Ray Tracing I: Basics

Transcription

1 Ray Tracing I: Basics Today Basic algorithms Overview of pbrt Ray-surface intersection Next lecture Techniques to accelerate ray tracing of large numbers of geometric primitives

2 Light Rays Three ideas about light rays 1. Light travels in straight lines (mostly) 2. Light rays do not interfere with each other if they cross (light is invisible!) 3. Light rays travel from the light sources to the eye (but the physics is invariant under path reversal - reciprocity).

3 Ray Tracing in Computer Graphics Appel Ray casting 1. Generate an image by casting one ray per pixel 2. Check for shadows by sending a ray to the light

4 Ray Tracing in Computer Graphics An improved Illumination model for shaded display T. Whitted, CACM x 512 Time: - VAX 11/780 (1979) 74m - PC (2006) 6s - GPU (2012) 1/30s Spheres and Checkerboard, T. Whitted, 1979

5 Ray Tracing Video

6 Spheres-over-plane.pbrt (depth=1)

7 Spheres-over-plane.pbrt (mirror depth=2)

8 Spheres-over-plane.pbrt (mirror depth=3)

9 Spheres-over-plane.pbrt (mirror depth=10)

10 Spheres-over-plane.pbrt (glass depth=1)

11 Spheres-over-plane.pbrt (glass depth=2)

12 Spheres-over-plane.pbrt (glass depth=3)

13 Spheres-over-plane.pbrt (g/m depth=10)

14

15 PBRT Render Loop

16 PBRT Intersection Methods

17 PBRT Intergrater

18 Ray-Surface Intersection

19 Ray-Plane Intersection Ray:!!! P = O + t D 0 t < O! D! t P! N!

20 Ray-Plane Intersection Ray: Plane:!!! P = O + t D 0 t <!!! ( P P" ) N = 0 ax + by + cz + d = O! D! 0 t P!! P! N!

21 Ray-Plane Intersection Ray: Plane:!!! P = O + t D 0 t <!!! ( P P" ) N = 0 ax + by + cz + d = O! D! 0 t P!! P! N! Solve for intersection!!!!!!! ( P P" ) N = ( O + td P" ) N = 0!!! ( O P" ) N t =!! D N

22 Optimize Ray-Convex Polyhedra? Polyhedra defined as the intersection of N half-planes

23 Triangles s 2 P! P! 3 s1 s 3 P! 1 Barycentric coordinates!!!! P = s P + s P + s P Inside triangle criteria P! 2 s 1 = area(4pp 2 P 3 )/area(4p 1 P 2 P 3 ) s 2 = area(4p 1 PP 3 )/area(4p 1 P 2 P 3 ) s 3 = area(4p 1 P 2 P)/area(4P 1 P 2 P 3 ) 0 s s s 1 3 s + s + s =

24 Triangle Inside Test Test whether a point is inside the triangle!!!! P = s P + s P + s P ! s1 " # $ # $ = #% s $ 3 & [ P P P ] s [ P] " s1 # $ % $ s % = $ & s % 3 ' 1 [ P P P ] [ P]

25 Moller-Trumbore Algorithm!!!!! O + td = (1 b b ) P + b P + b P!!! t "! S " 2 E2 # $ 1 #!! $ # b $ =!! 1 # S1 S S!! $ 1 E1 #% b $ & # $ 2 % S 2 D & Cost = (1 div, 27 mul, 17 add) Where:!!! E1 = P1 P0!!! E = P P!!! S = O P 0!!! S1 = D E2!!! S = S E

26 Ray-Sphere Intersection O! D! P! C!!!! Ray: P = O + td P! C! R = Sphere: ( ) 2 2 0

27 Ray-Sphere Intersection D! O! C! Ray: Sphere:!!! P = O + td P! C! R = P! ( ) 2 2 0!!! + = ( ) 2 2 O t D C R 0 t = ± 2 b b 4ac 2a 2 at + bt + c = 0 2 a = D!!!! b = 2( O C) D!! 2 2 c = ( O C) R

28 Ray-Sphere Intersection D! O! C! Ray: Sphere:!!! P = O + td P! C! R = P! ( ) 2 2 0!!! + = ( ) 2 2 O t D C R 0 t = ± 2 b b 4ac 2a 2 at + bt + c = 0 2 a = D!!!! b = 2( O C) D!! 2 2 c = ( O C) R

29 Shapes

30 Shapes in pbrt class Shape { public: // Shape interface virtual BBox ObjectBound() const = 0; virtual BBox WorldBound() const; virtual bool CanIntersect() const; virtual void Refine(vector<Reference<Shape>> &refined) const; virtual bool Intersect(const Ray &ray, float *thit, float *rayepsilon, DifferentialGeometry *dg) const; virtual bool IntersectP(const Ray &ray) const; virtual void GetShadingGeometry(const Transform &obj2world, const DifferentialGeometry &dg, DifferentialGeometry *dgshading) const { *dgshading = dg; } virtual float Area() const; }; // Shape Public Data const Transform *ObjectToWorld, *WorldToObject;

31 Quadrics CS348b Lecture 2 Pat Hanrahan, Spring 2016

32 Geometric Methods: Normals e.g. Sphere N! = P! C! x = sinθ y = sinθ cosφ sinφ z = cosθ! P = ( x, y, z) P!, u, v S! N!! T! P = (cosθ cos φ,cosθ sin φ, sin θ ) θ! P = ( sinθ sin φ,sinθ cos φ,0) φ!!! P P N = θ φ

33 Geometric Methods: Parameters e.g. Sphere x y z = = = sinθ cosφ sinθ sinφ cosθ φ θ = = 1 tan ( x, y) cos 1 z θ = θ + v( θ θ ) φ = uφ min max min max v = ( θ θ ) /( θ θ ) u = φ / φ max min max min

34 Differential Geometry Representation struct DifferentialGeometry { // Filled in by Shape::Intersect() Point p; Normal nn; float u, v; const Shape *shape; Vector dpdu, dpdv; Normal dndu, dndv; };

35 Ray-Implicit Surface Intersection f ( x, y, z ) = 0 * ( ) 0 f t = x = x + x t 0 1 y = y + y t 0 1 z = z + z t Substitute ray equation 2. Find positive, real roots Univariate root finding Newton s method Regula-falsi Interval methods

36 Ray-Algebraic Surface Intersection pn( x, y, z ) = 0 p * ( t ) = 0 n x = x + x t 0 1 y = y + y t 0 1 z = z + z t 0 1 Degree n Linear: Plane Quadric: Spheres, Quartic: Tori Polynomial root finding Quadratic, cubic, quartic