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Engineering a Robust 2D Quality Mesh Generator and Delaunay Triangulator

This document outlines the construction of a 2D mesh generator and Delaunay triangulator based on Ruppert's Delaunay Refinement Algorithm. It emphasizes avoiding small angles and ensuring the mesh conforms to input constraints using efficient data structures and algorithms. Key steps include constructing Delaunay triangulation, incorporating input segments for constrained triangulation, and refining the mesh to meet angle and area constraints. The implementation focuses on robustness and adaptive algorithms, providing detailed procedural guidelines and algorithmic complexity.

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Engineering a Robust 2D Quality Mesh Generator and Delaunay Triangulator

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  1. UMass Lowell Computer Science 91.504Advanced AlgorithmsComputational GeometryProf. Karen DanielsSpring, 2010 Shewchuck 2D Triangular Meshing

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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