Geometry Introduction

1. Geometry Introduction

2. Topic • Introduction • Two lines Intersection Test • Point inside polygon • Convex hull • Line Segments Intersection Algorithm

3. Geometry Components • Scalar (S) • Point (P) • Free vector (V) • Allowed operations • S * V  V • V + V  V • P – P  V • P + V  P

4. Examples u+v Vector addition v u p p-q Point subtraction q p+v v Point-vector addition p

5. a = 0.5 a = 1.25 q p จุดใดๆบนเส้นตรงที่ลากผ่านจุด p และ q สามารถสร้างได้จาก affine combination:

6. Euclidean Geometry • In affine geometry, angle and distance are not defined. • Euclidean geometry is an extension providing an additional operation called “inner product” • There are other types of geometry that extends affine geometry such as projective geometry, hyperbolic geometry…

7. Dot product is a mapping from two vectors to a real number. Dot product Distance Length Angle Orthogonality: u and v are orthogonal when

8. Topic • Introduction • Two lines Intersection Test • Point inside polygon • Convex hull • Line Segments Intersection Algorithm

9. p2 p3 p3 p2 p4 p1 p2 p1 p0 p1 (2) (3) Cross-Product-Based Geometric Primitives Some fundamental geometric questions: source: 91.503 textbook Cormen et al. (1)

10. p2 p1 p0 (1) Cross-Product-Based Geometric Primitives: (1) 33.1 Advantage: less sensitive to accumulated round-off error source: 91.503 textbook Cormen et al.

11. p2 p1 p0 (2) Cross-Product-Based Geometric Primitives: (2) 33.2 source: 91.503 textbook Cormen et al.

12. isLeft() // isLeft(): tests if a point is Left|On|Right of an infinite line.//    Input:  three points P0, P1, and P2//    Return: >0 for P2 left of the line through P0 and P1//            =0 for P2 on the line//            <0 for P2 right of the lineinline intisLeft( Point P0, Point P1, Point P2 ){    return ( (P1.x - P0.x) * (P2.y - P0.y)            - (P2.x - P0.x) * (P1.y - P0.y) );}

13. Segment-Segment Intersection • Finding the actual intersection point • Approach: parametric vs. slope/intercept • parametric generalizes to more complex intersections • e.g. segment/triangle • Parameterize each segment Lab c Lcd C=d-c b Lab c q(t)=c+tC Lcd b d A=b-a a d a p(s)=a+sA Intersection: values of s, t such that p(s) =q(t) : a+sA=c+tC 2 equations in unknowns s, t : 1 for x, 1 for y source: O’Rourke, Computational Geometry in C

14. Assume that a = (x1,y1) b = (x2,y2) c = (x3,y3) d = (x4,y4)

15. Code typedefstruct point { double x; double y;} point; typedefstruct line { point p1; point p2;} line; intcheck_lines(line *line1, line *line2, point *hitp) { double d = (line2->p2.y - line2->p1.y)*(line1->p2.x-line1->p1.x) - (line2->p2.x - line2->p1.x)*(line1->p2.y-line1->p1.y); double ns = (line2->p2.x - line2->p1.x)*(line1->p1.y-line2->p1.y) - (line2->p2.y - line2->p1.y)*(line1->p1.x-line2->p1.x); double nt = (line1->p2.x - line1->p1.x)*(line1->p1.y - line2->p1.y) - (line1->p2.y - line1->p1.y)*(line1->p1.x - line2->p1.x); if(d == 0) return 0; double s= ns/d; double t = nt/d; return (s >=0 && s <= 1 && t >= 0 && t <= 1)); }

16. p3 p2 p4 p1 (3) Intersection of 2 Line Segments Step 1: Bounding Box Test p3 and p4 on opposite sides of p1p2 33.3 Step 2: Does each segment straddle the line containing the other? source: 91.503 textbook Cormen et al.

17. Topic • Introduction • Two lines Intersection Test • Point inside polygon • Convex hull • Line Segments Intersection Algorithm

18. Point Inside Polygon Test • Given a point, determine if it lies inside a polygon or not

19. Ray Test • Fire ray from point • Count intersections • Odd = inside polygon • Even = outside polygon

20. Problems With Rays • Fire ray from point • Count intersections • Odd = inside polygon • Even = outside polygon • Problems • Ray through vertex

21. Problems With Rays • Fire ray from point • Count intersections • Odd = inside polygon • Even = outside polygon • Problems • Ray through vertex

22. Problems With Rays • Fire ray from point • Count intersections • Odd = inside polygon • Even = outside polygon • Problems • Ray through vertex • Ray parallel to edge

23. Solution • Edge Crossing Rule • an upward edge includes its starting endpoint, and excludes its final endpoint; • a downward edge excludes its starting endpoint, and includes its final endpoint; • horizontal edges are excluded; and • the edge-ray intersection point must be strictly right of the point P. • Use horizontal ray for simplicity in computation

24. Code //cn_PnPoly(): crossing number test for a point in a polygon//      Input:   P = a point,//               V[] = vertex points of a polygon V[n+1] with V[n]=V[0]//      Return:  0 = outside, 1 = inside// This code is patterned after [Franklin, 2000]intcn_PnPoly( Point P, Point* V, int n ){intcn = 0;    // the crossing number counter    // loop through all edges of the polygon    for (inti=0; i<n; i++) {    // edge from V[i] to V[i+1]       if (((V[i].y <= P.y) && (V[i+1].y > P.y))    // an upward crossing        || ((V[i].y > P.y) && (V[i+1].y <= P.y))) { // a downward crossing            // compute the actual edge-ray intersect x-coordinate            float vt = (float)(P.y - V[i].y) / (V[i+1].y - V[i].y);            if (P.x < V[i].x + vt * (V[i+1].x - V[i].x)) // P.x < intersect                ++cn;   // a valid crossing of y=P.y right of P.x        }    }    return (cn&1);    // 0 if even (out), and 1 if odd (in)}

25. A Better Way

26. A Better Way

27. A Better Way

28. A Better Way

29. A Better Way

30. A Better Way

31. A Better Way

32. A Better Way

33. A Better Way

34. A Better Way

35. A Better Way

36. A Better Way • One winding = inside

37. A Better Way

38. A Better Way

39. A Better Way

40. A Better Way

41. A Better Way

42. A Better Way

43. A Better Way

44. A Better Way

45. A Better Way

46. A Better Way

47. A Better Way

48. A Better Way • zero winding = outside

49. Requirements • Oriented edges • Edges can be processed in any order

50. Advantages • Extends to 3D! • Numerically stable • Even works on models with holes: • Odd k: inside • Even k: outside • No ray casting