3 Mapping of Spatial Data - ifis.cs.tu-bs.de · •Challenges –Projection of the 3D surface on two dimensions (paper, film, screen) –Selection of the spatial objects and their

3.1 Properties of Maps

3.2 Signatures, Text, Color

3.3 Geometric Generalization

3.4 Label and Symbol

Placement

3.5 Summary

Spatial Databases and GIS – Karl Neumann, Sarah Tauscher– Ifis – TU Braunschweig 200

3 Mapping of Spatial Data

http://homepage.univie.ac.at/.../Janschitz_Text.pdf

• Mapping (visualization) of spatial objects by

transforming them into representation objects

(map objects)

• Map

– Generalized model in a reduced scale for representing

selected spatial information



http://www.xerokampos.eu/

• Topographic map

– Geometrically and positionally accurate representation of landscape objects drawn to scale (topography, water network, land use, transportation routes)



http://www.wiedenbruegge.net/

• Thematic map

– Emphasis is

on subject-

specific

information

(geological

map,

vegetation

map, biotope

map)



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• Challenges

– Projection of the 3D surface on two dimensions

(paper, film, screen)

– Selection of the spatial objects and their attributes to be displayed

– Generalization of geometric and thematic properties (simplify, omit depending on scale)

– Exaggeration and displacement (e.g. river valley with roads and railway lines)



http://maps.google.de/

• Graticule

– Mapping the earth (sphere,

ellipsoid) to a plane

– Two tasks

• Conversion of geographic

coordinates (longitude

and latitude) to cartesian

coordinates (x, y; easting,

northing)

• Scaling of the map



www.klett.de

• Projections on

– Cone

– Plane

(azimuthal

projection)

– Cylinder



[HGM02]

• Desired properties

– Length preservation: with plane maps only limitedly

attainable (in certain directions or at certain points)

– Equivalent (equal area): preserve area measurements

shape, angle,

and scale

may be

strongly

distorted



http://www.informatik.uni-leipzig.de/~sosna/

– Conformal:

important

for

navigation

in shipping

and air

transport



http://flightnavigation.de/

• Areas and angles can not be preserved at the

same time, therefore

– Compromise projections minimize overall distortion



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• The Gauss-Krüger coordinate system

– Used in Germany and Austria

– Cartesian coordinate system to represent small areas

– Divides the surface of the earth into zones

• Each 3° of longitude in width

• The zones are projected onto a cylinder with the earths

diameter



www.geogr.uni-jena.de/.../GEO142_3b.ppt http://homepage.univie.ac.at/.../Janschitz_Text.pdf

– Origin of the coordinate system:

intersection point of the central meridian and the

equator

– X coordinate from the

origin positive towards east,

y coordinate from the origin

positive towards north

– X and y values given in meters



http://www.gerhard-tropp.de/Troppo/gauss_krueger.html

www.geogr.uni-jena.de/.../GEO142_3b.ppt

– To avoid negative x coordinates the central meridian

is set to 500,000 m (false easting)

– Each projection zone (3°, 6°, ..., 180°) is identified

by a number (1, 2, …, 60)

– This number is placed prior to the x value

– Example: the Gauss-Krüger coordinates

x: 3,567,780.339 and y: 5,929,989.731

refer to the following location

x: 67,780.339 m East of the 9° meridian

y: 5,929,989.731 m North of the equator



• Transforming geographic coordinates into

Gauss-Krüger coordinates

– Given geographic coordinates lon, lat

(e.g. from GPS, reference ellipsoid: World Geodetic

System 1984 (WGS 84))

– Transformation into

Cartesian coordinates x, y, z



https://www.univie.ac.at/

– 7 parameter Helmert transformation into Cartesian

coordinates x’, y’, z’ referring to the Bessel ellipsoid

– Re-transformation into geographic coordinates

– Transformation

into Gauss-Krüger

coordinates



X Y Z

WGS84 gc 10.529 52.273 ---

cc 3845123.169 714709.117 5021445.455

Bessel cc 3844489.704 714681.311 5020993.750

gc 10.531 52.274 ---

Gauss-Krüger 3604481.733 5794379.689 ---

• The UTM coordinate system

– Similar to the Gauss-Krüger

system

– Divides the surface of the

earth into 6° zones

– Covers the area

between 84° North

and 80° South



https://www.e-education.psu.edu/

http://earth-info.nga.mil/

– The diameter of the

transverse cylinder is

slightly smaller than the

diameter of the Earth

secant projection with

two lines of true scale

(about 180 km on each

side of the central

meridian)



https://www.e-education.psu.edu/

• Common mapping errors



interchanging of x- and y-coordinates

no trans-formation into Cartesian coordinates

ignoring the negative direction of the canvas` y-axis

• Structure of maps

– Body of the map presentation of the actual map

– Map margin contains name, map scale, legend

– Map frame borders map, contains numbering of the respective coordinate system



http://www.lbeg.niedersachsen.de/

• Graphical design elements of maps are

– Symbols

(signatures)

for

• Points

• Lines

• Areas

– Text



http://lehrer.schule.at/Ecole/

• The use of point, line, area symbols depends on

– Spatial scale of a map

– Purpose of the map

– Convention

• Point signatures are (composite) symbols for the representation of spatial objects with point geometry



[HGM02]

• Line signatures represent objects with line geometry

• Area signatures represent objects with area geometry

• Texts are needed in various fonts for the labeling of maps



[HGM02]

• Color

– Important design element

– Additive mixture

• Most relevant for displaying maps

on screen

• Common color space: RGB

– Subtractive mixture

• Most relevant for printing maps

• Common color space: CMYK



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– There exist

various

color tables

Example:

color table for

the official

German real

estate map

1:1000



• For maps and map series the signatures on-hand,

fonts, and colors are specified in signature

catalogs and color

tables

– Example:

topographical map

1:25.000 (TK25)



http://www.lgn.niedersachsen.de/

• Signature catalog "SK25" describes how

the TK25 is derived from the digital

landscape model DLM25

• The catalog consists of derivation rules and signatures

(about

300

forms)

• Some

derivation

rules:



TK25

DLM25

SK25

• Some signatures:



– Example:

real estate map 1:1000

• The ALKIS project (official property cadastre information

system, Amtliches Liegenschaftskataster Informationssystem)

specifies a signature library and derivation rules

• Approximately 670 signatures

• Approximately 850 derivation rules



• A signature

specification:



• Section of a real estate map



• The direct geometric derivation (coordinate

transformation) of map objects from spatial

objects works only with large-scale maps

(e.g. real estate

map)

• For small-scale

maps there are

too many details



http://www.geobasis-bb.de/GeoPortal1/

• Generalization simplifies map content for the

preservation of readability and comprehensibility

• Replacement of scale preserving mappings

through simplified mappings, symbols, and

signatures

• Select and summarize information

• Maintain important objects, leave

out unimportant ones



http://www.kartographie.info/

• Eight elementary operations

– To simplify

– To enlarge

– To displace

– To merge



– To select

– To symbolize

– To typify

– To classify



www.ikg.uni-hannover.de



Base map 1:5000 Topographical map 1:25.000

[HGM02] [HGM02]



Topographical map 1:25.000 Topographical map 1:50.000

[HGM02] [HGM02]



Topographical map 1:50.000 Topographical map 1:100.000

[HGM02] [HGM02]

• Typical operations for generalization

(alternative view)



[SX08]

• Smoothing of polylines by means of a simple

low-pass filter:

y(n) = 1/3(x(n)+x(n−1)+x(n−2))



• Electronic filter that passes low-frequency signals

but attenuates high-frequency signals

• Further common filter types

– High-pass

– Band-pass

– Band elimination


3.3 Low-pass Filter

• Passive electronic realization

– Simple circuit consists of a resistor and a capacitor

(RC filter, first order filter)

– More advanced circuit with

additional inductor

(RLC filter,

second order

filter)


3.3 Low-pass Filter

http://upload.wikimedia.org/

http://ecee.colorado.edu/~mathys/

• Active electronic realization

– For example RC filter with an operational amplifier

– An operational amplifier has a high input impedance

and a low output

impedance

– Voltage gain is

determined by

resistors R1, R2


3.3 Low-pass Filter

http://www.electronics-tutorials.ws/

• Digital filters are realized as

– Finite impulse response

filter (FIR)

• Impulse response lasts for

n+1 samples and settles to zero

• Output is a weighted sum of

the current and a finite

number of previous values

of the input

– Infinite impulse response

filter (IIR)


3.3 Low-pass Filter


• A moving average filter is a very simple FIR filter

– All n filter coefficients

are set to 1/n

– Realization by means of

• Delay units

• Ring buffer


3.3 Low-pass Filter


http://blog.avangardo.com/wp-content/

• In offline mode (no real-time constraints)

a moving average filter results in a simple loop


3.3 Low-pass Filter

x[1...n] input values y[1...n] output values for (int i=2; i<n; i++) { y[i] = 0.333*(x[i-1]+x[i]+x[i+1]); }; y[1] = x[1]; y[n] = x[n];

• Adaptation of simple low-pass filter to

two-dimensional

geometry

x′i = ⅓(xi−1 +xi+ xi+1)

y′i = ⅓(yi−1 +yi + yi+1)

+ very efficiently to compute

− coordinates are changed, no reduction of points



– Example (polyline with 41 points):



• Simplification of polylines

– Reduction of points

– No change of coordinates

• Douglas/Peucker algorithm [DP73]

– Given: polyline L, threshold g

– g is very small: L remains unchanged

g is very large: L is changed to

one single line



• Douglas/Peucker algorithm

1. given: polyline L, threshold g

2. determine line between the start and end point of L,

3. determine the point of L that is furthest from the line segment

4. if distance > g then the point is significant, repeat procedure for both sub-lines, otherwise remove all the points between the start and end point of L



Examples:





Properties:

+ simple base operation: distance measurement

− runtime of naive implementation: O(n2)

+ runtime of optimized implementation: O(n log n),

using two convex hulls

+ good results even with a strong reduction of points

+ extendable for polygons

− "outliers" are not

eliminated



• Polygon to polyline conversion

– Presentation of rivers and roads

– Text placement within polygons



http://www.almenrausch.at/bergtouren/ http://www.naturathlon2006.de/

• Schoppmeyer/Heisser procedure [SH95]

1. Given: elongate polygon P

2. Determine the longitudinal axis of P

3. Drop the perpendicular from all edge points to the

longitudinal axis

4. Determine the intersection points of the extended

perpendicular lines

5. Determine the centre of the resulting axes

6. Connect adjacent centers

7. Determine suitable start and end segment



Example

1. Given: elongate polygon P

2. Determine the longitudinal axis of P



3. Drop the perpendicular from all edge points to the

longitudinal axis

4. Determine the intersection points of the extended

perpendicular lines



5. Determine the centre of the resulting axes

6. Connect adjacent centers



7. Determine suitable start and end segment

Result



Using the result for text placement



Properties:

+ relatively short runtime

+ quite good results with "good-natured" polygons

− determining the start and end segment

− procedure fails for

non-elongate

polygons



• Procedure for more polygon classes Petzold/Plümer [PP97]

1. given: Polygon P (set of points S)

2. determine Voronoi diagram of S

3. determine intersection points of the Voronoi edges and the polygon edges

4. consider resulting Voronoi skeleton:

5. choose an appropriate sequence of edges

Voronoi diagram:

assigns each point Pi ∈ S the points of the plane that are closer to Pi , than to each Pj ∈ S, i≠j



Example

1. given: Polygon P (set of points S)

2. determine Voronoi diagram of S



3. determine intersection points of the Voronoi edges

and the polygon edges

4. consider resulting Voronoi skeleton:

5. choose an appropriate sequence of edges



Properties:

− computation of the Voronoi diagram

− selection of an appropriate polyline from the

skeleton

+ results for

more

polygon classes



Example (polygon with 2000 points):



visualization tool: [Me11]

• Procedure with rasterization, thinning, and re-vectorization

1. given: polygon P (set of points S)

2. rasterization of P

3. topological thinning of P (building a skeleton)

4. line following and re-vectorization

5. determine the longest path

6. line simplification



Example

1. given: polygon P (set of points S)



2. rasterization of P



visualization tool: [Bu11]

3. topological thinning of P




4. line following and re-vectorization




5. determine the longest path




6. line simplification




• Simplification of polygons

– Low-pass filter

• Coordinates are changed

• No reduction of points

• Shapes might be drastically changed



– Douglas/Peucker (adapted)

• Decompose polygon P into two polylines L1, L2 e.g. at the

points PL1, PL2 ∈ P, with maximum distance between each

other

• Simplify L1 and L2

• Recombine

resulting polylines

at PL1 and PL2



• Example



• Further

example





• Cartographically

desired

generalization

• Results with

Douglas/Peucker

(adapted)

• Douglas/Peucker reduces the number of polygon points

• Characteristic shapes are preserved (within certain limits)

• Good results with "natural" geometries (bogs, lakes, forests)

• Less satisfactory results with polygons with predominantly

right angles (buildings), therefore modification to preserve

right angles and long edges (e.g. [NSe01])



• Geometric generalization (and displacement) of

building sketches is a challenging task

• Convenient results achieved by programs

• E.g. CHANGE, PUSH, TYPIFY [Se07]



http://www.ikg.uni-hannover.de/

• Placement of all labels in the same way results

with high probability in overlappings

• Therefore, it is

necessary to move,

scale down, rotate,

or omit texts

• In the general case

this is a NP-hard

optimization problem


3.4 Label and Symbol Placement

http://www.laum.uni-hannover.de/ilr/lehre/Isv/

• Objectives of labeling

– Easily readable

– Unambiguity: each label must be easily identified with exactly one graphical feature

– Same facts are represented in the same way

– Different facts are represented differently

– Important facts are emphasized

– Important objects are never covered

– No uniform pattern (avoidance of raster effect)



http://www.mdc.tu-dresden.de/

• Text or label placement is divided into

– Point labeling

• Positions to the right are preferred to those on the left

• Labels above a point are preferred to those below

• E.g. cities with a horizontal label

– Line labeling

• Labels should be placed as straight as possible

• E.g. rivers with names

– Area labeling

• It must be clear what the total area is

• E.g. forest areas containing their names



http://www.sprachkurs-sprachschule.com/


• For the three classes there are many specialized

algorithms, based on

– Greedy algorithm

• Labels are placed in sequence

• Each position is chosen

according to minimal

overlapping

• Acceptable results only for

very simple problems

• Very fast



• Example :

placement of

75 city names

priorities for

a single

placement:



– Local optimization

• The labels are checked

several times

• On each pass a single

label is repositioned

and tested

• The position is kept, if the

overall result improves

• Stop, if a local optimum is

reached

• Suitable only for maps

which relatively few labels



– Simulated annealing

• Similar to local optimization, however yielding better results

• A placement of a label can be kept even though it (initially)

downgrades the overall result

• At first "high temperature", thus leaving local optima is

possible

• Later on ("low temperature") only

small changes are possible

• Challenges: a good evaluation function,

and a good annealing schedule



http://www.hs-augsburg.de/informatik/

• Name and inspiration from annealing in metallurgy: controlled cooling of a material until it changes from liquid to solid

• Structure of the solid depends on "the cooling schedule"

– Fast cooling results in • Unordered solid

• Internal stresses

– Slow cooling results in • Ordered solid

• Low internal energy

• Macroscopic crystal lattice


3.4 Simulated Annealing

http://webuser.hs-furtwangen.de/~neutron/

• Example: SiO2

– Short range order: tetrahedron

– Crystalline form: Cristobalit

– Without long range order: Silica glass



http://de.wikipedia.org/wiki/Glas

Si O

• Elements

– Initial solution (liquid)

– Modifications (vibrations

of molecules)

– Cooling schedule: change

of temperature over time

• Initial temperature

• Freezing point

– Weighting function (internal energy)



• Pseudocode:

– Usually the temperature is decreased after multiple changes



T = initialTemperature; currentSolution = InitialSolution; while (T > freezingPoint){ newSolution = CHANGE(currentSolution); if (ACCEPT){ currentSolution = newSolution; } ANNEAL(T); }

– Decision if the new solution is accepted

• A better solution is always accepted

• Probability of accepting a worse solution depends on the

temperature and its cost

• Boltzmann factor: e-E/k*T ; k = 1.3806504(24) · 10−23



ACCEPT{ Δ = EVALUATE(newSolution) – EVALUATE(currentSolution) if((Δ < 0) or RANDOM(0,1) < e-Δ/T)){ return true; } return false; }

• Example: label placement



http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.2327

• Initial solution: random label placement

• Weighting function

– Number of covered (or deleted) labels

– Consideration of cartographic preferences by

weighting of possible positions for point labels

• Modifications

– Move an arbitrary or covered label to a new position

– If cartographic preferences are considered an

arbitrary label should be moved



2

4

1 5

6

8

7 3

• Cooling schedule:

– Initial temperature ~2.47 → the probability of accepting that a solution whose cost are 1 higher is accepted is 2/3, i.e.: e-1/T = 2/3

– T = 0.1 * T

– T is decreased as soon as more than 5*n new solutions have been accepted (n is the number of objects)

– Search ends

• As soon as T has been decreased 50 times

• If none of 20*n new solutions in a row has been accepted



• Result



http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.2327

• Symbol placement (point signatures) is just as

complex as text placement

• In the following two examples for special cases

– Displacement and placement

of tree symbols in TK25-like

presentation graphics

[NPW06]

– Placement of point signatures

in polygons of buildings for

real estate maps [NKP08]



• Displacement and placement of tree row symbols

– Roads as well as tree rows are given as polylines only

– Placement of the tree symbols on the points of the

tree rows does not result in an equidistant pattern

– The visualization of roads is much

wider than the actual street width

– The tree symbols are often hidden

by the line signatures of the streets

– An alternative placement and a

displacement is needed



• One single road from the DLM25

(example, XML/GML encoding)



<AtkisMember> <Strasse> <gml:name> Badstrasse </gml:name> <AtkisOID> 86118065 </AtkisOID> <gml:centerLineOf> <gml:coord><gml:X>4437952.980</gml:X><gml:Y>5331812.550</gml:Y></gml:coord> <gml:coord><gml:X>4437960.070</gml:X><gml:Y>5331818.450</gml:Y></gml:coord> <gml:coord><gml:X>4437967.200</gml:X><gml:Y>5331825.410</gml:Y> </gml:coord> </gml:centerLineOf> <Attribute> <Zustand> in Betrieb </Zustand> <AnzahlDerFahrstreifen Bedeutung=“tatsaechliche Anzahl”> 2 </AnzahlDerFahrstreifen> <Funktion> Strassenverkehr </Funktion> <VerkehrsbedeutungInneroertlich> Anliegerverkehr </VerkehrsbedeutungInneroertlich> <Widmung> Gemeindestrasse </Widmung> </Attribute> </Strasse> </AtkisMember>

• A tree row from the DLM25



<AtkisMember> <Baumreihe> <gml:centerLineOf> <gml:coord><gml:X>3524258.170</gml:X><gml:Y>5800238.690</gml:Y> </gml:coord> <gml:coord><gml:X>3524256.190</gml:X><gml:Y>5800220.270</gml:Y> </gml:coord> <gml:coord><gml:X>3524255.240</gml:X><gml:Y>5800196.070</gml:Y> </gml:coord> ... <gml:coord><gml:X>3524581.650</gml:X><gml:Y>5799674.000</gml:Y> </gml:coord> </gml:centerLineOf> <Attribute> <Vegetationsmerkmal> Laubholz </Vegetationsmerkmal> </Attribute> </Baumreihe> </AtkisMember>

• Direct visualization



• Simple displacement procedure



for all t : treeRow do begin if exists s : street (distance(t,s) ≤ minDistance(s.dedication)) then for all p : t.coord do begin r := refSegment(s,p); move(p,r) end do end if end do;

t

s

• Visualization with displacement



• Placement procedure



for all t : treeRow do begin l := length(t.centerLineOf); n := ⌊l/distanceConst⌋ + 1; t’ : new treeRow; for i=1 to n do begin computePoint (t’.coord[i], t.centerLineOf, distanceConst) end do end do;

• Visualization with displacement and placement



• Placement of point signatures in polygons of buildings

– Derivation of the real estate map (1:1.000) from ALKIS inventory data extracts relatively straight forward

– No generalization and no displacement is needed

– Representation of buildings, parcels, border points, etc., with the given signature library and the derivation rules

– E.g. symbolisation of buildings as colored polygons with a boundary line and a typical point signature depending on the attribute building function



• A building from an ALKIS inventory data extract



<gml:featureMember> <AX_Gebaeude gml:id="DEHHSERV00001FN1"> ... <position><gml:Polygon> <gml:exterior><gml:Ring> ... <gml:pos>3567807.047 5930017.550</gml:pos> <gml:pos>3567810.850 5930024.755</gml:pos> ... <gml:pos>3567807.047 5930017.550</gml:pos> ... </gml:Ring></gml:exterior> </gml:Polygon></position> <gebaeudefunktion>2000</gebaeudefunktion> <weitereGebaeudefunktion>1170</weitereGebaeudefunktion> <bauweise>2100</bauweise> <anzahlDerOberirdischenGeschosse>1 </anzahlDerOberirdischenGeschosse> </AX_Gebaeude> </gml:featureMember>

• For many point signatures which are related to buildings so-called presentation objects are supplied in the inventory data, defining the optimal position of the respective signature (given in "world coordinates")



<gml:featureMember> <AP_PPO gml:id="DEBWL00100000fAW"> <lebenszeitintervall>... </lebenszeitintervall> <modellart>... </modellart> <anlass>000000</anlass> <position> <gml:Point><gml:pos>3540847.175 5805897.864</gml:pos></gml:Point> </position> <signaturnummer>3316</signaturnummer> <dientZurDarstellungVon xlink:href="urn:adv:oid:DEBWL00100000jwR"/> <drehwinkel>67.000</drehwinkel> </AP_PPO> </gml:featureMember>

• But some buildings lack the presentation objects

• An obvious, easily determined position for the

signature:

– Choose the center of the smallest axis parallel

rectangle, which encloses the polygon of the building

– min(x1, ..., xn)+((max(x1, ..., xn)−min(x1, ..., xn))/2),

– min(y1, ..., yn)+((max(y1, ..., yn)−min(y1, ..., yn))/2)



• Unfortunately, the results are not always satisfactory

• Therefore, heuristic procedure, based on

– Convexity

– (approximate) symmetry points

– (approximate) symmetry axis

– Various quality criteria

– Discrete increments





if polygon of building convex: choose centroid if signature frame fits completely in polygon of building: place there otherwise choose appropriate point with the smallest distance to the centroid else if symmetry point in polygon of building: place there otherwise further procedure with symmetry axes

⇒ generation of a new presentation object with "optimal" positioning coordinates

centroid

best symmetry point

best symmetry axis

finally chosen point

• The procedure is not suited for the placement of

signatures for churches/chapels

– Besides the determination of an appropriate position

– Also the signatures alignment to the shape of the

building’s polygon is needed

– An alignment to the north south axis is rarely optimal

– Crosses for churches as parallel as possible to the

churches naves

– Crossbrace of cross should be

placed in the transept



• Heuristic procedure

– Determine a preferable large cross that just fits in the

building’s polygon

– Proportions of the large cross and the church

signature are the same

– If the large cross is found, place the signature just in

the intersection point

• For this purpose first simplify the building’s

polygon



• In several rotation angles: look for a preferable large, well placed cross

– Restriction of the potential rotation angles and positioning points (e.g. consider minimum distance to boundaries of the building’s polygon)

– Evaluate all appropriate crosses within a rotation angle

– Evaluate all the best crosses (e.g. lengths of the crosses, alignments)



• Placement of the signature in the best cross of all

rotation angles

⇒ Generation of a new presentation object with

"optimal" positioning coordinates and

"optimal" rotation angle



• Both methods applied to inventory data extracts



• Mapping of spatial data

– Topographic map

– Thematic map

• Properties of maps

– Graticule

– Projections

– Gauß-Krüger coordinate system

– Body of map, map frame, map margin


3.5 Summary

• Signatures, text, color

– Point signatures

– Line signatures

– Area signatures

– Derivation rules

– TK25, real estate map

• Geometric generalization

– Smoothing of polylines

– Simplification of polylines


3.5 Summary

– Douglas/Peucker algorithm

– Polygon to polyline conversion

– Simplification of polygons

– Geometric generalization of building’s ground plans

• Text and symbol placement

– Methods for text placement

simulated annealing

– Displacement and placement of symbols for tree rows

– Placement of point signatures in building’s polygons


3.5 Summary

2

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3.5 Summary

GIS graticule

map

text signatures

color

collect

manage

analyse

display

generalization

area labelling

simplifying lines

polygon→ line

design elements

placement

topographic

thematic

building signatures

tree rows as optimization problem

Documents

3 Mapping of Spatial Data - ifis.cs.tu-bs.de · •Challenges –Projection of the 3D surface on two dimensions (paper, film, screen) –Selection of the spatial objects and their