Terrain Domain Height Paraboloid terrain Catenoid Com S

Terrain Domain Height Paraboloid (terrain) Catenoid (Com S 477/577) (not a terrain) Landscape (https: //github. com/Gruftikus/lltool)

Height Interpolation Naïve approach: Not natural (stairlike) Not continuous

Better Interpolation Point Lift every sample point to its height. Polyhedral terrain Triangulation Which triangulation is the most appropriate for approximating a terrain?

Triangulation of a Planar Point Set

Existence of a Triangulation Point Any polygon can be triangulated! Unbounded face is the complement of the convex hull (i. e. , the remaining region in the plane).

Complexity of Triangulation Point Every triangle has three edges. Every edge is incident to two faces.

Complexity (cont’d) Euler’s formula: Point

Angle Vector Point Order these angles into an angle vector:

Thales’ Theorem Point

Illegal Edge Point edge flip We can locally increase the smallest angle by flipping edges.

Edge Flip Point

Determining an Illegal Edge subtended by

Cont’d We need only consider the remaining eight angles, four from each triangulation.

Special Case and Legal Triangulation Point A legal triangulation has no illegal edge.

Generating a Legal Triangulation Point #triangulations is finite. The while loop will terminate with a legal triangulation. Too slow!