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Classroom Examples of Robustness Problems in Geometric Computations

Classroom Examples of Robustness Problems in Geometric Computations. Lutz Kettner , Kurt Mehlhorn MPI f ü r Informatik, Saarbr ü cken Stefan Schirra Otto-von-Guericke-Universit ät, Magdeburg. Sylvain Pion INRIA, Sophia Antipolis Chee Yap New York University. Overview. Motivation

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Classroom Examples of Robustness Problems in Geometric Computations

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  1. Classroom Examples of Robustness Problems in Geometric Computations Lutz Kettner, Kurt Mehlhorn MPI für Informatik, Saarbrücken Stefan Schirra Otto-von-Guericke-Universität, Magdeburg Sylvain Pion INRIA, Sophia Antipolis Chee Yap New York University

  2. Overview • Motivation • 2D Convex Hull Algorithm • Constructing Bad Input Examples • Analysis Classroom Examples ESA'04

  3. Motivation • The algorithms of computational geometry are designed for a machine model with exact real arithmetic. • Substituting floating point arithmetic for the assumed real arithmetic may cause implementations to fail. • Although this is well known, it is not common knowledge. • There is no good material for the classroom to show the inadequacy of floating point arithmetic. Classroom Examples ESA'04

  4. Example • Arrangements of circular arcs Seen in Antibes, France, during SGP’04 Classroom Examples ESA'04

  5. Example • Arrangements of circular arcs, with double arithmetic Classroom Examples ESA'04

  6. Example • Arrangements of circular arcs, with float arithmetic Classroom Examples ESA'04

  7. Motivation • Let’s be more serious … Classroom Examples ESA'04

  8. Intersection of Four Simple Solids • Rhino3D: • ((( s1∩ s2) ∩ c2) ∩ c1) → successful • ((( c1∩ c2) ∩ s1) ∩ s2) → ``Boolean operation failed'' Classroom Examples ESA'04

  9. Intersection of Four Simple Solids • Rhino3D: • ((( s1∩ s2) ∩ c2) ∩ c1) → successful • ((( c1∩ c2) ∩ s1) ∩ s2) → ``Boolean operation failed'' • output is a combinatorial object plus coordinates (not a point set) Classroom Examples ESA'04

  10. Motivation • Let’s be more serious … • …, but that is hard to explain in all detail in class. • Let’s do a really simple 2D convex hull algorithm. Classroom Examples ESA'04

  11. 2D-Orientation of Three Points Orientation( p, q, r) = sign Orientation( p, q, r) = sign((qx-px)(ry-py) – (qy-py)(rx-px)) Implemented with IEEE 784 double precision  float_orient (p, q, r) Classroom Examples ESA'04

  12. Convex Hulls in the Plane v3 • maintain current hull as a circular list L=(v0,v1,...,vk-1) of its extreme points in counter-clockwise order • start with three non-collinear points in S. • consider the remaining points r one by one. v1 r v2 Classroom Examples ESA'04

  13. Convex Hulls in the Plane v4 • maintain current hull as a circular list L=(v0,v1,...,vk-1) of its extreme points in counter-clockwise order • start with three non-collinear points in S. • consider the remaining points r one by one. v1 v3 v2 Classroom Examples ESA'04

  14. Convex Hulls in the Plane v4 • maintain current hull as a circular list L=(v0,v1,...,vk-1) of its extreme points in counter-clockwise order • start with three non-collinear points in S. • consider the remaining points r one by one. v1 v3 v2 Classroom Examples ESA'04

  15. Convex Hulls in the Plane v4 • maintain current hull as a circular list L=(v0,v1,...,vk-1) of its extreme points in counter-clockwise order • start with three non-collinear points in S. • consider the remaining points r one by one. r v1 v3 v2 Classroom Examples ESA'04

  16. Convex Hulls in the Plane v4 • [Property A] A point r is outside CH iff there is an i such that the edge (vi, vi+1) is visible for r. (orientation(vi,vi+1,r) > 0) r v1 v3 v2 Classroom Examples ESA'04

  17. Convex Hulls in the Plane v4 • [Property A] A point r is outside CH iff there is an i such that the edge (vi, vi+1) is visible for r. (orientation(vi,vi+1,r) > 0) r v1 v3 v2 Classroom Examples ESA'04

  18. Convex Hulls in the Plane v4 • [Property A] A point r is outside CH iff there is an i such that the edge (vi, vi+1) is visible for r. (orientation(vi,vi+1,r) > 0) r v1 v3 v2 Classroom Examples ESA'04

  19. Convex Hulls in the Plane v4 • [Property A] A point r is outside CH iff there is an i such that the edge (vi, vi+1) is visible for r. (orientation(vi,vi+1,r) > 0) • [Property B] If r is outside CH, then the set of edges that are weakly visible (= orientation is non-negative) from r forms a non-empty consecutive subchain; so does the set of edges that are not weakly visible from r. r v1 v3 v2 Classroom Examples ESA'04

  20. Convex Hulls in the Plane v4 • [Property A] A point r is outside CH iff there is an i such that the edge (vi, vi+1) is visible for r. (orientation(vi,vi+1,r) > 0) • [Property B] If r is outside CH, then the set of edges that are weakly visible (= orientation is non-negative) from r forms a non-empty consecutive subchain; so does the set of edges that are not weakly visible from r. r v1 v3 v2 Classroom Examples ESA'04

  21. Convex Hulls in the Plane v4 • [Property A] A point r is outside CH iff there is an i such that the edge (vi, vi+1) is visible for r. (orientation(vi,vi+1,r) > 0) • [Property B] If r is outside CH, then the set of edges that are weakly visible (= orientation is non-negative) from r forms a non-empty consecutive subchain; so does the set of edges that are not weakly visible from r. r v1 v3 v2 Classroom Examples ESA'04

  22. Convex Hulls in the Plane v4 • [Property A] A point r is outside CH iff there is an i such that the edge (vi, vi+1) is visible for r. (orientation(vi,vi+1,r) > 0) • [Property B] If r is outside CH, then the set of edges that are weakly visible (= orientation is non-negative) from r forms a non-empty consecutive subchain; so does the set of edges that are not weakly visible from r. r v1 v3 v2 Classroom Examples ESA'04

  23. Convex Hulls in the Plane v4 • [Property A] A point r is outside CH iff there is an i such that the edge (vi, vi+1) is visible for r. (orientation(vi,vi+1,r) > 0) • [Property B] If r is outside CH, then the set of edges that are weakly visible (= orientation is non-negative) from r forms a non-empty consecutive subchain; so does the set of edges that are not weakly visible from r. r v1 v3 v2 Classroom Examples ESA'04

  24. Convex Hulls in the Plane v4 • [Property A] A point r is outside CH iff there is an i such that the edge (vi, vi+1) is visible for r. (orientation(vi,vi+1,r) > 0) • [Property B] If r is outside CH, then the set of edges that are weakly visible (= orientation is non-negative) from r forms a non-empty consecutive subchain; so does the set of edges that are not weakly visible from r. r v1 v3 v2 Classroom Examples ESA'04

  25. Convex Hulls in the Plane v4 • [Property A] A point r is outside CH iff there is an i such that the edge (vi, vi+1) is visible for r. (orientation(vi,vi+1,r) > 0) • [Property B] If r is outside CH, then the set of edges that are weakly visible (= orientation is non-negative) from r forms a non-empty consecutive subchain; so does the set of edges that are not weakly visible from r. v3 v1 v2 Classroom Examples ESA'04

  26. Devil’s Advocate • Systematic construction of instances leading to violations of properties (A) and (B) when executed with double’s • and in all possible ways • a point outside sees no edge • a point inside sees an edge • a point outside sees all edges • a point outside sees a non-contiguous set of edges • examples involve nearly collinear points, of course • examples are realistic as many real-life instances contain collinear points (which become nearly collinear by conversion to double’s) Classroom Examples ESA'04

  27. Global Consequences • A point outside sees no edge of the current hull. p1=(7.3000000000000194, 7.3000000000000167) p2=(24.000000000000068, 24.000000000000071) p3=(24.00000000000005, 24.000000000000053) p4=(0.50000000000001621, 0.50000000000001243) p5=(8, 4) p6=(4, 9) p7=(15, 27) p8=(26, 25) p9=(19, 11) float_orient(p1,p2,p3) > 0 float_orient(p1,p2,p4) > 0 float_orient(p2,p3,p4) > 0 float_orient(p3,p1,p4) > 0 Classroom Examples ESA'04

  28. Global Consequences • A point outside sees all edges of the current hull. p1=(200, 49.200000000000003) p2=(100, 49.600000000000001) p3=(-233.33333333333334, 50.93333333333333) p4=(166.66666666666669, 49.333333333333336) float_orient(p1,p2,p3) > 0 float_orient(p1,p2,p4) < 0 float_orient(p2,p3,p4) < 0 float_orient(p3,p1,p4) < 0 Classroom Examples ESA'04

  29. Global Consequences • A point outside sees all edges of the current hull. p1=(200, 49.200000000000003) p2=(100, 49.600000000000001) p3=(-233.33333333333334, 50.93333333333333) p4=(166.66666666666669, 49.333333333333336) float_orient(p1,p2,p3) > 0 float_orient(p1,p2,p4) < 0 float_orient(p2,p3,p4) < 0 float_orient(p3,p1,p4) < 0 Depending on the implementation: • Algorithm does not terminate! • Algorithm crashes! Classroom Examples ESA'04

  30. Global Consequences • A point inside sees an edge of the current hull. p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 6.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) (p1,p2,p3,p4) form a convex quadrilateral p4 is truly inside this quadrilateral, but float_orient(p4,p1,p5) > 0 p4 p5 p1 p2 p3 Classroom Examples ESA'04

  31. Global Consequences p4 • A point inside sees an edge of the current hull. p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 6.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) (p1,p2,p3,p4) form a convex quadrilateral p4 is truly inside this quadrilateral, but float_orient(p4,p1,p5) > 0 p5 p1 p4 p2 p5 p1 p2 p3 Classroom Examples ESA'04

  32. Global Consequences • A point outside sees a non-contiguous set of edges p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 6.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) p6=(6, 6) p4 p5 p1 p6 p2 p3 Classroom Examples ESA'04

  33. Global Consequences p4 • A point outside sees a non-contiguous set of edges p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 6.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) p6=(6, 6) p5 p1 p4 p2 p5 p1 p6 p2 p3 Classroom Examples ESA'04

  34. Global Consequences p4 • A point outside sees a non-contiguous set of edges p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 6.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) p6=(6, 6) float_orient(p4,p5,p6) < 0 float_orient(p5,p1,p6) > 0 float_orient(p1,p2,p6) < 0 p5 p1 p4 p2 p5 p1 p6 p2 p3 Classroom Examples ESA'04

  35. Global Consequences p4 • A point outside sees a non-contiguous set of edges p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 10.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) p6=(6, 6) p5 p1 p4 p2 p5 p1 p6 p2 p3 Classroom Examples ESA'04

  36. Global Consequences • A point outside sees a non-contiguous set of edges p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 10.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) p6=(6, 6) p4 p5 p1 p6 p2 p3 Classroom Examples ESA'04

  37. Global Consequences p4 • A point outside sees a non-contiguous set of edges p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 10.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) p6=(6, 6) p5 p1 p6 p4 p5 p1 p6 p2 p3 Classroom Examples ESA'04

  38. Global Consequences Classroom Examples ESA'04

  39. Global Consequences p4 • A point outside sees a non-contiguous set of edges p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 6.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) p6=(6, 6) p5 p1 p4 p2 p5 p1 p6 p2 p3 Classroom Examples ESA'04

  40. Global Consequences • A point outside sees a non-contiguous set of edges p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 6.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) p6=(6, 6) p4 p5 p1 p6 p2 p3 Classroom Examples ESA'04

  41. Global Consequences p4 • A point outside sees a non-contiguous set of edges p1=(24.00000000000005, 24.000000000000053) p2=(24.0, 6.0) p3=(54.85, 6.0) p4= (54.850000000000357, 61.000000000000121) p5=(24.000000000000068, 24.000000000000071) p6=(6, 6) p6 p5 p1 p4 p6 p5 p1 p6 p2 p3 Classroom Examples ESA'04

  42. Global Consequences Classroom Examples ESA'04

  43. 2D-Orientation of Three Points Orientation( p, q, r) = sign((qx-px)(ry-py) – (qy-py)(rx-px)) Classroom Examples ESA'04

  44. 2D-Orientation of Three Points Orientation( p, q, r) = sign((qx-px)(ry-py) – (qy-py)(rx-px)) 256x256 pixel image red=pos., yellow=0, blue=neg. orientation evaluated with double Classroom Examples ESA'04

  45. 2D-Orientation of Three Points Orientation( p, q, r) = sign((qx-px)(ry-py) – (qy-py)(rx-px)) 256x256 pixel image red=pos., yellow=0, blue=neg. orientation evaluated with double Classroom Examples ESA'04

  46. 2D-Orientation of Three Points Orientation( p, q, r) = sign((qx-px)(ry-py) – (qy-py)(rx-px)) Keep y-coordinate fix Classroom Examples ESA'04

  47. 2D-Orientation of Three Points Orientation( p, q, r) = sign((qx-px)(ry-py) – (qy-py)(rx-px)) Keep y-coordinate fix: block sizes of 25 and 24 Classroom Examples ESA'04

  48. 2D-Orientation of Three Points Orientation( p, q, r) = sign((qx-px)(ry-py) – (qy-py)(rx-px)) 256x256 pixel image red=pos., yellow=0, blue=neg. orientation evaluated with double Classroom Examples ESA'04

  49. 2D-Orientation of Three Points Orientation( p, q, r) = sign((qx-px)(ry-py) – (qy-py)(rx-px)) 256x256 pixel image red=pos., yellow=0, blue=neg. orientation evaluated with double Classroom Examples ESA'04

  50. 2D-Orientation of Three Points Orientation( p, q, r) = sign((qx-px)(ry-py) – (qy-py)(rx-px)) 256x256 pixel image red=pos., yellow=0, blue=neg. orientation evaluated with ext double Classroom Examples ESA'04

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