1. UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2007 Geometric Modeling

2. 2D Mesh Generation Aspect Shewchuck Title Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator Source Conf:1 st Workshop on Applied CG, 1996 Application Areas geometric modeling; graphics Input Objects PSLG of object Aspect Shewchuck Dimensional ity 2D Problem/ Task (constrained) Delaunay triangulation; robustness Theory? Implementat ion? implementation ADTs & Data Structures triangular mesh; (constrained) Delaunay triangulation; Voronoi diagram; convex hulls; Guibas/Stolfi quad-edge; triangular data structure; PSLG; splaytree; heap Algorithmic Paradigms & Techniques sweep-line; geometric divide- and-conquer; incremental insertion Math Topics duality

3. Geometric Modeling “Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator” Jonathan Richard Shewchuck http://www.cs.cmu.edu/~jrs/jrspapers.html

4. Goals ä Construct 2D mesh of triangles for geometric modeling that: ä avoids small angles ä constrained Delaunay triangulation ä is efficient in time and space ä careful choice of data structures & algorithm ä is robust ä adaptive exact arithmetic C code at http://www.cs.cmu.edu/~quake/triangle.html

5. Approach: Overview ä Based on Ruppert’s Delaunay Refinement Algorithm ä Input: Planar Straight Line Graph (PSLG) ä collection of vertices and segments ä Step 1: Construct Delaunay triangulation of point set

6. Approach: Overview (continued) ä Step 2: ä Start with the Delaunay triangulation of the point set ä Add input segments ä segments become constraints ä constrained Delaunay triangulation some differences

7. Approach: Overview (continued) ä Step 3: (not in Ruppert’s algorithm) ä Remove triangles from concavities ä “triangle-eating virus” ä Step 4: ä Refine mesh to satisfy additional constraints on triangle’s minimum ä angle size ä area

8. Step 1: Construct Delaunay Triangulation of Point Set ä Delaunay Triangulation Algorithms: ä O(nlogn) expected time: ä Randomized incremental insertion ä Edge flipping restores empty circle property ä O(nlogn) worst-case time: ä Compute Voronoi diagram, then dualize ä Fortune’s plane sweep (parabolic front) ä O(nlogn) worst-case time: ä Divide-and-Conquer ä alternating cuts Shewchuck experimental comparison [speed, correctness] fastest slowest [point location bottleneck] deBerg handout

9. Experimental Delaunay Triangulation Timings Note: Robust versions require more execution time.

10. Delaunay Triangulation Algorithms: Divide-and-Conquer ä O(nlogn) worst-case time ä Recursively halve input vertex set ä Stop when size = 2 or 3 ä Triangulate small set ä forms edge(s) or triangle ä Merge 2 triangulations ä Ghost triangles allow fast convex hull traversal ä Fit together like gear teeth

11. Step 2: Constrained Delaunay Triangulation ä Force mesh to conform to input line segments ä User Chooses Approach: ä Recursive segment subdivision ä Insert segment midpoint ä Flip edges to restore Delaunay (empty circle) property ä Constrained Delaunay triangulation (default) ä Insert entire segment ä Delete triangles it overlaps ä Retriangulate regions on each side of segment ä No new vertices are inserted

12. Step 4: Mesh Refinement ä Refine mesh to satisfy additional constraints on minimum triangle ä angle size ä area ä Insert new vertices ä Flip edges to restore Delaunay (empty circle) property ä Halting Issue: ä Halts for angle constraint <= 20.7 o ä May not halt for angle constraint >= 33.9 o

13. Step 4: Mesh Refinement (continued) ä Vertex Insertion Rules: ä Segment’s Diametral Circle ä smallest circle containing segment ä any point in the circle encroaches on segment ä split encroached segment ä insert vertex at midpoint ä Triangle’s Circumcircle ä circle through all 3 vertices ä bad triangle: ä angle too small ä area too large ä split bad triangle ä insert vertex at circumcenter

14. Step 4: Mesh Refinement (continued) Note: Figures are in left- to-right, top-to- bottom order

15. Implementation Issues: Representation ä Ghost triangles: ä connected in ring about a “vertex at infinity” ä facilitate convex hull traversal + Topologically richer + Elegant - Slower - More memory edge-based representation tradeoffs triangle-based - Topologically less rich - Longer code + Faster + Less memory Shewchuck preference

16. Implementation Issues: Robustness ä Tests ä Can influence program flow of control ä Can classify entities (e.g. sweep-line events) ä Depend on correctness of geometric predicates ä Orientation (left/right/on) ä In-Circle (in/out/on) ä Each computes sign of a determinant ä Constructions ä Represent geometric objects incorrectness can be serious some incorrectness can sometimes be tolerated

17. Implementation Issues: Robustness (continued) ä Ideal Goal: real arithmetic for some operations ä Challenge: compounded roundoff error in floating- point arithmetic calculations: ä Tests: can cause program ä to hang ä to crash ä to produce incorrect output ä wrong topology ä Constructions: ä can cause approximate results What causes incorrectness?

18. Implementation Issues: Robustness (continued) ä Arithmetic Alternatives to Floating-Point: ä Integer or rational exact arithmetic ä fixed precision ä extended precision ä Floating point +   -testing ä robust topological decisions ä filter: ä identify adequate precision for an operation (bit complexity) ä if expressible as multivariate polynomial, degree gives clue ä floating-point comparisons except when correctness is threatened ä Shewchuck adaptive precision (logical extreme of filter): ä compute quantity (e.g. sign of determinant) via successively more accurate approximations ä stop when uncertainty in result is small No single solution fits all needs. Collection of techniques is needed. slow but sure fast but loose exact floating-pt hybrid time vs. error tradeoff

19. Implementation Issues: Robustness (continued) ä Shewchuck uses: ä multi-stage adaptive precision for geometric primitives ä Orientation (left/right/on) ä In-Circle (in/out/on) ä Each ä computes sign of a determinant ä takes floating-point inputs ä stops when uncertainty in result is small ä can reuse previous, less accurate approximations ä fast arbitrary precision arithmetic ä for small (yet extended) precision values For general discussion of robustness issues and alternatives, see Strategic Directions in Computational Geometry Working Group Report Also see research by Prof. Victor Milenkovic: http://www.cs.miami.edu/~vjm.