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. 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

9. 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

10. 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

11. 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

12. 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