kieffer (1987). In the scope

of this paper, the word "transformation" means the conversion of real, measured data into

a geographically referenced set of discrete points.

Rosenfeld (1974) also presented a number of theorems, formulas and proofs that

transform straight lines, curves, circles, and other objects onto a hexagonal grid1. Li-De Wu

(1982), and S. Pham (1986, 1987) used Freeman Chain Codes to confirm that Rosenfeld's

main theorem can indeed represent linear objects onto a hexagonal grid. Rosenfeld's ideas

are more inclined to fit a three-dimensional model than Wuthrich and Stuck.i's Nearest

Neighbor transformation scheme.

The actual Grid Point Transformation process, where all the points belonging

within the map extent2 ( or map boundary) are transformed from the square to the hexagonal

1. Rosenfeld's Theorem: "Consider the straight line of the plane of equation y • mx+ q, and suppose that its slope m belongs to the interval [· l ,l ]. Then all the points of its grid point transfonnation can be found by computing only its inteniections with the vertical lines of the grid and approximating them to the nearest grid point. as the intersections with the horizontal lines of the grid do not identify any additional points of the [transformation]" (Wuthrich and Stucki, 1991, p. 327)

2. Map Extent: For all lattice points A, let the map extent Il.U. (witlµQ the lattice base 0") equal all points (y) belonging to the set of real numbers R2, where (y) e<Jllals all the lattice points l A) , plus the vectors vn (with the corresponding x­coordinate also belonging to the lattice) (Wuthrtch & Stucld, 1991, p. 325).

12

lattice, is shown in Figure 2.7 (Wuthrich and Stucki, 1991, p. 329, Figure 7). The main

difference between the two grids lies in the layout formula on the hexagonal grid's diagonal

axis, where y = ( l3/3) i instead of y = 0 .

• • • •

B • • • •

-X

• • • •

Figure 2.7: Results of a Grid Transfo ation (T 1).

This process of transforming the map extent's discrete points from one type of

lattice to another is similar to the implementation of the planetary relief model, where the

ETOP05 dataset is transformed from a square projection into a triangular lattice. The

square-to-triangle transformation will not introduce map generalization errors, since the

dataset will retain its original elevation and latitude/longitude coordinates. Indeed, it

provides supporting evidence to the grid point transformation process discussed above.

Section 2.3: Quadtrees for Storage of Global Datasets

Efficiency of the method used for storing and retrieving related information is

essential in a multilateral, planetary model. Not only must the storage method be

hierarchical, it must be able to reference different tessellation levels, with each level having

a higher resolution (Tobler and Chen, 1986). Also, the method selected must be able to

store and retrieve data in both raster and vector formats to fit the proposed global model

(Chen and Peuquet, 1985; Mark and Lauzon, 1985). Based on these requirements, the

proposed model will be referenced based on a hexagonal/triangular coordinate system, to

be stored in a modified quadtree structure.

13

The quadtree is considered a hierarchical, quaternary data structure where the map

extent is tessellated into nested squares (Clarke, 1990, p. 115). At the first level of

tessellation, the entire map may be thought of as being a single square, to be subdivided

into four quadrants, or descendants. Each quadrant is further subdivided only if the

quadrant's level of detail is too fine for that quadrant to represent a single, homogeneous

attribute (Chen and Peuquet, 1985). The quadtree, being a hierarchical structure, allows for

fast search and access to one-to-many relations within a given dataset (Martin, 1982).

A quadtree is also known for the names assigned to its components. In a binary

image, for example, the root node corresponds to the entire array, and each son of a node

corresponds to a quadrant (labeled NW, NE, SW, and SE [or 0, 1, 2, 3]). Leaf nodes

represent sections for which no further subdivision is needed. Leaf nodes are also either

black (contains only O's) or white (contains only 1 •s). On the other hand, non-leaf nodes

are labeled grey (Samet, 1990a).

Several methods have been proposed to index areas within quadtree structures,

including the B+-Tree (Abel, 1984), Morton Sequencing (Morton, 1966), PM3' quadtree

for vector data (Samet and Webber, 1985), and the Quaternary Triangular Mesh (Dutton,

1989b). Each of these quadtree variations is reviewed in order to determine their feasibility

and implementation for global modeling.

2.3.a: The B+-Tree

David Abel (1984) modified the quadtree structure to fit large databases, where

each tessellation level needs referencing. Instead of the base4 representation in Morton

sequencing and Q1M, the B+-Tree uses a base5 system to implicitly indicate the level of

the tree (Abel, 1984). This method thus facilitates data searches for examination of adjacent

nodes (a certain level of topology). On the other hand, if the programming language is C

(in which bit manipulations are possible), the Morton Sequence is more appropriate

(Lauzon, et al., 1984). Although the B+-Tree was designed for large, global databases such

14

as the one discussed in this study, its base5 system is unnecessary because the tessellation

process implemented in this study eliminates the need for a tree level indicator as in the B+­

Tree. Therefore, this option will add data redundancy into the selected database

management system selected, meaning that the B+-Tree does not fit the specified

requirements.

2.3.b: Morton Sequencing I Two-Dimensional Run Encoding

The original Morton Sequence is "defined as the assignment of consecutive

numbers to the cells on a grid, beginning with 0, such that in the number's binary

representation, the odd-numbered bits denote the coordinate position of the [x-axis and the

even numbered bits denote the coordinate position of y-axis], where the coordinates begin

at (0,0)" (Lauzon et al., p. 57). Figure 2.8 shows the Morton addressing sequence and order

of the numbers themselves.

Since the Morton method is symmetric, there is no need to know the maximum

coordinate values. "The position of each element in the ordering (termed its key) can be

determined by interleaving the bits of the x and y coordinates of the element" (Samet,

1990b, p. 25). This characteristic makes Morton sequencing an excellent choice when an

area has been aggregated into squares. Because of its simplicity and versatility, Morton

Sequencing was chosen for this research implementation. The Morton technique was

NW

0

2

s ,w

1

3

NE

SE

0

1

2

3

6

7

x~ 0 1234567

0 1 4 5 16 17 20 21

2 3 6 7 18 19 22 23

8 9 12 13 24 25 28 29

0 II 14 15 26 27 30 31

132 33 36 37 48 49 52 53

34 35 . 38 39 50 51 54 55

40 41 44 45 156 57 60 61

42 43 46 47 158 59 62 63

Lauzon, et. al., 1985 Figure 2.8: Morton Sequence and Number Ordering

15

improved by Lauzon, et. al. (1985), who developed a two-dimensional run encoded

technique (2DRE) that compresses data to an even greater degree by labeling areas of equal

value with the Morton Number for the last region within that area.

In this two-dimensional run encoding technique (Figure 2.9), the top left region

within quadrangle zero is labeled "15," meaning that there are 16 subregions of equal value

within that quadrangle. Since in the original Morton sequence each subregion is given a

label, the 2DRE technique has improved upon the original Morton method by reducing the

number of Morton addresses from 256 Morton numbers to the 43 2DRE addresses shown

below. This indicates an 83 percent reduction of used addresses by using the 2DRE method.

0 1 ..----(Illa---...... - ... ---.... 67 71

127

223

I 17:! 191 239 = w

L_,fr_' _ _:J Figure 2.9: A 2-Dimensional Run Encoded quadtree Using Morton Sequencing (Lauzon, et aL, 1985, p. 60).

16

2.3.c: The PM3' Quadtree

The PM3' quadtree for vector data was originally presented by Samet and Webber

(1985). This quadtree uses floating point data instead of integer data to store coordinates.

This makes it possible to ignore fractional problems attributed to pixel size on raster

structures. In addition, when only vector data is associated to the quadtree (no raster data),

"the area processed by the quadtree algorithms need not be of any particular size or shape,"

although they must remain somewhat quadrilateral in shape (Ibbs and Stevens, 1987).

Although Ibbs and Stevens implemented the PM3' quadtree in the C language, the PM3'

quadtree's use of floating points was rejected because its structure is exceedingly

complicated, besides being a less effective compression tool.

2.4 The Quaternary Triangular Mesh (OTM)

2.4.a: Introduction & Significance

The Quaternary Triangular Mesh (QTM) is a spatial data model developed by

Geoffrey Dutton for global data modeling and measurement of locational uncertainty

(Dutton, 1988). Even though the QTM data structure is similar to the previously discussed

quadtrees in that it is based in a two-bit hierarchical structure, similarities end there.

Dutton's spatial data model is unique because it uses a triangulated, numerical scheme for

geocode addressing within a spherical rather than a planar basis (Dutton, 1984).

The QTM is especially important to this study because it matches the author's

original desire for a nontraditional data structure, and serves as a model not yet fully

implemented - in need of further research and development. Although other quadtree

variations exist for storage of global data - such as the cubic quadtree structure discussed

by Tobler & Chen (1986) and Mark & Lauzon (1986) - these models tend to ignore the

modelling of global phenomena in favor of accessing and retrieving map and image data

17

(Dutton. 1989a). Goodchild (1988) also stated the need for a spherical model for point

pattern analysis based on Theissen polygons and the tessellation of polyhedral skeletons.

The advantage of tessellating a polyhedral skeleton into finer and higher resolutions

lies in its description of the planet using a set of consistent facets. The chosen polyhedron

must be able to produce similar. recursive tessellations and retain the geometric shape of

its original facets. And. since the length of the QTM address expands by a factor of four as

the number of tessellation levels increase. QTM addresses can also be stored within a

quadtree structure. Therefore, Dutton selected an octahedron because four of its eight

triangular octants can be used to represent each hemisphere into a separate QTM quadtree

(Figure 2.10).

Figure 2.10: Tirree-dimensional Triangular Tessellation Levels

Level Three

Level One

2.4.b: Triangulation & 2-Bit Addressing

Triangle facets are used because they always keep their original geometric form up

to an infinite number of tessellations (Figure 2.1 la). In QTM addressing, every triangle on

the surface of an octahedron is tessellated by connecting the midpoints of every edge into

four similar subtriangles (or children). All triangles are known as tiles. The four subtiles

derived from the primitive (or parent) triangle are labeled O through 3, where each tile "can

be identified by a 2-bit binary number. so that 2L bits (or L/4 bytes) are needed to specify

18

a QTM address at L levels of detail." (Dutton, 1988, p. 6). Binary notation, though, is not

the optimal data storage notation available.

Figure 2.lla: A tessellated triangle produces more triangles.

Figure 2,llb: Facet Breakdown Numbering. (Dutton, 1988)

To further improve upon the storage of QTM addresses, Dutton suggests using

hexadecimal rather than binary notation. For example, a representative base4 (2-bit) QTM

string may be 0311021223013032 which, if translated to hexadecimal notation, becomes

the 32-bit number 3526B lCE, the same resolution (30 meters) as that found in Landsat

pixels (Dutton, 1988, p. 7).

2.4.c: Octahedral geometry & QTM Resolution

Geometry is then used to determine the exact location of a QTM address. Since a

QTM tessellation always produces four subtriangles, it is possible to follow any QTM

address from a chosen global quadrant down to any level of resolution desired. Each global

octant, after the first tessellation, will always have its center triangle labeled O and its apex

labeled 1, while the other vertices are arbitrarily assigned 2 and 3 (see Figure 2.1 lb).

Note also that the central triangle in Figure 2.12a (next page) is slightly larger than

the vertices as a result of the cylindrical nature of a sphere, and that it points in no particular

spatial direction, meaning that the level of tessellation can be inscribed directly into the

QTM address by appending zeroes without affecting geographic position (Dutton, 1988).

If the triangle were equilateral, the center facet would instead be identical. In Figure 2.12a,

19

only one digit is used to depict Level 1. Figure 2.12b, on the other hand, shows Levels 2

and 3 depicting two and three digits to indicate resolution, respectively.

(a) (b)

Figure 2.12: a) Octahedron face with four Level-one triangles b) Triangular tessellation, tiling and locators.

(Laurini,1992, p. 246).

2.4.d: Triangle Orientation After Tessellation

Example of Second level

Example of Third Level

Another property of triangular tessellations claimed by Csillag (1993) to be a

liability in developing nontraditional data structures is that triangles resulting from a

tessellation always change apex orientation; they point either up or down (upright or

inverted). Although this would truly be a setback if regular quadtrees were used, it proves

to be a boon in QTM addressing. Note that those global octants on the northern hemisphere

are upright triangles, while the southern hemisphere octants are inverted triangles (vertices

always have the same orientation of the parent triangular facet, while the center (or zero)

facet always has the opposite orientation (Dutton, 1988). Upright triangles mean that the

triangle has a horizontal base with its apex on top, while an inverted triangle has its apex

below the central facet and its base is above. Dutton applied this orientation anomaly to the

more complex of the following data structure versions.

20

2.4.e: QTM Addressing Methods

Once the basis for QTM addressing was determined, Dutton suggested using either

of two quadtree data structures for storing QTM addresses: (1) A regular quadtree structure

linking triangular tessellation (QTM) addresses to a set of global coordinate points; and (2)

an alternative hierarchical technique based on hexagon attraction, resampling and aliasing.

The first method suggested involves assigning QTM addresses to every point on a specific

coordinate projection, followed by the graphical rendition of the octahedral skeleton based

on the coordinates for each QTM address. Although this method offers simplicity and

geolocational precision, it lacks the ability to measure scale throughout the octahedron's

surf ace. As the levels of tessellation change, so does the length of every triangular segment,

meaning that scale continues to change throughout the globe's surface and becomes

impossible to measure (Dutton, 1989a). In other words, while a coordinate may tell the user

where an object is, it fails to inform the actual size of this object.

The second method is to average the data for every point that falls within each facet

and assign this average to the QTM address of the center (zero) triangle - instead of

assigning coordinates to each node on the octahedral structure. Given that every location

Figure 2.13: Organizational structure of triangles, showing the location of attractors and null-cells. Also note the hexagons that form within the triangular structure, and that zero-cells are covered by lower orders of null cells.

Attractor

Source: Dutton.. 1989b, Figure 1.

21

(node) on the structure is georeferenced to the selected projection, each averaged facet

location may be thought of as an "attractor," where each attractor serves as the focal point

for the specific facet (Figure 2.13, previous page). It must also be noted that since the zero­

value triangles surrounding the QTM attractor form hexagons, it may be possible to

transform the QTM between square and hexagon grids, using the modified Nearest

Neighbor algorithms listed by Wuthrich and Stucki (1991), and described in Chapter 2.2.

If the triangular subsets are aliased with the coordinate nodes, this modified

quadtree structure becomes "relatively stable under translation and rotation," and spatial

generalization occurs (Dutton, 1988, pp. 134). This is the main advantage found in Dutton's

QTM structure. Of the quadtree structures reviewed, the QTM structure is the only one that

allows for seamless data generalization, no matter what point of origin is selected for the

tessellation. If any of the other quadtrees is used, the leaf-node tree structure will change

whenever different starting coordinates are selected for the original octant. A simple and

yet effective mode of data access and storage can then be achieved by generalizing the

spatial data structure into a discrete set of geocoded nodes.

In the QTM structure, nodes that fall within null-cells (the white, center tiles seen

in Figure 2.13) are thought of as belonging to a particular attractor. Values surrounding this

attractor node are aliased (averaged), and this new value is assigned to the attractor. As

resolution increases, those tiles surrounding the attractor form hexagons. Then, as the QTM

facets become smaller, the hexagonal geocode addresses grow longer and features become

more unique. Since these nodal points lie at the centerpoint of the hexagons, they "may be

thought of as a locus of attraction, or attractor, to which nearby observations gravitate"

(Dutton, p. 8, 1989a).

These attractors can later be tracked based on the order of QTM addresses and order

of QTM cells dominated by attractors using the spiral method, as shown in Figure 2.14,

next page. The addresses for the attractors themselves are shown in Table 2.1 (also next

page).

22

Figure 2.14: Spiral Method for Ordering QTM Attractors (Based on Dutton. 1989, Figure 4).

O Level 1

8 Level 2

Level 3: 24 Attractors Level 4: 45 Attractors

TABLE 2.1: Addn:ssts and Qrdtrs Qf OTM Ctlls DQminattd b~ Attra,ctQrs, Sgiral E.Qnn

Source: Dutton. 1989, p. 7. ATT# QTMl QTM2 QTM3 QTM4 QTM5 QTM6 ORDl ORD2 ORD3 ORD4 ORD5 ORD6

1 1 1 22 2 33 3 411 4 502 12 32 5 11 19 633 6 701 21 31 7 14 18 822 8 903 13 23 9 12 16

10111 10 11102112 132 11 43 51 12033133 233 12 52 68 13201221 231 13 62 66 14222 14 15203213 223 15 60 64 16011211 311 16 58 74 17302312 332 17 75 83 18333 18 19301321 331 19 78 82 20002122 322 20 47 79 21103113 123 21 44 48 22001021 031 101 121 131 22 30 34 38 46 50 23002012 032 202 212 232 23 27 35 55 59 67 24003013 023 303 313 323 24 28 32 72 76 80

23

2.4.f: Concluding Statements on the Quaternary Triangular Mesh,

In conclusion, the QTM data structure, using the attractor method, has the potential

ability to measure locational uncertainty and local scale by averaging values surrounding

specific nodes instead of assigning arbitrary coordinates to these points. On the other hand,

this alternative quadtree structure introduces a certain degree of error to produce a map at

the cost of more generalization, lower resolution, and less accurate detection of geographic

location.

Dutton (1989d) has also developed a Zenithial Orthotriangular Projection (ZOT),

which is a simplified version of the QTM tessellation. This map projection is azimuthal,

equidistant and orthotriangular; like the QTM, it is projected unto an octahedron rather than

a sphere.

24

CHAPTER3

Triangular Tessellation and Quadtree Modifications

3.1: Triangular Tessellation of the Octahedron

The most significant difference between the proposed planetary model and most

other global data models is the use of triangles to georeference spatial data. While most

coordinate projections use a square-based lattice structure, this triangular tessellation

model recursively decomposes an octahedron into triangles (up to any level of resolution

desired) and georeferences it to a global relief dataset.

Each of the facets on the octahedron's surface is tessellated into four geometrically

identical triangles. At the beginning, an octahedron contains eight octants, or four

quadrants for each hemisphere. Then, after each new tessellation subdivides a facet into

four more subfacets, the facet resolution increases until the octahedron begins to resemble

a sphere (see Figure 2.10). These patterns can be stored later in the hierarchical quadtree

structure presented and implemented in Chapter 4.

Triangles, unlike squares, have corners that match important surface nodes, and in

spherical coordinate systems, their edges touch the global great circles. Therefore, triangle

edges can represent linear features that are easily networked into a topological vector

structure (e.g., the Triangulated Irregular Network (Peucker, 1978; Peucker, et al, 1986).

Tsai and Vonderohe (1993) also suggested Delaunay triangles for the GIS application of

tetrahedral data for three-dimensional modeling. On the other hand, triangular tessellations

are oriented to line features as well as points, making them more complicated than square­

based proximal areas (Laurini and Thompson, 1992). Although this statement suggests a

more complex set of problems, it is beneficial in this study because, as previously stated,

changes in orientation can be used to determine north-south orientation and other spatial

properties, such as topology.

Finally, note that a hexagon, unlike a triangle, is not as easily stored in a database

25

because its geometric structure cannot be tessellated without losing its original shape

(Dutton, 1984); it is a nonisohedral figure. Even though hexagons are not directly applied

in this study, they are still useful since hexagons are automatically formed within the

tessellated triangular structure.

3.2: The TrianiYlated Ouadtree Structure (TOS)

The proposed three-dimensional model will need a quadtree structure capable of

georeferencing coordinates and elevation data to the triangle nodes within the octahedral

structure, up to any level of tessellation. The selected quadtree, if necessary, must also

allow the use of a hexagonal coordinate grid after the original square-based projection has

been mathematically transformed into a triangular map projection. Such a quadtree must

also be capable of linking large numbers of entities, attributes and relationships between

the dataset and the octahedral facets.

The proposed quadtree structure is a combination of Morton sequencing and the

Quaternary Triangular Mesh structure, modified to fit the previously defined

specifications. Each tessellation level can then be referenced using a polygon attribute table

attached to the quadtree's elevation database. In order to store the outline of the world in

the octahedral structure of the Interrupted Collignon Projection, the geocode given to each

triangular quadrant must follow a static pattern similar to a rectangular Morton sequence

(see Figure 3.1, page 28). With a static pattern it is then possible to automatically assign

Morton numbers and QTM addresses simultaneously.

Three steps must be followed to achieve a static addressing scheme. First, the center

facet of each tessellated triangle must be labeled zero [0] (Dutton, 1988); its north-south

orientation is always opposite to the location of the central triangle's base. Second, the apex

(labeled one [1]) shares the same base with the center facet (note that both triangles face in

opposite directions). Finally, the remaining facets are labeled by assigning the numbers two

26

[2] and three [3] in a counterclockwise direction beginning at the apex. This step differs

from Dutton's labeling scheme, where he said that facets [2] and [3] may be arbitrarily

labeled (Dutton, 1988). The result is a static addressing pattern that allows a triangular

tessellation to be stored directly into a quadtree structure and vice versa, and which can also

be stored using either Morton or QTM numbers (see Chapter V for a full description of the

implementation process).

AB seen in Figure 3.1 (next page), a raster image was originally stored using Morton

sequences in a two-dimensional run encoding (2DRE) format (see previous chapter for full

explanation of these terms). In the original square quadtree (3. la), areas with identical

values are assigned identical Morton numbers, while areas with different values are

recursively decomposed until values are identical. White areas indicated no data (all O's)

and grey areas indicate data (all l's). In Figure 3.lb, the same image has been directly

transposed into an equilateral triangle structure. Note that the sequencing scheme used is

the same as the Morton/2DRE addressing method in Figure 3.la. The only difference is the

applied geometric base. The next step is to apply the QTM addressing method. Once the

raster image has been transferred from a square to a triangular lattice, it is a simple matter

of substituting the Morton numbers for QTM addressing numbers. The result is shown in

Figure 3. lc.

The resulting Triangulated Quadtree Structure, labeled IQS., is an excellent global

data storage method because it contains the hierarchical structure of a quadtree, the

addressing simplicity of Morton Sequencing, and the ability to translate these latitude/

longitude coordinates into TQS addresses, which will, in turn, be applied for data

translation and rotation.

27

Figure 3.1: Although both a) and b) and c) are identical tessellations, a) is a 2DRE/Morton Quadtree. b) is a Triangular Quadtree Structure (TQS), and c) is a QTM triangular structure.

The original iml_lge shown in 3. la was directly imposed on 3.1 b following the addressing pattern between Figures 3. la and 3.1 b. Each number shown in every one of these :figures labels the highest value assigned to each particular quadrant, and based on the original image.

0 1

15 95

47 127

14.3 223

175 191 239

2 a) 2DRFJMorton Sequencing 3

NWEffiNE -~a~ex -S E ase

b) 2DRFJMorton Sequencing (TQS) QTM Addressing

28

CHAPTER4 Applied Map Projections

4.1: Geodesic Concerns in Choosini a Map PrQjection

A major concern in developing a planetary relief model from an octahedron is the

apparent error that may be introduced if the model does not account for the Earth being a

geoid or ellipsoid rather than a perfect sphere. The Earth is about 42 km (0.1 % ) longer in

diameter around the Equator than along the average meridian. As previously discussed,

every time an octahedron is sequentially tessellated it more closely resembles a multilateral

globe. Dutton, in his Geodesic Elevation Model (1983; before it evolved into the QTM

model), proposed to solve this dilemma by first tessellating an octahedron to several

thousand facets, then project it onto a geodesic projection. He did not suggest a specific

projection in any of his papers (from 1983 to 1991); he claimed that more research was

needed to determine a feasible solution. Dutton also stated that researchers know little

about how to optimize data structures or processing, or how to take advantage of the facet­

node duality of his QTM model.

To solve the riddle of what projection gives a feasible solution, over one hundred

projections were examined, using An Album of Map PrQjections (Snyder and Voxland,

1989). As a result, it was determined that only a pseudocylindrical, triangular projection

with straight parallels and meridians - such as the Collignon Projection - permits the

world's outline to be projected onto a three-dimensional octahedron. Straight parallels and

meridians are necessary to join the results of this research into the octahedral skeleton of a

future three-dimensional model. After several modifications, the Collignon projection was

used to georeference both the Digital Chart of the World and the ETOP05 Global Relief

Dataset into eight triangular lattice structures (octants). AC language program was written

to modify the Collignon algorithm, then fit the resulting triangular structures on the surface

of a three-dimensional octahedral skeleton. The C source code is listed in Appendix A.

29

4.2: The Collignon Projection

To test the effectiveness of the algorithms involved, it was decided that the digital

outline of the world be displayed using the original Collignon formulas. Using a map scale

of 1:50,000,000 and a global circumference of 6,378,137meters, the World Databank-I

(WD-0 dataset was displayed using the Collignon Projection formulas, such that they could

later be modified to project each individual octant.

The Collignon projection is a pseudocylindrical, equal-area projection, where

meridians are equally spaced straight lines converging at the North Pole. On the other hand,

parallels are unequally spaced straight lines, farthest apart near the South Pole, closest near

the North Pole, and perpendicular to the central meridian. Scale is true along latitudes

15°51 'N (Snyder and Voxland, 1989). The WO-I dataset must be transformed from degrees

to radians, followed by the use of the Collignon algorithms. The following are the original

Collignon formulas (Snyder and Voxland, 1989):

Collignon: X = 2RS('A.-'A.0) x (1-s~cj,)1/2

7t

Y = 1tl/lRS [ 1 - (1 - sincj,) 1/2]

where A • longitude, }..0 • central meridian, + • latitude, R • radius and S • scale.

These algorithms were multiplied by 0.000,000,05 to transform them into a scale of

1:50,000,000. Figure 4.1 (next page) shows the resulting Collignon projection, displayed

using the UNIX-SUN OKS environment.

30

l.,.l -

Figure 4.1: Collignon Projection.

Al&ortthm: Snyder and Voxland, 1989. C code: Clarke, 1994.

4.3: The Interrupted Collignon Equal Area Projection

The final step is to modify the Collignon algorithm such that each octant of the

octahedron may be projected into its own, individual triangular projection. This way, each

hemisphere may be stored into four triangular quadrants; Clarke (1994) called it the

Interrupted Collignon Equal Area Projection, or "Clarke's Butterfly Projection." Each

hemispheric quadrant is projected individually, depending on the central meridian chosen

for the projection.

The modifications involved on the Collignon require that the user choose the central

meridian ( A0) to be displayed, which gives a small degree of rotation, although no

translation among the quadrants and hemispheres. Next, each quadrant must be tested to

determine whether it belongs to the northern or southern hemisphere. If it falls below the

Equator, the quadrant is inversely projected (all points Y k are multiplied by -1.0). The

chosen central meridian is then used to determine the coordinates that define the

longitudinal range of the quadrant's base, or equator. Once these steps are accomplished,

the chosen quadrant is projected through the following algorithms:

Interrupted Collignon (modified):

(2.0RSx (A-Ao) x (1.0- sin(cp))o.s) For Xk = 1 to k, fie ,

For yk = 1 to k, cos (30.0 x 3:o> x Jii. x RS x [ 1.0 - ( 1.0- sin (<I>)) 0·5

]

where ,. • longitude, "'o • central meridian, + • latitude, and R • radius.

For the southern hemisphere, are reflected about axis of central meridian, meaning

that southern hemisphere points are first treated as belonging in the northern hemisphere,

then inverted into position.

32

The most important modification to this projection is the rescaling of the original

projection to fit an equilateral triangle; the Interrupted Collignon is no longer equal-area.

As a result, the Interrupted Collignon will provide the specific Collignon location for any

given latitude/longitude (x,y) coordinate in an equilateral triangle lattice. First, the user

provides a specific (x,y) coordinate. The C program responds by fitting the latitude and

longitude of each point through the Interrupted Collignon algorithm and assigns it to one

of eight triangular octants. This feature not only locates all coordinate points on Earth, but

serves as the method by which QTM and TQS addresses can be issued to every node on the

octahedral skeleton.

The quadrants can be seen in Clarke's Butterfly Projection (see Figure 4.2), where

each quadrant is automatically arranged to resemble a butterfly. This projection can also be

cut out and glued into the shape of a three-dimensional paper octahedron.

Figure 4.2: The Interrupted Collignon(Buttertly) Projection.

© Copyright Keith C. Clarke. 1994

33

CHAPTERS

Implementation and Tessellation of the Triangulated Quadtree Structure (TQS)

5 .1 Introduction

The tessellation and geocoding portion of the Triangulated Quadtree Structure

(TQS) was tested and implemented using the ETOP05 Global Relief Dataset (see

Appendix C for a detailed description of ETOP05). Each data point in the ETOP05 dataset

is first read in from its ERDAS raster file, transformed into the Interrupted Collignon, given

a geocode similar to Dutton's Quaternary Triangular Mesh (QTM), and finally stored into

its corresponding octant (see Figure 4.2, previous page).

5 .2 Recursive Tessellation for Determination of a TOS Address

First, each point is transformed from its original (x,y) coordinate (squared) lattice

into a triangular lattice with the modified Collignon algorithm. The original rectangular

image was transformed and stored into a set of eight separate quadtrees ( one octant per

quadtree). Each of the quadtrees in turn represents each of the facets of the global

octahedron. After each (x,y) coordinate was called in from the original ERDAS ".Ian"

image and transformed into a triangular Collignon coordinate, the C program called

Get_TQS.c assigned a QTM address to each point within ETOP05. The Collignon

coordinate was then tested recursively to determine where within each triangle - facets 0,

l, 2, or 3 - did the point belong. Even so, the maximum number of tessellations needed

had to be determined before a TQS address could be assigned for each point.

Given a global circumference of 40,000 kilometers (at the Equator), an ETOP05

resolution of 5 minutes, and an Erdas raster image consisting of 4,320 columns x 2,160

rows, the pixel distance at the Equator= 6,378,137 meters/ 4,320 columns= 9,259 meters

per pixel. In other words, 1 minute = 40,000 kilometers/ 360 degrees (360 degrees = 21,600

34

minutes)= 1851.85 meters. Therefore, the center of each ETOP05 point lies at a distance

of 5 minutes, or 9,259 meters from the other adjacent points (excluding the diagonal

neighbors). Table 5 .1 shows the resolution of each pixel at different tessellation levels 1:

Table 5.1: Sea;ment distances at each Tessellation Leyel <and Depth}.

~ l&l21!l 23 * 1 22 2 21 3 20 4 19 5 18 6 17 7 16 8 15 9 14 10 13 11 12** 12 11 13 10 14 09 15 08 16 07 17 06 18 05 19 04 20 03 21 02 22 01 23 00 24

Disram·& <meters) 10,000,000.00 5,000,000.00 2,500,000.00 1,250,000.00 625,000.00 312,500.00 156,250.00 78,125.00 39,062.50 19,531.25 9,765.62 4,882.81 2,441.41 1,220.70 610.35 305.18 152.59 76.29 38.15 19.07 9.54 4.77 2.38 1.19

Circumference C • 21t (radius) • 2.0nr = (2.0(3.14) x 6,378,137 meters)= 40,000,000 meters = 40,000 kilometers.

• Octant distance at the Equator •• Depth required to represent the ETOP05 dataset.

Once the desired resolution was known, each point was transformed from a square­

based lattice to the Collignon triangular lattice. Figure 5 .1 shows the boundaries of an

octant after a point is fitted through the Collignon algorithm. The most prominent feature

is the grouping of points as one moves from the triangle base up to the apex (this is more

clearly in the Collignon Projection, as seen in Figure 4.1).

1. Nme: While depth indicates how many times an image is tessellated, the root node is at level n, while a node at level O corresponds to a single image pixel (Samet, 1990a).

35

Figure 5.1: Octant boundary for map coordinates.

max_y

xrange

In Figure 5.2a, the ETOP05 data points are located in a rectangular, latitude­

longitude model. Each rectangular octant was then projected into the model shown in 5 .2b,

which is later arranged into its proper position within 5 .2c through eight separate rotation

algorithms (found in Collignon.c, Appendix C). Note that the octant numbering scheme

was chosen in order to more easily project each point into its respective octant within

Figure 5.2c.

Figure 5.2: Transformation from a) square-based raster image. to b) trtangle-based lattice structure. to c) Butterfly Projection.

a) (1 80901 f270 901 1090l 90,90

1 2 3 4 (1 8( ,0) (271 ,0) (00) (90 0)

8 6 7 5

1180 90)

(18( • 0)

(180,-90) (270,-90) (90,-90) (180,-90)

b)

c)

The implementation of the Triangulated Quadtree Structure (TQS) itself began with

the assignment of QTM addresses to each ETOP05 point. The addressing scheme used

36

differs from Dutton's model (Chapter 2.4) in that triangle addresses are always assigned in

a counterclockwise order - Dutton's schema varied from triangle to triangle (Dutton,

1984, 1991). The order of facets through each tessellation is shown in Figure 5.3. This

method was selected for two reasons: 1) a regulated order is imperative if TQS is to be

linked to any type of hierarchical structure, and 2) the computer algorithm method used to

detect triangle location is based on this ordering scheme (see Get_TQS.c, Appendix C).

a) QTM: Address Ordering b) QTM: Bit-Addressing

Figure 5.3: TQS Addressing Scheme.

The get_ TQS() function assigns TQS addresses to ETOP05, point by point, by

determining whether they-value falls between the midpoint's y-range and the maximum y­

value. If so, this point is 1) assigned a qtm number representing its triangle location, and

then 2) its x- and y-values are divided in half, along with the x- and y-ranges. This

coordinate division allows for the subsequent tessellation of this point's location until the

full qtm address is determined up to the level of resolution desired, or maximum depth is

achieved; see Figures 5.4, next page.

If the point fails to fall above the midpoint range, then the point is continually

rotated 240° counterclockwise until the correct corner triangle location is determined. And,

if it again fails to fall in a corner triangle, then by default it must fall in the center triangle.

Therefore, by continually rotating and tessellating each x- and y-value, this method assigns

a quaternary hierarchical address to each and every point in the dataset.

37

Each time a point gets translated to determine whether it belongs to the facet that is

currently above the midpoint range, it must be rotated counterclockwise 2400, its x­

coordinate gets substracted one fourth of the xrange value, and its y-coordinate gets added

one half the yrange value. These additions and subtractions translate the octant back to its

original position, except that the next triangle in the tessellation order is now at the apex

(above the midpoint range). Note that, while it seems that a full triangle gets translated, only

one point gets translated at a time. Triangles are used to help visualize the area being

processed.

Figure 5.4: Triangle Rotations and Translations Triangle rotation and translation for a point falling in the center facet. Note that the last translation inverts the point, then reduces its value by half (literally moving the point to the next tessellation level).

base

t Note: The last rotation yields this position such that, if the point falls in the center facet, it is automatically readied for the next tessellation.

Since geometric translation and rotation of points requires that all calculation be

performed in radians, each point is first translated in radians, and then rotated using the

formulas included in Figure 5.5:

Figure 5.5: Clockwise and Counterclockwise Rotation Formulas

y' ~-

-~

x • x'cos8-y'sin8

y • x'sin8+y'cos8

x' • xcos8+ysin8

y' • -xsin8+ycos8

38

Counterclockwise Rotation

Clockwise Rotation

F_igure 5.6: Toe actual translation and Rotation Process. along with r~e coordinates for trianwedetection algorithm. If the point falls in the shaded region then it belongs to that facet.

rotate ... u a)

Y translate ... ~x

Y translate. .. y

iy

d)

Figure 5.6 shows the rotation process before, during and after the transformation.

In Figure 5 .6a, a point ( ~) is found not to fall in the top facet of a triangular area. Therefore,

the point gets rotated 240°. Figure 5.6b shows the location of the point relative to the rest

of the triangular area after the translation. Figures 5.6c and 5.6d translate the point to the

location where the triangle area is in its correct location. Now, the point falls above the

midpoint range and is ready for the next tessellation. Recall that each point goes through

this process twelve times at increasing resolution to determine the full TQS address (12

tessellations).

5.3: Discussion of the TQS Addressina Method

Dutton's Quaternary Triangular Mesh provided the means to develop and assign a

TQS address for every coordinate point on the ETOP05 Global Relief Dataset. Each TQS

address can be stored into a maximum of three bytes (24 bits), for a global resolution of

39

approximately 4,883 meters, or 5 minutes. This is accomplished by assigning a 2-bit value

(from Oto 3) each time a point gets tessellated. This 2-bit value gets bit-shifted to the left

(twice) and then appended to the previous value until the TQS address consists of twenty

four binary digits (0 or 1). Therefore, TQS addressing eliminates the need for a pointer to

each tessellation level since each pair of binary digits represent each level - with the

leftmost pair indicating the root node. As a result, this method allows the user to recursively

select any level of resolution desired. The TQS address is now explained.

Beginning from the left (root node) of the TQS number, each pair of binary digits

represents the image's tessellation level up to the maximum pixel resolution; the two

rightmost digits indicate the tessellation depth with the highest resolution. The TQS

number is compressed even further by storing it in hexadecimal, rather than binary

notation. This method introduces another source of compression into the geocoding

process, though. When the binary number gets translated into hexadecimal notation, all

leading zeroes get truncated from the number. The result is a much smaller number than

originally anticipated. For example, lets say that a point falls in TQS binary number

[00 00 00 01 01 11 10 01 00 00 001 1

Translated into hexadecimal notation, this number becomes

[5E40]

If this number is again translated into a binary location, it becomes

1 01 11 10 01 00 00 00

Note that, instead of the original 24 bits, the number now consists of only 15 bits.

The missing levels are computed by adding the number of zeroes missing to the left of the

binary address, until the number of binary pairs is equal to the maximum depth desired by

the user (24 bits in this example).

1. Note: Spaces were added after every two binary number for ease and clarity of the TQS concept

40

5.4 Results from the Ap_plication of the TOS Addressing Method

The TQS, by storing an exact locational code for each point in hexadecimal

notation, reduced the amount of storage needed from 52,254,720 bytes (world.Ian: ERDAS

image of ETOP05) to 48,540,279 bytes (23 percent of the original size), The file itself was

originally 214,617,600 bytes, before all points below sea level were excluded. Again, the

TQS method further reduced this file to 48,540,279 bytes, and stored it into eight individual

files (one for each octant>1. Since the original coordinates were excluded from the octant

files, the resulting files contain implicit location, instead of explicit location ( octant

number, elevation, and TQS address; no coordinates), and use binary form, not ASCII.

Instead of storing the floating point elevation value (plus its link) required for

representing each point in ASCII format at a 5-minute resolution, the TQS method stored

the same point into a 24-bit binary value (three bytes instead of sixteen bytes); the

following representations will help explain this concept. At a 5-minute resolution, the

original coordinate and elevation values (in ASCII format) were stored as

± D D • MM ± D D D • MM h h h h \ u {D=degrees, M=minutes, h=elevation, \u=Null}, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 {bytai)

or the equivalent of twenty bytes. The same value in TQS format becomes

h h h h 8 8 8 \ u {h=elevation, e,,,, one byte, and \u=Null}, 1 2 3 4 5 6 7 8 9 (bytea)

or nine bytes.

ETOP05 was further simplified by eliminating all those points with identical x- and

y-coordinates within the ERDAS image file. Elimination of data redundancy within a raster

image was crucial because the original raster image adds identical x, y and elevation values

as the image reaches the poles to account for the stretch caused by transposing the Earth's

1. File size per Octant (in bytes): Octant 1 = 10,939,457; Octant 2 = 6,737,606; Octant 3 = 4,734,016; Octant 4 = 5,340,595; Octant 5 = 5,840,493; Octant 6 = 6,269,712; Octant 7 = 6,075,360; and Octant 8 = 2,603,040, for a total of 48540.279 bytes.

41

surf ace onto a flat grid. For example, the x, y and elevation values are identical for every

pixel along the latitudes for the north and south poles, while every point is unique along the

Equator.

The Interrupted Collignon projection, unlike the original image, transposed the

(x,y) coordinates closer and closer together as the points reached the poles, thus

compressing the high latitudes into the apex of each triangle. Although it may seem as if

accuracy is lost by concentrating more points in smaller and smaller areas, actually there is

no more space at the poles than anywhere else on Earth. The error of reduction in areal

resolution can be solved by implementing the three-dimensional planetary relief model

proposed in the next chapter.

Unfortunately, the ETOP05 dataset was not stored into the Triangulated Quadtree

Structure itself; if so, the resulting database would have been compressed even further. This

will be the next step in further research.

42

CHAPTER6

Future Applications of the Triangulated Quadtree Structure

6.1: Conditions and Data Reg,uirements

The elevation dataset, ETOP05 is available on CD-ROM from the National

Geophysical Data Center, NOAA; a full description is found in Appendix C: ETOP05

Documentation. After geographically referencing the ETOP05 dataset to a set of latitude/

longitude coordinates, it may be used to locate each node on the tessellation structure. Note

that, since elevation data are irregularly spread, they will produce irregularly shaped

triangle facets. This problem calls for more complex algorithms than if the nodes were

fitted to the surface of a perfect sphere. A perfect sphere is used because it permits the

application of a simple isohedral tesse3llation, such as the one already developed by Dutton

(1984). To optimize the model's data structure, the elevation data was stored on the TQS

structure - using a quaternary tessellation similar to the QTM structure. A quadtree

structure is again recommended because it is a hierarchical database that fits the

requirements for a global data structure (Tobler and Chen, 1986).

The next question is then how to retrieve elevation data from a raster-based

quadtree structure and reference it to each node on the vector-oriented triangular

tessellation. This can be solved either with Morton Sequencing (Cebrian et. al., 1985;

Lauzon et al., 1985; Burrough, 1986), or with the Triangulated Quadtree Structure (TQS).

Each quadrant level on the quadtree is given a binary digit based on the quadtree level and

its relative location along the coordinate plane. Morton Sequencing then allows the

computer to read the actual quadtree's location directly from the binary string.

43

6.2: The Application Process

The application process that portrays the finalized data structure onto a three­

dimensional graphical interface - called a Three-Dimensional Planetary Relief Model -

is now introduced. The first step is to fit an isolateral octahedron inside a perfect sphere

(Figure 6.1). The points where the octahedron touches the sphere along its great circles are

called the Primary Intersect Nodes, or PINs, for short. These PINs will then constitute the

octahedron's six (6) cardinal points in Fig. 18 (N, E, S, W, NP, SP). These PINs correspond

to the o0 , 90°, 180°, 90°W meridians, the north pole (NP) and the south pole (SP) (Laurini

and Thompson, 1992, pp. 246-7, Figure 6.22).

SP Figure 6.1:

Primary Intersect Nodes (PINs), Midpoints, and Primary Angles.

, • • .,;,,.~ X, ~ , .-. ,.?. ,•,:.-.-X•?. ... ' • , X ?.',,:-•

44

Once all pins are connected to the nodes representing the poles (NP, SP), the first

level of the octahedral sphere is complete. To reduce storage and search procedures, each

node will be automatically georeferenced to the elevation database. It is thus possible to use

a Triangulated Irregular Network (TIN) to visualize the octahedral nodes, especially since

a tessellated triangle inherently contains topologically oriented data (Dutton, 1988).

The steps involved below will produce each successive tessellation level:

1. Split all arcs at their midpoints.

2. Place a georeferenced node at each of the new midpoints.

3. Elevate (or lower) the midpoint nodes from the octahedron's center, based on the corresponding

elevation for each node.

4. The arcs and nodes belonging to polygon areas of low relief are left intact. Areas with high relief

will be tagged as further subdivision areas. Low and high relief values are computed using

statistical elevation variance (spread of points away from the mean).

5. Save all P/Ns and newly elevated nodes under a new tessellation level.

6. Repeat process until desired tessellation level is achieved.

This procedure, once completed, will be capable of tessellating the octahedron to

almost any level of tessellation, up to the resolution of the dataset. Once the source code

has been developed, tested and implemented, output becomes the next problem to

overcome. Hardware will limit the output's quality to an n-dimensional virtual projection

on the monitor. The model will need further modification to display only those areas

directly facing the user. This may be solved by fitting the octahedral skeleton into a TIN

structure, and transform it using any of various "watershed analysis" algorithms.

In addition, the user must be able to select those areas he/she wishes to display.

Since the dataset is georeferenced using individual quadrants, the user may select any area

of the globe, provided that it all belongs within a specific quadrant at the corresponding

tessellation level. The user delimits the map extent based on the east and west coordinates

45

of the desired octant, along with a choice between the northern and southern hemispheres.

These corners will be the map's ground control points.

For final display, the desired area may eventually be transformed from a three­

dimensional structure to either 1) a two-dimensional digital elevation model (DEM), or 2)

a one-dimensional map projection. For a two-dimensional DEM, Cebrian et. al.,(1985)

proposed that "digital elevation models be integrated into the linear quadtree GIS by using

Morton numbers as addresses for cell elevation data in a one-dimensional array or file,"

(Cebrian, 1985, p. 58). These dimensional transformations may serve as both the display

methods and testing tools for determining the accuracy of the model.

On the other hand, to produce a one-dimensional map projection, the ground control

points may be used to statistically transform the three-dimensional model to a one­

dimensional map. This statistical method is known as rubber sheeting (Clarke, 1990, p.

147). Although this technique adds a large degree of error by stretching the map extent into

a flat surface, it produces aesthetically pleasing maps for any part of the world at a scale

limited only by the tessellation levels and resolution of the data. The dataset must also be

large enough for a true global approximation, although it may still be tessellated to any

resolution desired. If the dataset is too small, it is then recommended that the data analysis

be performed on individual octants.

6.3: Future Applications

Future applications involve substituting the global relief dataset with other types of

ordinal data, such as population density, crime distribution, income, spread of disease,

temperature, and so forth. If the desired dataset is not sequentially ordered, it must be

modified and fitted to the planetary model, such that it can be hierarchically stored.

46

CHAPTER 7 Conclusion, Summary & Importance

In conclusion, the Triangulated Quadtree Structure addressing algorithm

successfully linked the land portion of the ETOP05 dataset to the Interrupted Collignon

Equal Area Projection. Therefore, the proposed planetary relief geographic information

system is now a feasible model for the application global spatial data. A 2.5-dimensional

elevation model was developed using a triangular tessellation to achieve any resolution

desired, the first step on developing an isohedral quadtree structure for the storage of

elevation data was presented, and a set of grid points were georeferenced into a triangular

coordinate grid.

Each chapter reviewed the application of square- triangle-based grids, algorithms,

triangular tessellations, and quadtrees. The accuracy and precision of the TQS addressing

numbers increased with every tessellation, until each address represented each elevation

point up to the resolution of the original dataset. Once the georeferenced data is stored in

the triangulated quadtree structure, areas with little relief will be left intact, while areas with

high relief are further subdivided to provide higher resolutions in those areas. The result has

been a model that fitted the Earth's surf ace onto the surf ace of a map projection that is

easily transformed into a paper octahedron. This accomplishment will eventually lead to a

three-dimensional model capable of providing a three-dimensional image of the globe,

two-dimensional digital elevation models of medium- to small scale areas, or one­

dimensional map projections given the desired level of tessellation, map scale and a

coordinate range.

Furthermore, the Earth's elevation was the original data type proposed (and used)

to define the surf ace of the geodesic octahedron, it is not the only attribute applicable to this

model. Other attributes may be statistically fitted to the planetary model include population

density, income, meteorological data, incidence of a particular disease, and others. In other

47

words, this model could be viewed as a three-dimensional choropleth map for the entire

world. Other global datasets include biodiversity distribution and soil types (Csillag, 1994).

On the other hand, while this method offers simplicity and geolocational precision, it lacks

the ability to measure scale throughout the octahedron's surface. As the levels of

tessellation change, so does the length of every triangular segment, meaning that scale

continues to change throughout the globe's surface and becomes impossible to measure

(Dutton, 1989b). In other words, while a coordinate may tell the user where an object is, it

fails to inform the actual size of this object.

Thus, despite this weakness in scale measurement, the recent acquisition of the

necessary technology to achieve the stated goals makes the proposed application likely to

be ready in the near future. Since squares began their incursion and conquest of the

cartographic world, people have taken for granted that a coordinate system must be a

discrete, rather than continuous, set of points. Therefore, most spherical projections are

subjected to inaccuracies and precision problems developing from trying to fit a circle into

a square. This is precisely the type of error that the proposed global relief geographic

information systems was designed to address.

48

REFERENCES

Abel, D. J. (1984). "A B+-Tree Structure for Large Quadtrees." Computer Vision, Graphics and Image Processing, v. 27, pp. 1-18.

Burrough, P. A. (1986). Principles of Geo~hical Information Systems For Land Resources Assessment. New York: Clarendon Press.

Cebrian, Juan A. James E. Mower, and David M. Mark (1985). "Analysis and Display of Digital Elevation Models Within a Quadtree-Based Geographic Information System." Proceedings Auto-Carto 7, Falls Church, VA: ASPRS/ACSM, pp. 55-64.

Chen, Zi-Tan and Donna Peuquet (1985). "Quadtree Spatial Spectra Guide: A Fast Spatial Heuristic Search in a Large GIS." Proceedings Auto-Carto 7, Falls Church, VA: ASPRS/ ACSM.

Clarke, K. C. (1990) Analytical and Computer Cartography. New Jersey: Prentice Hall, pp. 15, 147.

Clarke, K. C. (1994). Interview. New .York: Department of Geology and Geography, Hunter College of the City University of New York.

Csillag, Ferenc (1993). "Variations and Hierarchies: Toward Linking and Integrating Structures." NCGIA: 2nd International Conference/Workshop on Integrating Geographic Information Syst~ins and Environmental Modeling. Breckenridge, Colorado, September 1993.

Dutton, Geoffrey. (1984). "Geodesic Modelling of Planetary Relief." Cartographica, v21, Numbers. 2 & 3, pp. 188-207.

Dutton, Geoffrey (1988). "Modeling Locational Uncertainty via Hierarchical Tessellation." ed. Goodchild Michael and Sucharita Gopal. Accuracy of Spatial Databases. New York: Taylor & Francis, pp. 123-140.

Dutton, Geoffrey (1989a). "Computational Aspects of Quaternary Triangular Meshes." Prime Park, MS: Prime Computer, Inc. (unpublished).

Dutton, Geoffrey (1989b). "The Fallacy of Coordinates." Prime Park, MS: Prime Computer, Inc. (unpublished).

Dutton, Geoffrey (1989c). "Glossary. of Basic, Obscure and Invented Geonomic Terms."Prime Park, MS: Prime Computer, Inc. (unpublished).

Dutton, Geoffrey (1989d). "Zenithial Orthotriangular Projection." Spatial Effects. Prime Park, MS: Prime Computer, Inc. (unpublished).

49

Dutton, Geoffrey (1991). "Improving Spatial Analysis in GIS Environments." Proceedings Auto-Carta 10, 1991, pp. 168-185.

Goodchild, Michael (1988). "The Issue Of Accuracy In Spatial Databases." Building Databases for Global Science, Mounsey, H., and Tomlinson, R. (editors). London: Taylor & Francis, pp. 31-48.

Hedley, Justin (1992). "3-D Migration via McClellan Transformations on Hexagonal Grids." Geophysics, v57, no. 8, pp. 1048-1053.

Ibbs, T. J. and A. Stevens (1987). "Quadtree Storage of Vector Data." Presented at GIS­LP'87, Lisbon, September 14-18.

Klette, Reinhard (1985). "The m-Dimensional Grid Point Space;'' Computer Vision, Graphics and Image Processing, v30, pp. 1-12.

Koffman, Elliot B. and Frank L. Friedman (1990). Problem Solvini and Strnctured Proifammini in FORTRAN 77. New York: Addison-Wesley Publishing Company, Inc.

Laurini R. and D. Thompson (1992). Fundamentals of Spatial Information Systems. New York: Academic Press Ltd.

Lauzon, J., D. Mark, L. Kikuchi, and A. Guevara (1985). "Two-Dimensional Run­Encoding for Quadtree Representation." Computer Vision, Graphics and Image Processing, v30, pp. 56-69.

Lillesand and Kieffer (1987). Remote Sensini and Imaie Interpretation. New York: John Wiley & Sons.

Mark, David M. and Jean Paul Lauzon (1985). "Approaches for Quadtree-based Geographic Informati9n Systems at Continental and Global S9ales." Proceedings Auto-Carto 7, Falls Church, VA: ASPRS/ACSM, pp. 355- 364.

The Towson State University Manual of GIS (1990). Department of Geography, Towson State University, pp. 1-13.

Martin, Johannes (1982). "Organization of Geographical Data with Quad Trees and Least Squares Approximation." Proceedings, IEEE Computer Society Conference on Pattern Recognition and Image Processing. Las Vegas, Nevada.

Morton, G. M. (1966). "A Computer Oriented Geodetic Data Base, and a New Technique in File Sequencing." IBM Canada Ltd.

Peucker, T. K. Fowler, R. J., Little, J.J., and Mark, D. M. (1986). Digital Representation of Three-djmensjonal Surfaces by Trianiulated Irre~ular Networks (TIN). Technical Report Number 10, United States Office of Navel Research, Geography Programs.

50

Peucker, Thomas K., et al (1978). "The Triangulated Irregular Network." Proceedings of the Digital Terrain Models Symposium of the American Society for Photogrammetry!American Congress on Surveying and Mapping. St. Louis, Missouri, USA, pp. 516-540.

Peuquet, D. J. (1984). "A Conceptual Framework and Comparison of Spatial Data Models." Cartographica 21, 4(1984), 66-113.

Peuquet, Donna J. ·and Duane F. Marble (1990). Introductory Readings in Geographic Infonnation Systems. New York: Taylor & Francis.

Pham, S. (1986). "On the boundary of Digital Straight Line Segments." Proceedings, Computer Graphics, Tokyo, 1986, in Advanced Computer Graphics (T. L. Kunni, Ed.), pp. 79-109, Springer-Verlag, Tokyo.

Pham, S. (1987). "Equations of digital straight lines," in Proceedings Computer Graphics, Tokyo, 1987 (T. L. Kunni, Ed.), pp. 221-248, Springer-Verlag, Tokyo.

Rogers, C. A. (1984). Packin~ and Coverin~. Boston: Cambridge University Press.

Rosenfeld, A. (1974). "Digital Straight Line Segments." IEEE Transactions, Computing, C-23, pp. 1264-1269.

Samet, Hanan (1990a). Design and Analysis of Spatial Data Structures. New York: Addison-Wesley Publishing Company.

Samet, Hanan (1990b). Applications of Spatial Data Strnctures. New York: Addison­Wesley Publishing Company.

Samet, Hanan and Robert Webber (1985). "Storing a Collection of Polygons Using Quadtrees." ACM Transactions on Graphics, Vol. 4, pp. 182-222.

Snyder, J. P. and P. M. Voxland (1989). An Album of Map Projections. Reston, VA: Department of the Interior, U.S. Geological Survey, Denver, CO.

Tobler, Waldo and Zi-tan Chen (1986). "A Quadtree for Global Information Storage." Geographical Analysis, v18(4), pp. 360-371.

Tsai, Victor J. D. and Alan p. Vonderohe (1993). "Delaunay Tetrahedral Data Modelling For 3-D GIS Applications." Proceedings, GIS!LIS'93, Minneapolis, MN.

Walker, Susan S. "An Introduction to Geographic Information System Terminology." Hunter College, Department of Geology and Geography, GEOG 790, Independent Research Project, December 22, 1987.

51

Wu, Li-De (1982). "On the Chain Code of a Line." IEEE Trans. Pattern Analysis, Machine Intelligence. PAMI-4(3), pp. 347-353.

Wuthrich C. A., and P. Stucki (1991). "An Algorithmic Comparison Between Square- and Hexagonal-Based Grids." Computer Vision, Graphics and Image Processing: Graphical Models and Image Processing. v53(4), pp. 324-339.

52

APPENDIX A: C Source Code

The following is a listing of the enclosed C source code files. Each file contains a

description header, which provides a brief introduction to each file, author(s), and functions

call~d. Each file was printed in a special format, which prints two file pages into one page.

File Name No, of Pa~es

1. Collignon.c 2

2. Main.c 1

3. Get_ TQS.c 3

4. Write_Etopo5.c 2

5. Swap.c 1

53

I* ************************************************************ *I /* Collignon.c REVISED TO DEAL WITH ONE OCTANT AND SCALED */ /* Function to transform a point to the Collignon Projection. •; /* The Collignon Projection is a pseudocylindrical, equal-area •; /* projection. Meridians are equally spaced straight lines •; /* converging at the North Pole, while parallels are unequally */ /* spaced straight lines, farthers apart near the North Pole, •; ;• closest near the South Pole, and perpendicular to the •; /* central meridian. Scale is true along latitudes 15deg5l'N. •; I* ************************************************************ */

finclude •extern.h" tinclude <math.h> #define PI 3.141592654 #define R 6378137 tdefine S 0.00000005 POINT

/* Radius of the Earth */ /*scale= 1:50,000,000 */

collignon(Point)

O;

POINT Point;

double sqrt_of_pi, two_pi_over_360, r_times_s, temp, lambda_sub

double mod_x, mod__y, rotate, Scos30, Ssin30; int i; float before x; POINT project; two_pi over 360 =PI/ 180.0; sqrt_of_pi ~ pow(PI, 0.5); r_times_s = (double) R * s;

Scos30 = xrange • 0.866025403; Ssin30 = xrange * 0.5; /* Assign the octant*/ if (Point.octant== 0)

else l Point.octant= 2;

if (Point.y > 0.0) l Point.octant= 4;

I else~

if (Point.x < 90.) Point.octant= 3;

if (Point.x < O.) Point.octant= 2;

if (Point.x < -90.) Point.octant= 1;

Point.octant= 5; if (Point.x < 90.)

Point.octant= 8; if (Point.x < 0.) •

Point.octant= 7; if (Point.x < -90.)

Point.octant= 6;

switch (Point.octant)

case 1: central meridian -135;

case 2:

case 3:

case 4:

case 5:

case 6:

case 7:

case B:

lat lo= 0.0; lat-hi= 90.0; long lo= -180.0; long-hi= -90.0; break;

central meridian= -45; lat lo~ 0.0; lat-hi= 90.0; long lo= -90.0; long-hi= 0.0; break;

central meridian= 45; lat lo~ 0.0; lat-hi= 90.0; long lo= 0.0; long-hi= 90.0; break;

central meridian= 135; lat lo~ o.o; lat-hi= 90.0; long lo= 90.0; long-hi= 180.0; break;

central meridian= 135; lat lo ~ -90.0; lat-hi= 0.0; long lo= 90.0; long-hi = 180. O; break;

central meridian= -135; lat lo~ -90.0; lat-hi= 0.0; long lo= -180.0; long-hi= -90.0; break;

central meridian= -45; lat lo~ -90.0; lat-hi= 0.0; long lo= -90.0; long-hi= 0.0; break;

central meridian= 45; lat lo~ -90.0; lat-hi= 0.0; long lo= 0.0; long-hi= 90.0; break;

/* Deal with the rotation of the central meridian*/ lambda_sub_O = two_pi over_360 * central_meridian; if ((Point.x - centraI meridian) > 180.0)

Point.x -= 360:o; else if ((Point.x - central_meridian) <= -180.0)

f_p1;

temp);

Point.x += 360.0;

/* Convert to radians*/

project.x = Point.x • two_pi_over_360; project.y = Point.y * two_pi_over_360; project.octant= Point.octant; project.elev= Point.elev;

1• * For southern hemisphere invert •1

if (Point.octant> 4) I project.y *= -1.0;

I /• Collignon transformations•/

temp= pow((l.O - sin((double) project.y)), 0.5); project.x = 2.0 * r_times_s * (project.x - lambda_sub_O) •temp/ sqrt_o

project.y = cos(30. * two_pi_over_360) * sqrt_of_pi * r_times_s * (1.0 -

1• • For southern hemisphere, reflect about axis of central meridian •1

if (Point.octant> 4) I project.x •= -1.0;

I return (project);

, .....•...•...•....•..•..••....•.......•....•.. , /* Main.c: */ /* - Read the ETOP05 dataset from its */ /* file into a structure. */ I* - Author: Keith C. Clarke */ I* - Last edited on 7-12-94. */ /******************************•***************/ tinclude <math.h> tinclude <stdio.h> linclude •extern.h" ldefine INPOT FILE "/home/popocatepetl/kclarke/world/etopo5/world.lan• main() -l

void POINT FILE

read erdas(); Point, collignon();

*fp;

fp = fopen(INPOT_FILE, "r");

(void) read erdas(fp); (void) fclose(fp);

/************************************************/ /* Get TQS.c: /* - - Author: Jaime Lugo /* - Last modified 07-12-94 /* - Contains functions: /* 1. Get TQS () I* 2. Tessellate 0

*I */ */ */ *I *I

/************************************************/ #include <stdio.h> tinclude <math.h> tinclude <ctype.h> linclude "header.h" tinclude •extern.h" tdefine FALSE 0 tdefine TRUE 1

1·····················································1 /* FUNCTION Get TQS(): */ /* - Calcuiates xrange, yrange for all octants. */ /* - Calls Function Tessellate(), which gets */ /* the TQS address for the erdas (x,y) point. */ /*****************************************************/ POINT Get TQS (Point)

- POINT Point;

POINT int int double

collignon(), Setup; tqs; Tessellate(); xrange, yrange;

/* get x-range value for whole octant*/ setup.x = -90.0; setup.y = o.o; Setup.octant= O; Setup = collignon(Setup); xrange = Setup.x; if (xrange < 0.0)

xrange *= -1.0; xrange *= 2.0; /* Original xrange = 0.565248 */

/* get y-range value for whole octant*/ Setup.x = -45.0; Setup.y = 90.0; Setup.octant = .O; Setup= collignon(Setup); yrange = Setup.y; /* Original yrange = 0.489519 */

/* printf("\n Original xrange = \f", xrange); */ /* printf("\n Original yrange = \f\n\n•, yrange); */

/* Transform Point.x and Point.y Point = collignon(Point); /* printf("x,y After Collignon: /* printf("gets placed in octant

into a Collignon coordinate. */

(\f \f) •, Point.x,Point.y); */ \d\n\n", Point.octant); */

/* Get the QTM address for that point. */ Point.qtm = Tessellate(Point, xrange, yrange);

/* printf("Tessellate() returns [\X] in Hex notation.\n\n",tqs); */

return (Point); I

/******************************************************/ /* FUNCTION Tessellate(): */ I* - Ftn tessellates each triangular octant up */ /* to the maximum depth specified. */ /* - MAXDEPTH has been assigned 12 recursions. */ I* - Finally, ftn returns a variable address */ /* up to 24-bits long (3 bytes) */ /******************************************************/ int Tessellate(Point,

POINT xrange, yrange)

Point; double xrange, yrange;

POINT int

short

double

Testyt; depth, new qtm, qtm~ MAXOEPTH end loop, do shift; before x, midpt y, cos 180, sin-180, cos-240, sin-240;

Test_J>t = Point; -cos 240 = 0.9973288; sin-240 = 0.0730431; cos-180 = 0.9984972; sin-180 = 0.0548037; qtm- = 00; new qtm = 00; do shift = TRUE; midpt_y = (0.5 • yrange);

12;

/* Temp storage for (x,y) */ /* counts depth level */

/* stores qtm triangle number*/ /* stores shifted QTM address*/

/* Maximum depth*/ /*flag*/ /*flag*/

/* midpoint of y-axis*/

I* cos 180 = cos(PI) /* sin-180 = sin(PI) /* stores the final qtm address in hex /* stores qtm address just assigned /* bit-shift if "do shift• is true

/*****************************************************/ /* TESSELLATION LOOP: Body of function Tessellate() */ /* Tessellate only to the max depth for the dataset: */

1·····················································1 for (depth= O; depth< MAXDEPTH; depth++) {

end loop= FALSE; /* -Check whether MAXOEPTH has been reached. */ if (depth== (MAXDEPTH - 1))

do shift= FALSE; /**********************************************/ /* if point falls in qtm triangle one= qtm.1 */ /**********************************************/ if ((Test_pt.y > midpt_y) && (Test_J>t.y <= yrange)) {

/* Assign point to triangle #1 and shift left */ /* while the tessellation hasn't reached MAXDEPTH. */ new qtm = 1; /* printf("Ol "); */ qtm = (new_qtm I qtm);

/*bit-parser*/ /* ... OR ... */

*I *I */ *I *I

I

end loop= TROE; if (do shift != FALSE)

/* tells computer to stop loop*/ /* Tessellate until MAXDEPTH is reached*/

I -/* The next section will run until MAXDEPTH has been reached. •; /* If MAXDEPTH hasn't been reached, then(Test_pt.x,Test_pt.y) */ /* is divided in half, along with the xrange, yrange and all •; /* the x,y boundaries. Thus, if the (x, y) pcint falls within•; /* qtm.1, then the base of the qtm.1 triangle gets set to •; /* y = O. The local loop ends here (skip rest of function). */ I* I\ • I ;• I \ midpt_y •; /* I 01 \ I */ I• I \ I --\ • I /* 1,---1, --/ /\ */ I• I \ 00 I \ I \ • I I* I 02 \ I 03 \ I 01 \ •; /* I \/ \ <-base-> I \ *I /***********~*******************'**************************/ qtm <<= 2; Test_pt.y xrange yrange midpt_y

*= *=

·=

midpt_y; 0.5; 0.5; o. 5;

else if (end loop != TROE) I -

;• Rotate Point (x,y) such that triangle QTM.2 is above midpt_y. •; , ......................................... , /* /\ /\ •; /* I \ I \ *I I• I 1 \ /-\ / 2 \ • / I• I \ \ \/ I \ • I /* 1,---/\ 1,---/\ */ /* I \ 0 I \ I \ 0 I \ */ /* I 2 \ I 3 \ / 3 \ / 1 \ */ I* I \/ \ I \/ \ *I /***********~************************/ before_x = Test_pt.x; Test_pt.x = (Test_pt.x • cos_240) - (Test_pt.y • cos_240); Test_pt.y = (before_x * sin_240) + (Test_pt.y • cos_240); Test_pt.x -= (0.25 • xrange); Test_pt.y += midpt_y;

/**********************************************/ /* if point· falls in qtm triangle two= qtm.2 •; /**********************************************/ if ((Test_pt.y > midpt_y)&&(Test_pt.y<=yrange)&&(end_loop != TRUE)) {

new qtm = 2; /* printf("lO "); */ qtm = (new qtm I qtm); end loop =-TROE; if (do shift != FALSE) { -

qtm <<= 2; Test_pt.y -= midpt_y; xrange •= 0.5; yrange •= 0.5; midpt_y *= 0.5;

/************************************************/

/* /\ */ /* I \ midpt_y •; /* I 2 \ I */ I* I \/ --\ */ /* 1,---1, --/ /\ */ /* I \ 0 I \ I \ */ /* I 3 \ / 1 \ / 2 \ */ I* I \/ \ I \ */ /****************'********************************/ I

I else if (end loop != TRUE) I -

/* Rotate Point (x,y) such that triangle */ /* QTM.3 1s above midpt_y. */ I* /\ /\ • I /* I \ I \ • I /* I 2 \ /-\ I 3 \ * / I* I \ \ \/ I \ • I I* /\ ___ /\ /\ ___ /\ */

/* I \ 0 I \ I \ 0 I \ */ /* I 3 \ / 1 \ / 1 \ / 2 \ */ I• I \/ \ I \/ \ * I /************************* .......... ********'******/ before_x = Test_pt.x; Test_pt.x = (Test_pt.x • cos_240) - (Test_pt.y • cos_240); Test_pt.y = (before_x • sin_240) + (Test_pt.y • cos_240); Test_pt.x -= (0.25 • xrange); Test_pt.y += midpt_y;

/************************************************/ ;• if point falls in qtm triangle three= qtm.3 •; /************************************************/ if ((Test_pt.y>=midpt_y)&&(Test_pt.y<=yrange)&&(end_loop!=TRUE))

l

new qtm = 3; ;• printf("ll "); •; qtm = (new qtm I qtm); end loop =-TRUE; if (do shift != FALSE) { -

qtm «= 2; Test_pt.y -= midpt_y; xrange •= 0.5; yrange •= 0.5; midpt y •= 0.5; /****************************************/ ;• I\ • I I* I \ midpt_y */ /* I 3 \ / */ I* I \/ --\ * I I* 1,---/\ --I /\ *I /* I \ 0 I \ I \ */ ;• I 1 \ / 2 \ / 3 \ •; I* I \/ \ I \ • I /*****************'*****************°******/

else if (end loop != TRUE) { -

return (qtm);

/*********************************************/ /* Rotate Point (x,y) such that triangle */ /* QTM.1 is above midpt_y once more. */ /*********************************************/ before_x = Test_pt.x; Test_pt.x = (Test_pt.x * cos_240) - (Test_pt.y • cos_240); Test_pt.y = (before_x * sin_240) + (Test_pt.y • cos_240); Test_pt.x -= (0.25 • xrange); Test_pt.y += midpt_y;

/*********************************************/ /* /\ /\ *I /* I \ I \ */ I• I 3 \ /-\ / 1 \ • / /* I \ \ \/ I \ *I /* 1,---1, 1,---1, */ /* I \ 0 I \ I \ 0 I \ *I /* I 1 \ / 2 \ / 2 \ / 3 \ */ /* I \I \ I \/ \ *I /*********************************************/

/******************************************************/ /* Rotate 180 degrees (invert center triangle):, qtm=O */ /* Then, raise the base of triangle Oto (y = 0.0). */ /******************************************************/ before_x = Test_pt.x; Test_pt.x = (Test_pt.x • cos_l80) - (Test__pt.y • cos_180); Test_pt.y = (before_x * sin_l80) + (Test_pt.y • cos_180);

Test_pt.y += midpt_y; /* elevate the base*/

/***********************************************/ /* QTM = qtm.O: Although the next two lines do*/ /* nothing, they help visualize the process. */ /***********************************************/ new qtm = 00; qtm - = (new qtm I qtm) ; /* printf("OO "); */ end loop= TRUE; if (do shift != FALSE) { -

qtm <<= 2; xrange *= 0.5; yrange *= 0.5; midpt_y *= 0.5; /************************************************/

·;• /\ *I /* I \ midpt_y */ /* I 3 \ / */ /* I \/ --\ * I /* /\ ___ /\ --/ \ I --\ /\ *I I* I \ 0 I \ \ 0 I --/ I \ * I I* I 1 \ / 2 \ \ / / 0 \ */ /* I \/ \ \/ I \ *I /*****************"*******************************/

/* Filename: write etopo5.c */ /************************************************/ /* read erdas(): */ I* - Inputs data from ERDAS format file */ /* containing the ETOP05 world relief */ /* database at 5 minute intervals. */ /* - Creates a structure indicating: */ /* 1. Octant */ I* 2. x-coordinate */ /* 3. y-coordinate */ /* 4. Elevation in meters */ /* 5. Triangulated Quadtree */ /* Structure (TQS) address, in */ /* hexadecimal notation. */ /* */ /* - Authors: Clarke, K., and J. Lugo */ /* - Date : 6-03-91 Last Change: 7-12-94 */ /************************************************/ tinclude "extern.h" tinclude <stdio.h> tinclude <ctype.h> tinclude "header.h" void read erdas(infile)

FILE *infile;

static union { unsigned char short int long int

byte[512]; half[256]; word[128];

POINT FILE FILE int void short short long long int double double

buffer;

Point, coll1gnon(), Get TQS(); *fpl, *fp2, *fp3, *fp4; - /* Northern Hemisphere*/ *fp5, *fp6, *fp7, *fp8; /* Southern Hemisphere*/ a, b, i, row, col, elevation; buzzswap (); ipack; nbands; irows; icols; sixteen bit; longituae, latitude, increment= 1.0 / 12.0; old_lat, old_long;

/* Read the first block, containing the header*/ fread(&buffer.byte[O], 1, 128, infile);

/* swap the first twelve half words, which contain the critical data*/ (void) buzzswap(&buffer, 12);

/* Read the byte size*/ ipack = buffer.half[3];

/* Next the number of bands*/ nbands = buffer.half[4]; if (nbands > 1) I

printf("ERDAS read is for single band files only\n"); /* return; */

if (ipack == 2) sixteen bit= l;

else I -sixteen bit= O;

/* Next the rows and columns*/ icols = (long) buffer.half[B]; irows = (long) buffer.half[lO];

if ( (irows >= MAXROW) 11 (icols >= MAXCOL)) I printf("Too many rows and/or columns. Change header.h - recompile \n"); exit();

l /* Skip remainder of buffer*/

/* open files to store each octant separately*/ fpl = fopen("/serverdisks/scratch/lugo tmp/octantl.dat•, "w"); fp2 = fopen("/serverdisks/scratch/lugo-tmp/octant2.dat•, "w"); fp3 = fopen("/serverdisks/scratch/lugo-tmp/octant3.dat•, "w"); fp4 = fopen("/serverdisks/scratch/lugo-tmp/octant4.dat•, "w"); fp5 = fopen("/serverdisks/scratch/lugo-tmp/octant5.dat•, •w•); fp6 = fopen("/serverdisks/scratch/lugo-tmp/octant6.dat•, •w•); fp7 = fopen("/serverdisks/scratch/lugo=tmp/octant7.dat•, •w•); fp8 = fopen("/serverdisks/scratch/lugo_tmp/octant8.dat•, "w");

/* Start the main data loop*/ nrows = (int) irows; ncols = (int) icols; row= nrows - i; col = O; latitude= 90.0; longitude= -180.0; old long= O.; old=lat = o.;

while (!feof(infile)) { fread(&buffer.byte[O], 1, 512, infile); if (sixteen bit) I

(void) buzzswap(&buffer, 256); for (i = O; i < 256; i++) I

elevation= (int) buffer.half[i]; /********************************************************/ I*

• Print to file those points with elevation • above zero *I

!********************************************************/ if (elevation> 0) I

Point.x = longitude; Point.y = latitude; Point= Get TQS(Point); Point.elev~ elevation;

/****************************************************/ I*

• if the next point is equal to the • previous, skip it. *I

/* * But first, each point must be

• checked as an integer. *I

/****************************************************/ a= (int) (1000. * (old_long - longitude)); b = (int) (1000. * (old_lat - latitude));

if ((a - b) != 0) { switch (Point.octant)

case 1: fprintf(fpl, "\ld •, Point.octant); fprintf(fpl, "\6d •, Point.elev); fprintf(fpl, "\X\n•, Point.qtm); break; case 2: fprintf(fp2, "\ld •, Point.octant); fprintf(fp2, "\6d •, Point.elev); fprintf(fp2, "tX\n•, Point.qtm); break; case 3: fprintf(fp3, "\ld •, Point.octant); fprintf(fp3, "\6d •, Point.elev); fprintf(fp3, "\X\n", Point.qtm); break; case 4: fprintf(fp4, "\ld •, Point.octant); fprintf(fp4, "\6d •, Point.elev); fprintf(fp4, "\X\n", Point.qtm); break; case 5: fprintf(fp5, "\ld •, Point.octant); fprintf(fp5, "\6d •, Point.elev); fprintf(fp5, "\X\n•, Point.qtm); break; case 6: fprintf(fp6, "\ld •, Point.octant); fprintf(fp6, "\6d •, Point.elev); fprintf(fp6, "\X\n•, Point.qtm); break; case 7: fprintf(fp7, "\ld •, Point.octant); fprintf(fp7, "\6d •, Point.elev); fprintf(fp7, "\X\n•, Point.qtm); break; case 8: fprintf(fpB, "\ld •, Point.octant); fprintf(fp8, "\6d •, Point.elev); fprintf(fpB, "\X\n", Point.qtm); break; /* End of switch*/

I /* End of old= new test */ f /* End of elev +ve test */ old long= longitude; old-lat= latitude; col++; longitude+= increment; if (col== ncols) {

col= O; longitude= -180.; latitude-= increment; row--;

} fclose(fpl) fclose (fp2) fclose (fp3) fclose(fp4) fclose(fp5) fclose(fp6) fclose(fp7) fclose(fpB) return;

/****************************************/ /* swap.c: */ /* swap the bytes of a 16 bit value. */ I* - Author: Keith Clarke, Ph.D. */ /* Hunter College-CONY */ /****************************************/ unsigned short wbyteswap(x)

unsigned short x;

return (((x & OxFF) << 8) (X » 8));

/* swap the bytes of a 32 bit value*/ long lbyteswap(x)

I

long x;

return ((long) wbyteswap((int) (x & OxFFFF))) << 16 I ((long) wbyteswap((int) x >> 16));

typedef union rtbuf I unsigned char unsigned short long int

b[512); h[256);

w[128]; rtbuf;

/* Buz's swap routine from Pascal at Ames 7-93 */ void buzzswap(buf, n)

rtbuf *buf; int

int unsigned j = 1; for (i

char

n;

i, j; save;

O; i < n; i++) save= buf->b[j - 1); buf->b[j - 1) = buf->b[j); buf->b[j] = save; j += 2;

/* * * language cartographic data structures after the/* * data standards kcc 5-88 /* * */

/* Define the constants*/ #define MAXCHAR 20 tdefine MAXPTS 999 tdefine MAXROW 20 tdefine MAXCOL 20 #define MAXHOLE 3 #define MAXSTRING 400 #define MAXARC 100 idefine MAXLINK 100 fdefine MAXCHAIN 100 #define MAXLABEL 20 typedef float coordinate; /* Zero dimensional objects */ typedef struct {

coordinate x; coordinate y;

POINT; struct ENTITY POINT

};

int POINT;

feature_type;

struct LABEL POINT char - label [MAXCHAR); POINT point;

}; struct AREA POINT {

int- polygon id; POINT point; -

}; struct NODE {

int POINT

};

node_identifier; node;

/* One-dimensional objects - Geometry Only*/ struct LINE SEGMENT {

POINT point[2); }; typedef struct

int POINT

} String; struct ARC {

int String int

}; struct RING {

int int POINT

};

number_of_points; point[MAXPTSJ;

arc identifier; string; -

*function;

ring identifier; number_of_points;

point[MAXPTSJ;

struct RING OF STRINGS { int- - number of strings; String string[MAXSTR!NG];

/* C digital cartographic

}; struct RING OF ARCS

int- -struct ARC

};

number of arcs; arc [MAXARCJ ;

/* One-dimensional cartographic objects - Geometry and topology*/ struct LINK {

struct LINE_SEGMENT link; }; struct DIRECTED LINK

char -struct NODE struct NODE

};

link identifier; from; to;

struct CHAIN { int struct String int int

chain identifier; DIRECTED LINK node;

string;

};

left_polygon; right _polygon;

struct RING OF CHAINS int- - number of chains;

chain[MAXCHAINJ; struct CHAIN }; struct RING OF LINKS

int- - number of links; link [MAXL!NKJ; struct LINK

}; struct AREA CHAIN {

String string; int left_polygon; int right_polygon;

}; struct NETWORK CHAIN {

struct-DIRECTED LINK node; String string;

}; /* Two-dimensional objects */ struct POLYGON

};

int char String

polygon_identifier; polygon_descriptor[MAXLABELJ;

boundary;

struct COMPLEX POLYGON {

};

int char String int String

struct GRID { char int int POINT int int

};

- polygon_identifier; *polygon descriptor;

outer boundary; - number of holes;

holes[MAXHOLE]; -

grid_descriptor[MAXLABELJ; nrows; ncols;

corners[4); datum; z[MAXROW) [MAXCOLJ;

(i)

/****************************************/ /* extern.h: */ /* - Header file containing the */ /* Point structure for ETOP05. */ /* - Authors: Clarke, K.C. */ /* J. Lugo */ /****************************************/ int central meridian; float lat lo, lat hi, long lo, long hi, xrange; typedef struct { - - -

float x; float y; int elev; int octant; int qtm;

POINT;

[i)

#include <stdio.h> #define MAXROW 2161 #define MAXCOL 4321 short int elev; int npts,

found, nrcws, ncols;

(i) /* Number of pts in x,y,z */

/* Number of rows and columns in zz[] [] */

APPENDIXB:

Glossary of Terms Source: Walker (1987), unless otherwise indicated.

Accuracy: "A measure describing the correctness, or accuracy of a condition, quality or quantity".

Address: (Synonymous with Location): A term generally referring to a location within a [computer's storage]. The term is normally used in the context of retrieving or storing Data.

Apex: 1) The uppermost point of an object (e.g., triangle); 2) The uppermost subset on a tessellated triangle, labeled zero, and used to determine spatial orientation (Webster's New World Dictionary).

Area: 1) The interior of a continuous two dimensional object. 2) A defined part, space, tract or region of the earth's surface.

Array: An orderly set of related attributes, entities, or elements in the form of a grid or matrix.

Attractor: A nodal region that dominates neighboring cells in a tessellation (Dutton, 1989c).

Attribute: 1) A characteristic, or property, associated with an observation or entity. 2) Non-spatial data.

B+-Tree: Modified the quadtree structure to fit large databases, where each tessellation level needs referencing (see Quadtree). Instead of the base4 representation in Morton sequencing and QTM, the B+-Tree uses a base5 system to implicitly indicate the level of the tree (Abel, 1984 ).

Binary Code: A code which is restricted to the binary digits O and 1.

Binary Digit (Bit): 1) The digit zero or one. 2) A means of representing information within the computer using the digits zero and one (indicating yes/no, true/false, on/off).

Bit manipulations: Programming techniques that allow bitwise control over linkage, location, addressing and manipulation of data. See: C.

Bit Matrix: A two-dimensional grid map where each cell is encoded using a binary digit.

C: A high level programming language originally developed for implementation with the UNIX operating system. C is the preferred programming language for geographic applications and computer graphics programmers.

Cartography, Analytical: Analytical cartography is the study of "theoretical and mathematical background behind cartography and the rules cartographers employ in the mapping process ... (Tobler, 1976)." (Qarke, 1990, p. 2).

Cartography, Computer: Computer cartography is "the particular set of methods and techniques which the current [computer] technology uses to produce maps." (Clarke, 1990, p. 2).

Cell: 1) Synonymous with Grid Cell). One square or rectangle formed by a matrix or grid. 2) An address or location in storage that [is] usually capable of holding one binary digit. 3) The principal means by which spatial data is represented by a raster data structure.

Centroid: 1) The center point of a single grid cell or polygon. 2) The center point, cell, or polygon of an area.

63

Coordinates: Two or more numbers that specify the absolute position of a point, line, area or entity in space. Numeric values of latitude and longitude are the coordinates most commonly used to locate features on the earth's surf ace.

Data Accuracy: A measure of accuracy that defines the accordance between 'real world' phenomena and their representation within a database.

Data Structure: A logical method of organizing, or representing spatial data within a database. Data structures control the means by which a computer stores, accesses and displays spatial patterns and features. The two most commonly recognized data structures are the raster data structure and the vector data structure.

Database: The collection of all machine readable files pertaining to a given location or entity. The database is defined accessed using the facilities of a database management system.

Database Management System (DBMS): A Software system that organizes information within the database for all phases of input, editing, retrieval, analysis, manipulation, and output.

Database Structure: (Synonymous with Data Model): A logical system of organizing and connecting data files within a database. The database structure controls the efficiency with which a computer answers a query. There are three principal types of database structures: the Hierarchical Database Structure, the Relational Database Structure, and the Network Database Structure.

Edge: A boundary between faces connecting vertices of a graph; a one-cell.; a line defining part of a polygon (Dutton, 1989c).

Entity: A real world phenomenon that cannot be subdivided into a group of elements.

Extent: The total study area represented in the database of a geographic information system.

FORTRAN (FORmula 1RANslation): First used in 1957, FOR1RAN was the first high-level programming language, primarily used for scientific computations. In the 1970s, it was modified to facilitate structured programs, which are needed for high-level graphics (Koffman and Friedman, 1990).

Geocode: A geographic code used to represent spatial data (See Code).

Geocoding: The process of assigning a geocode to a computer record.

Geographic Information System (GIS): A system designed to accept, store, manage, recall, analyze and display spatial and non-spatial data.

Graphical Kernel System (GKS): The first international standard for programming computer graphics. The principal purposes of this standard are: to facilitate the exchange of graphics programs between different computers, to aid the understanding and use of graphics programming and to provide hardware and software manufacturers with a guideline of useful combinations of graphic devices and software.

Graticule: A regular ruling across a model surface; a standard reference mesh based on a grid oriented in a manifold.

Grid: "The division of [an] area into square, or rectangular cells of equal size. (See Coordinates, Matrix, Raster).

Hexagon: A polygon with six angles and six sides.

64

Hierarchical Database Structure: A method of strucnrring files so that units of data are connected and stored via a hierarchically dependent pathway.

High Level Language: A computer programming language with a strucnrre that accommodates the requirements of the programmer rather than the architecnrre of a computer's data strucnrre and control unit [CPU].

Interpolation: To estimate the value of an attribute at an unsampled site by quantitatively calculating the relationship between the unsampled site and one or more sampled sites that lie within the same neighborhood. The technique is based upon the observation that points that are close together are more likely to have a similar attribute value than two points that are far apart. Interpolation allows irregularly sampled data to be represented at regularly sampled intervals. The most common interpolation techniques include: Weighted Averaging, Trend Surface Analysis, Fourier Analysis, Splines, Nearest Neighbor Analysis and Kriging.

Language: Any code used to communicate programs or simple instructions to a computer.

Lattice: A i.fmkft. A0

is a discrete number of points within the set of all real numbers in a two­dimensional space, such as a map projection, where CJ indicates the geometric base of the lattice. Therefore, the lattice strucnrres for both the square and hexagonal grids are A4 and A6, respectively (CJ = 4 for the square lattice A4, and CJ = 6 for the hexagonal lattice A6).

Line (Synonymous with Segment): 1) A one-dimensional object defined by being the connection between two points or coordinates. 2) The basic building block of a polygon.

Linear Array (Synonymous with Vector): A one-dimensional array of attribute values or elements.

Link: 1) A one-dimensional object that is a direct connection between two nodes.

Location: A place, or point in space. In computing, the term is synonymous with address.

Low Level Language: A programming language with a strucnrre that directly reflects the architecnrre of a computer's data strucnrre and [CPU] rather than the needs of the programmer, such as Assembly language.

Map Extent: See Extent. For all lattice points"'· let the map extent TI0

(within the lattice base CJ) equal all points (y) belonging to the set of real numbers R2, where (y) equals all the lattice points (A), plus the vectors Vn (with the corresponding x-coordinate also belonging to the lattice) (Wuthrich & Stucki, 1991, p. 325).

Mathematical Model: A set of formal, mathematical rules that define the exact relationships between the elements of a physical process or nanrral phenomenon.

Matrix: A two-dimensional, rectangular array of attribute values, or elements.

Model (Synonymous with Terrain Model): A representation of a nanrral phenomenon or physical process. The function of a model is to abstract, simplify and organize an aspect or process of the terrain in order to focus attention on the function of one or more elements or entities.

Morton Sequencing: (Morton, 1966). See Chapter 2.3.

Nearest Neighbor Analysis: 1) "A spatial analysis technique that compares the spacing of an observed point distribution with the spacing expected in a random point distribution." (Walker, 1987, p. 19). 2) The value of a picnrre element (pixel) "in the output matrix is determined on the basis of the pixel values which surround its transformed position in the original input matrix" (Lillesand and Kieffer, 1987, p. 614)

65

Node (Synonymous with Junction): a zero-dimensional object that is considered to be the topologic junction between two or more links.

Non-Spatial Data: Data which are one-dimensional, such as text. Object: An abstract digital representation of a feature.

Octahedron: A regular polyhedron consisting of 6 vertices, connected by 12 edges to form 8 triangular facets (Dutton, 1989c).

Picture Element (Pixel): The smallest, non-divisible, two-dimensional element of an image. Pixels are normally in the shape of a square.

Platonic Solids: Plato's basic geometric structures - Tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron- (Gasson, 1983; Tobler, 1986; Laurini, 1992).

PM3' quadtree: Section 2.3 (Samet and Webber, 1985).

Point: A zero-dimensional object that specifies a geometric location. Every point has a set of coordinates associated with it.

Pointer: A high-level programming operation that helps locate the specific address of data in computer storage.

Projection: 1) (Map Projection) A coordinate transformation used to represent part or all of the earth on a two-dimensional surface... 2) In computing, to combine selected files from two independent relations (See Appendix A: Collignon, UIM, and Butterfly Projections).

Quadtree: The quadtree is a variation of the hierarchical data structure, where the map extent is subdivided into four progressively smaller quadrants, until the smallest homogeneous unit of data is reached (Clarke, 1990, p. 115, and Walker, 1987, p. 23).

Quaternary Triangular Mesh (QTM): See Chapter 2.4 for a full description.

Raster Data Structure: A data structure for organizing or representing spatial data within a database. The form of a feature is explicitly constructed from a set of grid or raster cells. Each Cell contains a number that represents the type of Feature or Attribute being Mapped. The Address of the Cell is referenced by a row and column number. Raster-To-Vector: The process of converting from a raster data structure to a vector data structure.

Relation: An ordered set of attribute values within a relational database structure that are grouped together in a two dimensional table. Each relation is usually considered to be a separate File. (Burrough, 1986)." (Walker, 1987, p. 24).

Resolution: The smallest discernible unit of information that may be shown on a visual display unit or image.

Spatial Data: Any data with locational attributes such as a map, image, or aerial photograph.

Tessellation: 1) An area divided into tiles. 2) A tessellation is also described as the subdivision of every facet within a geometric figure (e.g. octahedron), such that each facet is divided into four subfacets without losing the original geometric shape. Each tessellation level yields a larger number of geometrically identical figures (that is, triangles), as seen in Figure 1. Multilateral tessellations are ideal for spherically oriented three-dimensional graphics.

66

Topology: "Topology is a mathematical procedure for explicitly defining [and storing] ... spatial relationships for map features: area of definition, connectivity and contiguity" (GIS/fowson University, 1990, p. 10).

Triangulated Irregular Network (TIN): Developed by Peucker (1978). "A digital terrain modelling product that represents spatial variations of elevation" (Walker, 1987, p. 29).

Transformation schemes: "The functions that transform the real plane into the discrete set of points which were chosen to represent the image space of a digital environment" (Wuthrich and Stucki, 1991, p. 326).

Universal Transverse Mercator Projection (UIM): A conformal, cylindrical map projection where bearings are straight lines and distances are correct.

UNIX: An all-purpose operating system produced by Bell Laboratories. It provides a simple, uniform, environment that allows a number of users to operate on a single system. (Walker, 1987, p. 29).

Vector: 1) (Synonymous with Linear Array; Compare with Matrix). 2) A Line with a specified angle, direction and length.

Vector Data Structure: A data structure for representing spatial data within a database. The form of a feature is implicitly constructed from a set of points and lines that form polygons. Each polygon is assigned a unique pointer that identifies the address of the feature's coordinates and attributes.

Vector-To-Raster: The process of converting a vector data structure into a raster data Structure.

Vertex: A mathematical point that serves to bound two or more faces in a graph, plays a role in a figure, or defines a node in a tessellation (Dutton, 1989c).

67

APPENDIX C: ETOP05

1: Introduction

ETOP05 is a global relief dataset, in CD-ROM format, from NOAA's National Geophysical Data Center and the United States Department of Commerce. ETOP05 contains global and local data at very high resolution. This CD-ROM contains ERDAS format global images, geographic data for coastlines and global geodynamic features, gazetteers of undersea features, satellite marine gravity data, topography and bathymetry data, and application software. Data is in IS0-9660 file format.

2: "README.lst." Enclosed documentation about the ETOP05 5-minute ifidded

elevation data.

1. Sources for ETOP05:

ETOP05 was generated from a digital data base of land and sea- floor elevations on a 5 minute latitude/longitude grid The resolution of the gridded data varies from true 5-minute for the ocean floors, the U.S.A., Europe, Japan, and Australia to 1 degree in data-deficient parts of Asia, South America, northern Canada, and Africa. Data sources are as follows: Ocean Areas: U.S. Naval Oceanographic Office; U.S.A., W.

Europe, Japan/Korea: U.S. Defense Mapping Agency; Australia: Bureau of Mineral Resources, Australia; New l.ealand: Department of Industrial and Scientific Research, New l.ealand; balance of world land masses: U.S. Navy Fleet Numerical Oceanographic Center.

2. Data Formats for ETOP05:

The data file is formatted as 16-bit BINARY IN1EGERS in two byte orders: Ef0P05.D0S is in IBM-PC/DEC-VAX "swapped," lo-byte-first order. The file ETOP05.DAT is in "normal," or hi-byte-first order. as used by Macintosh, Sun, and some other workstations. In both files, there are 2160x4320 data values, one for each five minutes of latitude and longitude, for a total of 9,331,200 points or 18,662,400 bytes. Data values are in whole meters, representing the elevation of the CENTER of each cell. The files "EfOP05.FMf;l" and ETOP05.IIDR;l" are placed in the ETOP05 directory for coovenient use of the "FREEFORM'' and "GEOVU2" utilities being developed at NGOC. (These programs were not available for inclusion on the CD -- watch for an announcement of their release.)

3. Data Order in the Files:

The file may be thought of as having a logical record size of 8640 bytes. The data start at the North Pole (90

N, 0 O' E) and are arranged in bands of 360 degrees x 12 points/degree= 4320 values (8640 bytes) ranging eastward from O O' East longitude to 359 55' East longitude (since it represents the North Pole, all possible longitudes still refer to a single point, thus the first band has 4320 identical values of -4290 m). The 8641st starts the latitude band for 89 55' N. and so on. There is NO record for the South Pole (elevation 2810 m.)

68

4. USING THE PC DOS ACCESS PROGRAMS FOR ETOPOS DATA:

(Macintosh users will fmd their own software in the Mac part of this CD-ROM)

There are three programs in this directory for accessing the ETOPOS data on a PC-compatible computer (Macintosh users will find their specific software in the "Macintosh Files: Public Software: NGOC Software" directory.). The PC programs are:

1) SELTOPl.EXE 2) SELTOPS.EXE 3) SELTOPSB.EXE

Each program copies a data selection from the ETOPOS data base to a user-specified file. The

programs are run by entering

set ETOPOS=<path to Ef0P05.D0S> SELTOPx /r 8640

where "x" is "1", "5", or "SB", and "path to Ef0P05DOS" is the pathname to access the file "Ef0P05DOS" from the CD-ROM, or from a copy on hard disk, e.g., "F:\TOPO\ETOP05\ETOP05.D0S". The command-line arguments "/r 8640" are necessary to set up a buffer for file access. As their names suggest, the three programs select data as follows: SELTOPl copies to ASCII (text) data by 1-degree squares, SELTOPS copies to ASCII data by 5-degree squares, and SELTOP5B copies to 16-bit binary data by 5-degree squares. Each program prompts you for the name of a file to receive the copied data (the file name must not already exist), followed by the latitude and longitude in whole degrees of the southwest comer of the data area. The program then asks for the number of 1- or 5- degree squares north and east of the origin to be copied -- an area 30 degrees wide (W-E) and 20 degrees high (S-N) would consist of 4 (5-degree) blocks northward and 6 (5-degree) blocks eastward of the origin (or 20x30 1-degree blocks). The ASCII output files contain this type of header:

"Data format: depths/elevations in whole Meters (+ = above sea level). Lat and Ion are given for South West corner of 1-degree square area covered by a 13xl3 matrix of 5-minute grid values starting at the NorthWest corner of the square and then given Eastward along increasing longitude for 13 values, then stepping 5 minutes South for the next row (last row, first entry= SW corner point of degree square)."

followed by the data. The binary files have no header.

69

UNIVERSITY OF MINNESOTA

Twin Citks Campus 0/Jke of Admissions Graduate School

MASTER'S HOLD REMOVAL FORM

309 Johnston Hall 101 PleasantStreetS.E. Minneapolis, MN 55455-0421

612-625-3014

A. If your master's degree has not been completed, or you have completed the degree, but do not have an official transcript showing the award of the degree, with the department's permission we will be able to remove the master's degree hold to allow registration.

This form must be signed by a member of the graduate faculty in the department to which the student was admitted. When completed, return this form to 309 Johnston Hall.

I understand that =m,...-'-r_._-s;_o...t_4

_m_.L ___ A_. __ ~ ___ ;:o.__, admitted to the Graduate School beginning ~ e:6&, \<';\C\'::\ , has not completed/has no proof of the award of the Master's degree.

/ ----- His/her admission was not contingent on the completion/proof of the master's degree. Please allow this student to register without completion/proof of degree award.

----- His/her admission~ contingent on the successful completion of the master's degree. I do not wish this student to register until proof of the award of the degree is available.

~~ Sigt{atureofGraduate Faculty

,g ... /&- i<f Date <¥· Phone No.

8. By bringing in my original Master's diploma for the Prospective Student Office to copy, I understand that the hold placed on my registration will be removed.

Student's Signature

Student's Name Date

HUNTER COLLEGE OF THE CITY UNIVERSITY OF NEW YORK DEGREE AUDIT/OFFICE OF THE REGISTRAR 217HN 695 PARK AVENUE, NEW YORK, NEW YORK 10021

~=~= of Gr5'tl.flY Degree __________________ _

Dear °([A,me A Preliminary/Ii-wt audit. of your academic record indicates that the following requirements have not been satisfied. item(s) checked below).

(See

_j_ 1. You :!lJSt currently be in attendance in order to graduate. Jome to Room 217 (Registration Area) in order to register for 'DA-file to Gcaduate' QC 'Maintaining Matriculation', 4C3 ft:>tz 5um rne.R J 99<./

2. You have listed Please contact an advisor in the _____________ de"',..artment for an evaluation. Return approved form to the aanissions office, Room 203HN.

3. An official transcript of work taken at is required. Send to the Records depar-:ment, Hunter College of the City University of New York, 695 Park Avenue 217HN, New York, N.Y. 10021.

4. Insufficient credits: _____ credits completed as of ________ semester. _____ credits are required for grac::i.;at ion.

A. c0 Grade !nC:?€:-QC. J'i9b5 · is'TNC _; _______ is __ ;_....,_ ____ is__;__,. ___ _ ___ ,s___::;J ,s is ___ on your academic record and therefore, have not been counted for degree credit.

S. C ) Do you expect to Cat;)lete the above courses? Yes __ No __ • Please check one, sign and ret:.irn to Grad:..:ate Audit.

Signature Date

C. C ) If you wish to be considered for_· __________ graduation, you must submit a letter from your instructor stating your woric was completed prior to ________ _

5. You have listed _________ on your certification form. This does not appear on your record sheet.

6. Permission is reG!Jired to substitute _____________ in Lieu of ____________ _ A written recomnendati on is needed from. ____________ _

_ ·1. Credit deducted tor _________ taken during this course. -----------

You have previously received credit for

8. ____ credits are reG!Jired for your major. Please list additional courses in your major on the enclosed copy of yo!Jr certification form and have it resigned by your department adviser and return it to me.

9. Your overall index is ___ _ A 3.0 is required for graduation. Please conta.ct !ii graduate adviser for advice.

_10. If you .ish credits toward your graduate degr~ for work tak~n prior to matriculation at Hunter College, a graduate transfer of credit form is reG!Jired. Please return the enclosed form to the Admissions Office, Room 203.

_11. Proof of: ( Thesis Completion; C ) Comprehensive Examination; C ) Language Examination, is required.

_12. You must obtain a waiver for ___________ _ Contact your adviser.

_13. The following office has placed a STOP on your record. Please contact ______ ....., ________ ------Official Office

All services from the College will be withheld and there will be no issuance of a diploma or transcript until all stops CS) have been cleared. '· -

__ 14. Please come in to the Degree Audit Division, Room 217HN, to discuss your candidacy for graduation.

_15. Condition Course(s) _________________ 111.Jst be taken or waived. Please contact your _Graduate advisor.

__ 16. Your time limit has expired. time.

Please contact your Graduate Advisor and the Divisional Dean regarding an extension of -·- - # -

PLEASE NOTE: THESE ITEMS 11.JST BE ATTEJIDEI> TO IN ORDER TO NAIIITAIJC TilJR CANDIDACY FOR GRADUATION. AJff DOCLMEIITATIOII SUPPORTING YWR CANDIDACY SHWU> IM'IEDIATELY BE BRaJGHT TO WR ATTEKTION. Pl.EASE CAL1. 77'2-4506 PRIOR TO ClJ!ING IN.

IT IS IMPERATIVE THAT YCIJ BRING THIS LETTER WITH Ya.I \IHEJI INQUIRING ABWT AJff OF THE ABOVE ITEMS.

JEH/dw

Sin~erely,

L1zzet~,~r1guez DEGREE AUDIT

') . -­~dc •c, ,A' 2.. t/ /