CMPUT 498 Terrains

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CMPUT 498Terrains

• Lecturer Sherif Ghali
• Department of Computing Science
• University of Alberta

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Terrain models

• Regular square grid
• Contour line model
• Triangulated Irregular Network
• Hierarchical model

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Terrain models

• Regular square grid
• Elevation at a regular square tesselation
• Storage 2-dimensional array
• Data represents
• Slab-like function
• Point-samples to be bilinearly interpolated
• Point-samples to be interpolated by splines

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Terrain models

• Contour line model
• Locations of constant elevations
• Set of (closed) polygonal chains
• Storage
• DCEL
• Contour Tree, a graph where
• nodes polygonal chains
• edges chains bound the same region

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Terrain models

• Triangulated Irregular Network
• Storage
• DCEL
• Quad-edge
• Variation shown

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Quad-edge data structure

source Guibas Stolfi
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Quad-edge data structure

source P. Heckbert
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Terrain models

• Hierarchical model
• Various scales/levels of details (and errors)
• Storage
• multiple levels? -expensive
• cross-linking to facilitate searching

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Grid to TIN conversion

• Incrementally eliminate vertices not introducing
  salient features (e.g. difference between
  interpolated height of neighbouring vertices and
  vertex is smallest)

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Grid to TIN conversiondrop heuristic

• Evaluate vertical distance between each vertex
  and the Delaunay (re)triangulated region built
  after its removal

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Grid to TIN conversiondrop heuristic

• How often does the error have to be recomputed?
• Use a balanced binary tree indexed by error(v)
• error difference w.r.t. retriangulation
• Cross-link
• nodes in tree reference vertices in TIN
• and vice-versa
• Assuming constant degree per vertex, an iteration
  takes O(log n)
• Algorithm takes O(n log n) time typically
• No guarantee on the error bound (why?)

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Grid to TIN conversionincremental refinement

• Input grid maximum allowed error
• maximum error is guaranteed

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Grid to TIN conversionincremental refinement
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Contour lines

• Treatment of horizontal triangles
• Only handle edges adjacent to a triangle with a
  higher vertex

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Computation of contour lines

• for each triangle
• for each desired height
• intersect height with triangle
• for n triangles, time is in O(n)
• Disadvantages
• Segments are disconnected

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Contour lines aspolygonal chains

• Method 1 (for a set of triangles)
• postprocess the segments to connect them
• for n triangles and k contour segments, time is
  O(n k log k)
• Method 2 (for a TIN)
• trace contour by traversing the TIN
• use mark bits to determine completion of a cycle
• takes O(n)

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Interval trees

• Given a set of intervals, build a data structure
  such that the query find the intervals
  containing a value can be answered efficiently

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Interval tree
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Interval trees for contour lines

• Why is the interval tree useful for contour line
  computation?
• Build interval tree on z-spans of triangles
• Time is in O(log n k)

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Generating chains

• Using postprocessing results in time in O(log n
  k log k)
• We can reduce the time to O(log n k) by
  traversing the TIN and marking the triangles
  traversed
• A second traversal is needed to reset the marks

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Reference
Chapter 3