Intersecting Simple Surfaces. Dr. Scott Schaefer

Transcription

1 Intersecting Simple Surfaces Dr. Scott Schaefer 1

2 Types of Surfaces Infinite Planes Polygons Convex Ray Shooting Winding Number Spheres Cylinders 2/66

3 Infinite Planes Defined by a unit normal n and a point o n( x o) 0 3/66

4 Infinite Planes Defined by a unit normal n and a point o n( x o) 0 L( t) p vt 4/66

5 Infinite Planes Defined by a unit normal n and a point o n( x o) 0 L( t) p vt n( p vt o) 0 5/66

6 Infinite Planes Defined by a unit normal n and a point o n( x o) 0 L( t) p vt nvt n ( o p) 6/66

7 Infinite Planes Defined by a unit normal n and a point o n( x o) 0 L( t) p vt n ( o p) t n v 7/66

8 Infinite Planes Defined by a unit normal n and a point o n( x o) 0 L( t) p vt n ( o p) p v n v 8/66

9 Polygons Intersect infinite plane containing polygon Determine if point is inside polygon 9/66

10 Polygons Intersect infinite plane containing polygon Determine if point is inside polygon How do we know if a point is inside a polygon? 10/66

11 Point Inside Convex Polygon 11/66

12 Point Inside Convex Polygon Check if point on same side of all edges 12/66

13 Point Inside Convex Polygon Check if point on same side of all edges 13/66

14 Point Inside Convex Polygon Check if point on same side of all edges 14/66

15 Point Inside Convex Polygon Check if point on same side of all edges 15/66

16 Point Inside Convex Polygon Check if point on same side of all edges 16/66

17 Point Inside Convex Polygon Check if point on same side of all edges 17/66

18 Point Inside Convex Polygon Check if point on same side of all edges 18/66

19 Point Inside Convex Polygon Check if point on same side of all edges 19/66

20 Point Inside Convex Polygon Check if point on same side of all edges 20/66

21 Point Inside Convex Polygon Check if point on same side of all edges 21/66

22 Point Inside Convex Polygon P i1 ( P ( P i i1 N T X ) X T ) T must be same sign X P i 22/66

23 Point Inside Polygon Test Given a point, determine if it lies inside a polygon or not 23/66

24 Ray Test Fire ray from point Count intersections Odd = inside polygon Even = outside polygon 24/66

25 Problems With Rays Fire ray from point Count intersections Odd = inside polygon Even = outside polygon Problems Ray through vertex 25/66

26 Problems With Rays Fire ray from point Count intersections Odd = inside polygon Even = outside polygon Problems Ray through vertex 26/66

27 Problems With Rays Fire ray from point Count intersections Odd = inside polygon Even = outside polygon Problems Ray through vertex Ray parallel to edge 27/66

28 A Better Way 28/66

29 A Better Way 29/66

30 A Better Way 30/66

31 A Better Way 31/66

32 A Better Way 32/66

33 A Better Way 33/66

34 A Better Way 34/66

35 A Better Way 35/66

36 A Better Way 36/66

37 A Better Way 37/66

38 A Better Way 38/66

39 A Better Way One winding = inside 39/66

40 A Better Way 40/66

41 A Better Way 41/66

42 A Better Way 42/66

43 A Better Way 43/66

44 A Better Way 44/66

45 A Better Way 45/66

46 A Better Way 46/66

47 A Better Way 47/66

48 A Better Way 48/66

49 A Better Way 49/66

50 A Better Way 50/66

51 A Better Way zero winding = outside 51/66

52 Requirements Oriented edges Edges can be processed in any order 52/66

53 Computing Winding Number Given unit normal n =0 For each edge (p 1, p 2 ) n(( p x) ( p2 x)) cos ( p x) ( p x) 1 ( p1 x) ( p2 x p1 x p2 x 1 1 ) 2 x p 2 p 1 If, then inside 53/66

54 Advantages Extends to 3D! Numerically stable Even works on models with holes (sort of) No ray casting 54/66

55 Intersecting Spheres Three possible cases Zero intersections: miss the sphere One intersection: hit tangent to sphere Two intersections: hit sphere on front and back side How do we distinguish these cases? 55/66

56 Intersecting Spheres F( x) ( x c) ( x c) r /66

57 57/66 Intersecting Spheres 0 ) ( ) ( ) ( 2 r c x c x x F 0 ) ( ) ( )) ( ( 2 r c vt p c vt p t L F

58 58/66 Intersecting Spheres 0 ) ( ) ( ) ( 2 r c x c x x F 0 ) ( ) ( )) ( ( 2 r c vt p c vt p t L F 0 ) ( ) ( ) ( 2 ) ( )) ( ( 2 2 r c p c p t c p v t v v t L F

59 59/66 Intersecting Spheres is quadratic in t 0 )) ( ( t L F 0 ) ( ) ( ) ( 2 ) ( )) ( ( 2 2 r c p c p t c p v t v v t L F a b c

60 Intersecting Spheres F( L( t)) 0 is quadratic in t F( L( t)) ( vv) t 2 2v ( p c) t ( p c) ( p c) r 2 0 a b c Solve for t using quadratic equation b t b 2 4ac 2a 2 If b 4ac0, no intersection 2 If b 4ac 0, one intersection Otherwise, two intersections 60/66

61 Normals of Spheres F( x) ( x c) ( x c) r 2 0 F( x) x c 61/66

62 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r A r C 62/66

63 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r A r C L(t) 63/66

64 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r 1.Perform an orthogonal projection to the plane A defined by C, A on the line L(t) and intersect with circle in 2D r C L ˆ( t ) 64/66

65 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r 2.Substitute t parameters from 2D intersection to 3D line equation A r C L(t) 65/66

66 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r 3.Normal of 2D circle is the same normal of A cylinder at point of intersection r C N L(t) 66/66

67 67/66

68 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r r A C P N 68/66

69 Infinite Cylinders Defined by a center point C, a unit axis direction A and a radius r A P N N P C (( P C) A) A r r C 69/66