Lecture 4: Planes, Interior Point Testing, Duality

Transcription

1 Lecture 4: Planes, Interior Point Testing, Duality Chris Tralie, Duke University 1/26/2016

2 Table of Contents Normals and Planes Interior Point Testing Duality

3 Announcements Everyone got Mini 1 Part 1 in on time! Part 2 Due Friday 11:55 PM Drop/Add Tomorrow! SIGGRAPH Student Volunteers application

4 Normals Vector in direction perpendicular to object in question

5 Perpendicular To A 2D Vector Negate y and swap: (a, b) ( b, a) y (-b, a) a b (a, b) -b a x

6 Normal Form of A Line Given point p and normal n, a point q is on line if ( q p) n = 0 y n p q-p q l 1 x

7 Normal Form of A Line Assume n = 1 (unit normal) ( q p) n = 0 q n = p n = d(l 1, origin) y n dist p q-p q l 1 x

8 Normal Form of A Line ( q p) n = 0 Let q = (x, y). Expanding and rewriting as implicit linear equation n x x + n y y d(l 1, origin) = 0 y n dist p q-p q l 1 x

9 Line: Degrees of Freedom n x x + n y y d(l 1, origin) = 0 Implicit form Ax + By + C = 0 How many degrees of freedom are there in a line?

10 Normal Form of a Line What s the line normal of the line y = mx + b?

11 2D Planes Given point p and normal n, a point q is on plane if ( q p) n = 0 = q n = p n = 0 (now 3D vectors) y n p q x z

12 2D Planes n x x + n y y + n z z d(p, origin) = 0 y n p q x z

13 2D Planes Ax + By + Cz + D = 0 y n p q x z

14 Ray Intersect Plane t v r Plane: ( q p) n = 0 Ray: r + t v, t 0 t = r n v n

15 Table of Contents Normals and Planes Interior Point Testing Duality

16 Convex Polygons YES NO Definition extends to 3D polytopes (and any geometric set)

17 Convex Or Not?

18 Convex Or Not?

19 Convex Or Not?

20 Point Inside Convex Polygon: Halfplane Method

21 Point Inside Convex Polygon: Halfplane Method

22 Point Inside Convex Polygon: Halfplane Method

23 Point Inside Convex Polygon: Halfplane Method

24 Point Inside Convex Polygon: Halfplane Method

25 Point Inside Convex Polygon: Hull Method Convex Hull (Segue)

26 Point Inside Convex Polygon: Hull Method Convex Hull (Segue)

27 Point Inside Convex Polygon: Hull Method Convex Hull Test YES NO

28 Point Inside Convex Polygon: Area Method Area( abc) = Area( abd) + Area( bcd) + Area( cad) c a d b

29 Point Inside Convex Polygon: Area Method Area( abc) < Area( abd) + Area( bcd) + Area( acd) d c a b

30 Point Inside Convex Polygon: Area Method

31 3D Ray Convex Polygon Intersection n r

32 Logarithmic Convex Polygon Test

33 Logarithmic Convex Polygon Test Segue: Binary Search

34 Logarithmic Convex Polygon Test

35 Logarithmic Convex Polygon Test

36 Logarithmic Convex Polygon Test

37 Logarithmic Convex Polygon Test

38 Logarithmic Convex Polygon Test

39 Nonconvex Polygons Inside or outside??

40 Nonconvex Polygons

41 Nonconvex Polygons: Ray Casting

42 Table of Contents Normals and Planes Interior Point Testing Duality

43 Points To Lines p : (a,b) p*: y = ax - b l: y = cx + d l*: (c,-d)

44 Points To Lines p > l l > p where > means above TODO: Verify this using vectors! p : (a,b) p*: y = ax - b l: y = cx + d l*: (c,-d)

45 Point Inside Convex Polygon: Halfplane Method What dual problem did we solve??

46 Point Inside Convex Polygon: Halfplane Method What dual problem did we solve??