Tessellations

1
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)

2
Tessellations versus entities
Entities - not a full tessellations
B
A
C
regular
D
irregular
3
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)

4
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

5
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

6
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

7
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)

8
Irregular tessellations based on triangles

• 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

9
Triangulation for surface modeling

• 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

10
TIN data

• 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

11
Preferred triangular structure

• 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

12
Benefits/ problems of triangular tessellations

• 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