Agenda. Ray(tracing(algorithm( Code(structure( Ray(intersec;ons Sphere Triangle Hints Transforma;ons Debugging How(to(start

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1 Magdalena(Schwarzl 1

2 Agenda Ray(tracing(algorithm( Code(structure( Ray(intersec;ons Sphere Triangle Hints Transforma;ons Debugging How(to(start Magdalena(Schwarzl 2

3 Basic algortihm For each pixel: 1. Construct ray 2. Test intersection with objects Nearest intersection point visible on image 3. Perform Shading 2

4 Components Scene: Objects, lights, viewpoint, materials,... xml scene format, see course website Camera: For ray construction Image: 2D Array of Vec3's (for r,g,b), represents pixels Rays: For intersection tests Shading: Illumination models 3

5 Ray Consists of origin + direction (both Vec3) dir Normalize direction! o Every point p on ray = o+ λ*dir Object Needs intersection routine, returning λ, normal, etc. Additional: Parameters for material properties,... For this lab: Spheres and triangle meshes Scene Arrange objects for easy intersection (for example in a list) 4

6 Implementation List<SceneObjects> objectlist for (y=0; y<width; y++) iterate over all pixels for (x=0; x<height; x++) ray = camera.constructrayforpixel(x,y) color = trace(ray,objectlist) image.setcolor(x,y,color) 5

7 Implementation List<SceneObjects> objectlist List<SceneObjects> objectlist for (y=0; y<width; y++) for (y=0; y<width; y++) for (x=0; x<height; x++) for (x=0; x<height; x++) ray = camera.constructrayforpixel(x,y) ray = camera.constructrayforpixel(x,y) color = trace(ray,objectlist) color = trace(ray,objectlist) image.setcolor(x,y,color) image.setcolor(x,y,color) trace(ray,objectlist) for(s=0; s < objectlist.size(); s++) object = objectlist.at(s) result = object.intersect(ray) //determine smallest result.t > 0! if (no intersection) return backgroundcolor else return result.color 6

8 So far: 7

9 Shading Extend trace method to use local illumination models: trace(ray,objectlist) for(s=0; s < objectlist.size(); s++) object = objectlist.at(s) result = object.intersect(ray) //determine smallest result.t > 0! if (no intersection) return backgroundcolor else //determine intersection point Vec3 ip = ray.origin + result.t*ray.direction Vec3 normal = result.normal Vec3 light Vec3 color = shade(ray, object, ip, normal, light) return color 8

10 Shading More than one light source: trace(ray,objectlist) for(s=0; s < objectlist.size(); s++) object = objectlist.at(s) result = object.intersect(ray) //determine smallest result.t > 0! if (no intersection) return backgroundcolor else //determine intersection point Vec3 ip = ray.origin + result.t*ray.direction Vec3 normal = result.normal Vec3 overallcolor; for (l = 0; l < lightsinscene; l++) Vec3 light overallcolor += shade(ray, object, ip, normal, light) return overallcolor 9

11 Shadows Point is in shadow if not directly illuminated by light Test by creating shadow ray from intersection point to light source Intersect with all objects again if (intersection in front of light) return darkcolor else //perform shading as before 10

12 So far, only direct illumination: 11

13 So far, only direct illumination: VS Recursion needed! 12

14 Recursion For every intersection point: Create new rays for reflection/refraction Trace those rays! Stop recursion after reaching max. bounces 13

15 Recursion trace(ray,objectlist, depth) //perform intersection and shading as before... if depth >= maxbounces return overallcolor stop after max bounces //depending on object's material properties, cast new rays if (object.transmittance > 0) ray.origin = ip ray.direction = calculatetransmittancedirection() colortrans = trace(ray,objectlist, depth+1) if (object.reflectance > 0) ray.origin = ip ray.direction = calculatereflectancedirection() colorrefl = trace(ray,objectlist, depth+1) return finalcolor = //combine colors 14

16 Combine colors Coefficients for reflectance, transmittance are defined in scene file Reflectance + Transmittance + own Color = 1 finalcolor = r*reflectedcolor + t*transmittedcolor + (1-r-t)*ownColor 15

17 Agenda Ray(tracing(algorithm( Code(structure( Ray(intersec;ons Sphere Triangle Hints Transforma;ons Debugging How(to(start Magdalena(Schwarzl 3

18 Ray(intersec;on A(ray(is(defined(as X = P + λ! d! d P Where(X(is(any(point(on(the(line( Magdalena(Schwarzl 4

19 Sphere A(sphere(is(defined(as r 2 = ( X C) 2 C r Where X is any point on the sphere Magdalena(Schwarzl 5

20 Intersect Ray-Sphere We(want(to(find(a(point(on(both! d r 2 = ( X C) 2 X = P + λ! d S 1 Compute lambda C r 2 = ((P + λ d) C! ) 2 P S 2 r r 2 = ((P + λ d) C! )((P + λ d) C! ) Magdalena(Schwarzl 6

21 Intersect Ray-Sphere r 2 = ((P + λ d) C! )((P + λ d) C! ) 0 = d 2 λ 2 + λ ( 2P d! 2C d! ) + ( P C) 2 r 2 A = d 2 B = ( 2P d! 2C d! ) C = ( P C) 2 r 2 λ 1,2 = B ± B2 4AC 2A Magdalena(Schwarzl 7

22 Intersect Ray-Sphere λ 1,2 = B ± B2 4AC 2A Magdalena(Schwarzl 8

23 Intersect Ray-Sphere λ 1,2 = B ± B2 4AC 2A <(0( no(intersec;on( =(0( tangent( >(0( two(intersec;on(points(! d S 1! d P r C P! d S 1 r C P S 2 r C Magdalena(Schwarzl 9

24 Barycentric coordinates V 2 A(triangle(is(defined(as e 12 V 1 e 23 e 13 V 3 Where X is any point inside the triangle Magdalena(Schwarzl 10

25 Barycentric coordinates V 2 A(triangle(is(defined(as 1 u v = A 1 e 12 u = A 2 A 1 + A 2 + A 3 A 1 + A 2 + A 3 V 1 A 3 A 2 X A 1 e 23 v = A 3 A 1 + A 2 + A 3 e 13 V 3 Where X is any point inside the triangle Magdalena(Schwarzl 11

26 Barycentric coordinates V 2 A(triangle(is(defined(as 1 u v = A 1 e 12 u = A 2 A 1 + A 2 + A 3 A 1 + A 2 + A 3 V 1 A 3 A 2 X A 1 e 23 v = A 3 A 1 + A 2 + A 3 e 13 X ( u, v) = ( 1 u v)v 1 + uv 2 + vv 3 u, v 0 V 3 Where X is any point inside the triangle u + v 1 Magdalena(Schwarzl 12

27 Triangle A triangle is defined by: X ( u, v) = ( 1 u v)v 1 + uv 2 + vv 3 u, v 0 u + v 1 V 1 e 12 V 2 e 13 e 23 V 3 3 points define a plane: Magdalena(Schwarzl 13

28 Triangle A triangle is defined by: X ( u, v) = ( 1 u v)v 1 + uv 2 + vv 3 u, v 0 u + v 1 V 1 e 12 V 2 e 13 e 23 V 3 3 points define a plane: n = e 13 e 12 e 13 e 12 nx = nv 1 Magdalena(Schwarzl 14

29 Intersect Ray-Triangle We(want(the(intersec;on ( ) = 1 u v X u, v ( )V 1 + uv 2 + vv 3 V 1 e 12 S 1 V 2 e 23 nx = nv 1 X = P + λ! d! d e 13 Compute lambda P V 3 n( P + λ d! ) = nv 1 Magdalena(Schwarzl 15

30 Intersect Triangle-Sphere 1) ray-plane λ = nv 1 np n! d Magdalena(Schwarzl 16

31 Intersect Triangle-Sphere 1) ray-plane λ = nv 1 np n! d!= 0 One intersection = 0 No intersection with plane S 1 S 1 = P + λ! d parallel Inside plane Magdalena(Schwarzl 17

32 Intersect Ray-Triangle 2) Inside triangle? P + λ d! = ( 1 u v)v 1 + uv 2 + vv 3 Use x,y,z to get linear system of 3 equations Solve for v & u Magdalena(Schwarzl 18

33 Intersect Ray-Triangle 2) Inside triangle? P + λ d! = ( 1 u v)v 1 + uv 2 + vv 3 Use x,y,z to get linear system of 3 equations Solve for v & u u,v <=1 AND u,v >=0 intersection u,v > 1 OR u,v < 0 no intersection Magdalena(Schwarzl 19

34 Agenda Ray(tracing(algorithm( Code(structure( Ray(intersec;ons Sphere Triangle Hints Transforma;ons Debugging How(to(start Magdalena(Schwarzl 20

35 Transformations Think about transformations early! Magdalena(Schwarzl 21

36 Transformations Think about transformations early! Two approaches: transform object inverse transform ray Precompute inv.-transfomations No translation on vectors Inverse-transpose for normals Magdalena(Schwarzl 22

37 Precision Problems? Magdalena(Schwarzl 23

38 Precision Problems? Shadow ray hit its source (collision at distance ==0) Offset the ray cast source a small amount in the direction of the ray. Do never test for == 0 Test for between 0 ± ε Magdalena(Schwarzl 24

39 Normals, Texures Visual debugging Output textures, normals, reflection vectors as color to see problems 0 ±ε Magdalena(Schwarzl 25

40 Agenda Ray(tracing(algorithm( Code(structure( Ray(intersec;ons Sphere Triangle Hints Transforma;ons Debugging How(to(start Magdalena(Schwarzl 26

41 How to start Plan code structure Classes, methods, Decide on libraries xml-parsing, images Error messages!! Create a black image Read in xml scene Compute intersections Use only object color Compute shading ambient, diffuse, specular Test on PC labs Include readme!! compile command If it might take longer ;) Magdalena(Schwarzl 27

42 Questions Any questions? Magdalena(Schwarzl 28