II Existence and complexity of a triangulation Outline III Legal Triangulation I Terrain Terrain Terrain 2dimensional 2D surface in 3D space such that every vertical line intersects ID: 1025390
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1. Point Set TriangulationI. Height interpolationII. Existence and complexity of a triangulationOutline: III. Legal Triangulation
2. I. Terrain TerrainTerrain: 2-dimensional (2D) surface in 3D space such that every vertical line intersects in point. Paraboloid (terrain)Catenoid (Com S 477/577) (not a terrain) DomainHeight Landscape (https://github.com/Gruftikus/lltool)
3. Height Interpolation Sample the value of at a finite set of points. Task: Approximate for every . Naïve approach: where for all (i.e., is the closest to among all points in ). Not natural (stairlike) Not continuous – using the Voronoi diagram of for fast location of .
4. Better InterpolationPoint Determine a triangulation of . Lift every sample point to its height.Polyhedral terrainTriangulation How to triangulate ? Which triangulation is the most appropriate for approximating a terrain?
5. II. Triangulation of a Planar Point Set A planar subdivision is maximal if no (straight) edge connecting two vertices can be added without destroying its planarity. A triangulation of a point setis themaximal planar subdivisionwhose vertex set is .
6. Existence of a TriangulationPoint Every bounded face of such a planar subdivision with vertex set is a polygon. Any polygon can be triangulated! Union of bounded faces is the convex hull of . Unbounded face is the complement of the convex hull (i.e., the remaining region in the plane).
7. Complexity of TriangulationPoint Assumption The points in are not all collinear. # points from on its convex hull Theorem Any triangulation of has triangles and edges. Proof is a triangulation with triangles. faces edges Every triangle has three edges.The unbounded face has edges. Every edge is incident to two faces.
8. Complexity (cont’d)Point Euler’s formula:
9. III. Angle VectorPoint : triangulation with triangles which have angles. ,…, Order these angles into an angle vector: is another triangulation with angle vector: if there exists with such that for all and (lexicographical order) is angle optimal if for all triangulations .
10. Thales’ TheoremPoint circle
11. Illegal EdgePoint An edge is incident to two triangles. is an illegal edge if We can locally increase the smallest angle by flipping edges.edge flip,,
12. Edge FlipPoint It can be shown that . Triangulation (before flipping ) Triangulation (after flipping ) Computing ,…, and is unnecessary!
13. Determining an Illegal EdgeTheorem Edge is illegal iff lies in the interior of circle determined by . Proof () Let and be the triangulations of the 4 points before and after the flip. arc subtended bysubtended bySimilarly, and are not the smallest angle in . and arenot the smallest angle in .
14. Cont’d We need only consider the remaining eight angles,four from each triangulation. two inscribed angles subtended by arc subtended by arc Thus, the minimum angle in is less than the minimum angle in . () Omitted.
15. Special Case and Legal TriangulationPoint are cocircular. Both and are legal! A legal triangulation has no illegal edge.
16. Generating a Legal TriangulationPoint LegalTriangulation()1. while contains an illegal edge do let and be the two triangles adjacent to remove add return Angle vector increases in every iteration. #triangulations is finite.The while loop will terminate with a legal triangulation. Too slow!