Transformations in Ray Tracing. MIT EECS 6.837, Durand and Cutler

Transcription

1 Transformations in Ray Tracing

2 Linear Algebra Review Session Tonight! 7:30 9 PM

3 Last Time: Simple Transformations Classes of Transformations Representation homogeneous coordinates Composition not commutative

4 Today Motivations Transformations in Modeling Adding Transformations to our Ray Tracer Constructive Solid Geometry (CSG) Assignment 2

5 Modeling Create / acquire objects Placing objects Placing lights Describe materials Choose camera position and camera parameters Specify animation...

6 Transformations in Modeling Position objects in a scene Change the shape of objects Create multiple copies of objects Projection for virtual cameras Animations

7 Today Motivations Transformations in Modeling Scene description Class Hierarchy Transformations in the Hierarchy Adding Transformations to our Ray Tracer Constructive Solid Geometry (CSG) Assignment 2

8 Scene Description Scene Camera Lights Background Materials (much more next week) Objects

9 Simple Scene Description File OrthographicCamera { center direction up size 5 } Lights { numlights 1 DirectionalLight { direction color } } Background { color } Materials { nummaterials <n> <MATERIALS> } Group { numobjects <n> <OBJECTS> }

10 Class Hierarchy Object3D Sphere Cone Cylinder Plane Triangle Group

11 Why is a Group an Object3D? Logical organization of scene

12 Simple Example with Groups Group { numobjects 3 Group { numobjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { numobjects 2 Group { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } Plane { <PLANE PARAMS> } }

13 Adding Materials Group { numobjects 3 Group { numobjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { numobjects 2 Group { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } Plane { <PLANE PARAMS> } }

14 Adding Materials Group { numobjects 3 Material { <BROWN> } Group { numobjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { numobjects 2 Material { <BLUE> } Group { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { Material { <GREEN> } Box { <BOX PARAMS> } Material { <RED> } Sphere { <SPHERE PARAMS> } Material { <ORANGE> } Sphere { <SPHERE PARAMS> } } } Material { <BLACK> } Plane { <PLANE PARAMS> } }

15 Adding Transformations

16 Class Hierarchy with Transformations Object3D Sphere Cone Cylinder Plane Triangle Group Transform

17 Why is a Transform an Object3D? To position the logical groupings of objects within the scene

18 Simple Example with Transforms Group { numobjects 3 Transform { ZRotate { 45 } Group { numobjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } } Transform { Translate { } Group { numobjects 2 Group { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } } Plane { <PLANE PARAMS> } }

19 Nested Transforms p' = T ( S p ) = TS p Translate Scale Sphere same as Translate Scale Sphere Transform { Translate { } Transform { Scale { } Sphere { center radius 1 } } } Transform { Translate { } Scale { } Sphere { center radius 1 } }

20 Questions?

21 Today Motivations Transformations in Modeling Adding Transformations to our Ray Tracer Transforming the Ray Handling the depth, t Transforming the Normal Constructive Solid Geometry (CSG) Assignment 2

22 Incorporating Transforms 1. Make each primitive handle any applied transformations Sphere { center radius 2 } 2. Transform the Rays Transform { Translate { } Scale { } Sphere { center radius 1 } }

23 Primitives handle Transforms Sphere { center z_rotation 30 r_major 2 r_minor 1 } r minor (x,y) r major Complicated for many primitives

24 Transform the Ray Move the ray from World Space to Object Space r minor (x,y) r major r = 1 World Space (0,0) Object Space p WS = M p OS p OS = M -1 p WS

25 Transform Ray New origin: origin OS = M -1 origin WS New direction: direction OS = M -1 (origin WS + 1 * direction WS ) - M -1 origin WS direction OS = M -1 direction WS origin WS direction WS q WS = origin WS + t WS * direction WS origin OS direction OS q OS = origin OS + t OS * direction OS World Space Object Space

26 Transforming Points & Directions Transform point Transform direction Homogeneous Coordinates: (x,y,z,w) W = 0 is a point at infinity (direction)

27 What to do about the depth, t If M includes scaling, direction OS will NOT be normalized 1. Normalize the direction 2. Don't normalize the direction

28 1. Normalize direction t OS t WS and must be rescaled after intersection t WS World Space t OS Object Space

29 2. Don't normalize direction t OS = t WS Don't rely on t OS being true distance during intersection routines (e.g. geometric ray-sphere intersection, a 1 in algebraic solution) t WS World Space t OS Object Space

30 Questions?

31 New component of the Hit class Surface Normal: unit vector that is locally perpendicular to the surface

32 Why is the Normal important? It's used for shading makes things look 3D! object color only (Assignment 1) Diffuse Shading (Assignment 2)

33 Visualization of Surface Normal ±x= Red ±y= Green ±z= Blue

34 How do we transform normals? n WS n OS World Space Object Space

35 Transform the Normal like the Ray? translation? rotation? isotropic scale? scale? reflection? shear? perspective?

36 Transform the Normal like the Ray? translation? rotation? isotropic scale? scale? reflection? shear? perspective?

37 What class of transforms? Projective Affine Similitudes Rigid / Euclidean Linear Translation Identity Rotation Isotropic Scaling Scaling Reflection Shear Perspective a.k.a. Orthogonal Transforms

38 Transformation for shear and scale Incorrect Normal Transformation Correct Normal Transformation

39 More Normal Visualizations Incorrect Normal Transformation Correct Normal Transformation

40 So how do we do it right? Think about transforming the tangent plane to the normal, not the normal vector n OS n WS v OS Original Incorrect Correct v WS Pick any vector v OS in the tangent plane, how is it transformed by matrix M? v WS = M v OS

41 Transform tangent vector v v is perpendicular to normal n: n OST v OS = 0 n OST (M -1 M) v OS = 0 (n OST M -1 ) (M v OS ) = 0 (n OST M -1 ) v WS = 0 v WS is perpendicular to normal n WS : n WS n WST = n OS (M -1 ) n OS v OS v WS n WS =(M -1 ) T n OS n WST v WS = 0

42 Comment So the correct way to transform normals is: n WS =(M -1 ) T n OS But why did n WS = M n OS work for similitudes? Because for similitude / similarity transforms, (M -1 ) T =λ M e.g. for orthonormal basis: ux uy uz xu xv xn M = vx vy vz M -1 = yu yv yn nx ny nz zu zv zn

43 Questions?

44 Today Motivations Transformations in Modeling World Space vs Object Space Adding Transformations to our Ray Tracer Constructive Solid Geometry (CSG) Assignment 2

45 Constructive Solid Geometry (CSG) Given overlapping shapes A and B: Union Intersection Subtraction

46 How can we implement CSG? Points on A, Outside of B Points on B, Outside of A Points on B, Inside of A Points on A, Inside of B Union Intersection Subtraction

47 Collect all the intersections Union Intersection Subtraction

48 Implementing CSG 1. Test "inside" intersections: Find intersections with A, test if they are inside/outside B Find intersections with B, test if they are inside/outside A 2. Overlapping intervals: Find the intervals of "inside" along the ray for A and B Compute union/intersection/subtraction of the intervals

49 "Fredo's First CSG Raytraced Image"

50 Questions?

51 Today Motivations Transformations in Modeling World Space vs Object Space Adding Transformations to our Ray Tracer Constructive Solid Geometry (CSG) Assignment 2 Due Wednesday Sept 24 th, 11:59pm

52 Simple Shading Single Directional Light Source Diffuse Shading Ambient Light c pixel = c ambient *c object + dir light n * c light * c object n dir light c light = (1,1,1) c ambient = (0,0,0) c light = (0.5,0.5,0.5) c ambient = (0.5,0.5,0.5)

53 Adding Perspective Camera up direction horizontal angle

54 Triangle Meshes (.obj) vertices triangles v v v v v v v v f f f f f f f f f f f f 2 8 6

55 Acquiring Geometry 3D Scanning (Images removed due to copyright considerations.)

56 Next Week: Ray Tracing Surface reflectance