Tessellations• Sets of connected discrete two-dimensional units -can be

irregular or regular– regular

• (infinitely) repeatable patter of regular polygon (can be 3D also)• every point is assigned to only one cell

– irregular• (infinitely) extending configuration of polygons of varied size and shape• representable as topological two-cells

• provide a way to deal with the occupation of space in contrast to dealing w/ identifiable entities

• some entity representations are also tessellations - e.g. land ownership (all locations are owned - at least in English law)

Tessellations versus entities

A

B

CD

regularirregular

Entities - not a full tessellations

Irregular tessellations

• “phenomenological” tessellations (i.e. real ones)– census units

– generally political/administrative units

– land parcels

– PLSS

• computational irregular tessellations– Triangulated irregular networks (TINs)

– wire frame models

– many 3D data structures (multiple triangles)

Regular tessellations

• all are computational in one sense– image data form remote sensing

– map grids

– data generated by photogrammetric systems as lattices of points

– regularly sampled data form continuous data

Attribute measurement and tesselations

• Tesselations provide a method for the referencing of entity locations but there is not a one-to-one relationship to geometric form. Because of the convenience of referencing, however, regular tesselations are often seen as “real”

• does value recorded for each two-cell reflect an average, sum, or ? of the attribute being observed

Lattices

• can be viewed as equivalent to the “intersections” of the grid lines in a tessellation

• or can be seen a “center” of the grid units– BTW different software does this differently

• lattices are “points” – the value at the point can either be seen as the value

“there” – or as the average of the two-cell that the point represents– or as a value “influenced” by other points nearby

Tessellation/lattice roles

• tessellations can be seen as as spatial units for recording data

• can also serve as basis for facilitating access to data distributed continuously in space– use of PLSS for property location

– use of USGS map units (w/ different name) to organize geographic data

• (NOTE - Skipping sections 6.2-6.5)

• creation of proximal regions• partitioning of space around “centers” such that

the boundaries associate the space with the nearest center– process:

• draw lines to connect all centers

• identify mid points of these lines

• connect these to form polygons

• Thiesen polygon, Voroni polygon, Dirichlet domain

Irregular tessellations based on triangles

• triangular irregular models (TIN)

• goals– facets tend to reflect actual slope

– corners represent important turning points (ridges, stream valleys etc.)

– linear features be represented by triangle edges

• process – choose data points

– connect points to create triangles

– store necessary data about triangle in DBM system

– avoid long narrow triangles

Triangulation for surface modeling

• gradient (slope) of each edge• aspect of each edge• planar and surface area of each triangle• slope (gradient) of each triangular facet• aspect of each triangular facet

TIN data

• many different triangular tessellations are possible• commonly preferred is Delaunay triangle

– produces triangles with low variance in edge length

– draw proposed triangle

– draw smallest circle that encompasses triangle

– if circle does not contain any data point then its accepted• if a data point is contained within the circle then there is a “superior” triangle to be drawn

“Preferred” triangular structure

• benefits– triangles can be stored/processed as irregular polygons

– they exhaust all space (no holes)

– planar enforcement (no overlaps)

– easy to process in certain software

• problems– creation computationally demanding

– many different possible triangulations for a given set of points

– can miss critical data characteristics unless properly formed

Benefits/ problems of triangular tessellations