Ray scene intersections

Transcription

1 Ray scene intersections Josef Pelikán CGG MFF UK Praha Intersection 2018 Josef Pelikán, 1 / 26

2 Ray scene intersection result 3D position: t, [x,y,z] (2D position: [u,v]) (Normal: n, n u n v ) u v [x,y,z] n v P 0 p 1 t n n u input Ray: P 0, p 1 Intersection 2018 Josef Pelikán, 2 / 26

3 Plane ray: P(t) = P 0 + t p 1 P 0 p 1 n plane: n = [x n,y n,z n ] x x n + y y n + z z n + D = 0 t intersection t = (n P 0 + D) / (n p 1 ) negative: 2±, 3*, positive: 5±, 6*, 1/ computation of [x,y,z]: 3±, 3* Intersection 2018 Josef Pelikán, 3 / 26

4 Inverse transformation on the plane plane: Pl(u,v) = Pl 0 + u U + v V U = [x U,y U,z U ], V = [x V,y V,z V ] n = U V input: Pl, U, V, [x,y,z] result: [u,v] v [x,y,z] U u n V Pl 0 linear system u x u + v x v = x - Pl 0x solution [u,v]: 5±, 5*, 2/ u y u + v y v = y - Pl 0y Intersection 2018 Josef Pelikán, 4 / 26

5 Parallelogram ray: P(t) = P 0 + t p 1 P 0 parallelogram: R(u,v) = R 0 + u U + v V 0 u,v 1 p 1 t, [x,y,z], [u,v] U V n R 0 computing t, [x,y,z], [u,v], tests of u,v positive case total: 13±, 14*, 3/, 4 Intersection 2018 Josef Pelikán, 5 / 26

6 Triangle ray: P(t) = P 0 + t p 1 P 0 triangle: R(u,v) = R 0 + u U + v V 0 u,v,u+v 1 p 1 U t, [x,y,z], [u,v] V n R 0 computing t, [x,y,z], [u,v], tests of u,v positive case total: 14±, 14*, 3/, 3 Intersection 2018 Josef Pelikán, 6 / 26

7 General planar polygon ray: P(t) = P 0 + t p 1 P 0 p 1... n V 4 polygon plane: n = [x n,y n,z n ] x x n + y y n + z z n + D = 0 t, [x,y,z] V 2 V 3 polygon vertices: V 1, V 2,... V M V 8 V 1 computing t, [x,y,z], planar test: point polygon intersection with the plane: 8±, 9*, 1/ Intersection 2018 Josef Pelikán, 7 / 26

8 Parallel planes ray: p P(t) = P 0 + t p P n parallel planes: n = [x n,y n,z n ] x x n + y y n + z z n + D i = 0 t 1 t 2 intersections t i = (n P 0 + D i ) / (n p 1 ) the 1 st plane: 5±, 6*, 1/, every next one: 1±, 1/ Intersection 2018 Josef Pelikán, 8 / 26

9 Convex polyhedron defined as an intersection of K halfspaces at most K intersections ray vs. plane parallelism of planes can be used e.g. cuboid variables t in, t out initialized to 0, ray vs. one halfspace: t, resp., t t in = max{ t in, t } resp. t out = min{ t out, t } early exit if t in > t out t out t in Intersection 2018 Josef Pelikán, 9 / 26

10 Implicit surface ray: P(t) = P 0 + t p 1 implicit surface: F(x,y,z) = 0 example: (c - cos ax) cos z + (y + a sin ax) sin z + + cos a(x+z) = 0 substitution P(t) into F: F*(t) = 0 finding roots of the function F*(t) sometimes only the smallest positive root is needed (the 1 st intersection), for CSG we need all roots Intersection 2018 Josef Pelikán, 10 / 26

11 Algebraic surface ray: P(t) = P 0 + t p 1 algebraic surface of degree d: i jk d i, j, k 0 A x, y, z a x y z ijk i j k 0 example (toroid with radii a, b): Tab x, y, z x y z a b a b z after substitution P(t) into A: A*(t) = 0 A* is a polynomial of degree d (at most) Intersection 2018 Josef Pelikán, 11 / 26

12 Quadric (d=2) general quadric: T x Qx 0 x x y, z 1 Q a b c d b e f g c f h i d g i j after substitution of P(t): a2t 2 a1t a0 0, 2 1 T T T 0 where kde a P QP, a 2P QP, a P QP Intersection 2018 Josef Pelikán, 12 / 26

13 Quadric of revolution quadric of revolution in standard position: x y az bz c sphere: x y z 1 0, after substitution of P(t): 2 t P P 2t P P P P Intersection 2018 Josef Pelikán, 13 / 26

14 Sphere (geometric solution) P(t) = P 0 + t p 1 p 1 P 0 center of the subtense t 0 = (v p 1 ) distance D 2 = (v v) - t 0 2 inclination t D2 = R 2 - D 2 v t 1 t 0 t 2 t D R D for t D 2 = 0 there is one tangent point P( t 0 ) for t D2 > 0 two intersections exist: P( t 0 ± t D ) negative case: 9±, 6*, 1<, positive addit.: 2±, 1 sqrt Intersection 2018 Josef Pelikán, 14 / 26

15 Inverse transformation on the sphere sphere: (x-x C ) 2 +(y-y C ) 2 +(z-z C ) 2 = R 2 pole dir: P, equator dir: E (P E) = 0 input: N, P, E result: [u,v] from [0,1] 2 v P u N E arccos N E sin arccos N P, 2 v, P E N 0 u, jinak else u 1 Intersection 2018 Josef Pelikán, 15 / 26

16 Cylinder and cone 2 unit cylinder and unit cone in basic position: x 2 2 y 1 0 x y z after substitution P(t) for the cylinder: t x y 2t x x y y x y after substitution P(t) for the cone: t x y z t x x y y z z 0 x y z Intersection 2018 Josef Pelikán, 16 / 26

17 Toroid z -a a x Two circles in the xz plane: x a z b x a z b x 2 z 2 a 2 b 2 2 4a 2 b 2 z 2 0 b After substitution r 2 = x 2 + y 2 for x 2 the 4 th degree equation: x 2 y 2 z 2 a 2 b 2 2 a 2 b 2 z Intersection 2018 Josef Pelikán, 17 / 26

18 Surface of revolution P 0 + t p 1 z z r x y equation of the ray in the rz plane: r x y x0 x1t y0 y1t z z z t r Intersection 2018 Josef Pelikán, 18 / 26

19 Ray in the rz plane 2 2 After elimination of t: ar bz cz d 0 (1) 2 a = z 1 b = x y 1 c = 2 z 1 e z 0 b 2 d = z 0 z 0 b 2 z 1 e f z 1 e = x 0 x 1 y 0 y 1 f = x y 0 after substitution of parametric curve K(s) into (1) we get an equation K*(s) = 0 K* has got a double degree (compared to K) Intersection 2018 Josef Pelikán, 19 / 26

20 CSG representation primitive solids are easy convex objects only two intersections set operations are performed in the 1D ray-space: distributivity: P (A-B) = (PA) - (PB) general ray-scene intersection is a collection of line segments (intervals in 1D ray-space) geometric transformations: inverse transformation applied to a ray Intersection 2018 Josef Pelikán, 20 / 26

21 Intersections PA, PB B PB P 0 p 1 A PA P 0 p 1 Intersection 2018 Josef Pelikán, 21 / 26

22 Intersection P(A-B) difference B P(A-B) A B P 0 p 1 A Intersection 2018 Josef Pelikán, 22 / 26

23 Implementation ray: origin P 0 and direction p 1 transforms with inverse matrices T i -1 (could not be efficient enough... 1 transformation: 15+, 18*) ray vs. scene intersection (partial & final): ordered list of t parameter in ray-space: [t 1, t 2, t 3,..] set operation: generalized merging of ordered lists [ t i ] transformation of normal vectors! Intersection 2018 Josef Pelikán, 23 / 26

24 Set operations on the ray A B AB AB AB Intersection 2018 Josef Pelikán, 24 / 26

25 Normal vector transformation n n n v n n u n = n u n v general afine transformation doesn't keep angles two tangent vectors instead a normal tangent vectors transformed by 3 3 submatrix only! alternative matrix for normal vectors: M n = (M -1 ) T Intersection 2018 Josef Pelikán, 25 / 26

26 References A. Glassner: An Introduction to Ray Tracing, Academic Press, London 1989, J. Foley, A. van Dam, S. Feiner, J. Hughes: Computer Graphics, Principles and Practice, Intersection 2018 Josef Pelikán, 26 / 26