1 Ray Casting

CLASS 1 Ray Casting 2

3 Overview of Today Ray Casting Basics Camera and Ray Generation Ray-Plane Intersection

4 Ray Casting For every pixel Construct a ray from the eye For every object in the scene Find intersection with the ray Keep if closest

5 Shading For every pixel Construct a ray from the eye For every object in the scene Find intersection with the ray Keep if closest Shade depending on light and normal vector

6 A Note on Shading Surface/Scene Characteristics: –surface normal –direction to light –viewpoint Material Properties –Diffuse (matte) –Specular (shiny) –… Much more soon! Diffuse sphereSpecular spheres N L V

7 Ray Tracing Secondary rays (shadows, reflection, refraction) In a couple of weeks reflection refraction

8 Ray Tracing

9 Ray Casting For every pixel Construct a ray from the eye For every object in the scene Find intersection with the ray Keep if closest Shade depending on light and normal vector N Finding the intersection and normal is the central part of ray casting

10 Ray Representation? Two vectors: –Origin –Direction (normalized is better) Parametric line –P(t) = origin + t * direction origin direction P(t)

11 Overview of Today Ray Casting Basics Camera and Ray Generation Ray-Plane Intersection

12 Cameras For every pixel Construct a ray from the eye For every object in the scene Find intersection with ray Keep if closest

13 Pinhole Camera Box with a tiny hole Inverted image Similar triangles Perfect image if hole infinitely small Pure geometric optics No depth of field issue

14 Simplified Pinhole Camera Eye-image pyramid (frustum) Note that the distance/size of image are arbitrary

15 Camera Description? Eye point e (center) Orthobasis u, v, w (horizontal, up, -direction) Field of view angle Image rectangle height, width

16 Perspective vs. Orthographic Parallel projection No foreshortening No vanishing point perspectiveorthographic

MIT EECS 6.837, Durand 17 Orthographic Camera Ray Generation? –Origin = center + (x-0.5)*size*horizontal + (y-0.5)*size*up –Direction is constant

18 Other Weird Cameras E.g. fish eye, omnimax, panorama

19 Overview of Today Ray Casting Basics Camera and Ray Generation Ray-Plane Intersection

MIT EECS 6.837, Durand 20 Ray Casting For every pixel Construct a ray from the eye For every object in the scene Find intersection with the ray Keep if closest First we will study ray-plane intersection

21 Recall: Ray Representation Parametric line P(t) = R o + t * R d Explicit representation RdRd RoRo origin direction P(t)

22 3D Plane Representation? Plane defined by –P o = (x,y,z) –n = (A,B,C) Implicit plane equation –H(P) = Ax+By+Cz+D = 0 = n·P + D = 0 Point-Plane distance? –If n is normalized, distance to plane, d = H(P) –d is the signed distance! H PoPo normal P P' H(p) = d < 0 H(p) = d > 0

23 Explicit vs. Implicit? Ray equation is explicit P(t) = R o + t * R d –Parametric –Generates points –Hard to verify that a point is on the ray Plane equation is implicit H(P) = n·P + D = 0 –Solution of an equation –Does not generate points –Verifies that a point is on the plane Exercise: Explicit plane and implicit ray

24 Ray-Plane Intersection Intersection means both are satisfied So, insert explicit equation of ray into implicit equation of plane & solve for t P(t) = R o + t * R d H(P) = n·P + D = 0 n·(R o + t * R d ) + D = 0 t = -(D + n·R o ) / n·R d P(t)

25 Additional Housekeeping Verify that intersection is closer than previous Verify that it is not out of range (behind eye) P(t) > t min P(t) < t current P(t)

26 Normal For shading –diffuse: dot product between light and normal Normal is constant normal

27 A moment of mathematical beauty Duality : points and planes are dual when you use homogeneous coordinates Point (x, y, z, 1) Plane (A, B, C, D) Plane equation  dot product You can map planes to points and points to planes in a dual space. Lots of cool equivalences –e.g. intersection of 3 planes define a point –  3 points define a plane!

CLASS 2 Ray Casting 28

29 Overview of Today Ray-Sphere Intersection Ray-Triangle Intersection Implementing CSG

30 Sphere Representation? Implicit sphere equation –Assume centered at origin (easy to translate) –H(P) = P·P - r 2 = 0 RdRd RoRo

31 Ray-Sphere Intersection Insert explicit equation of ray into implicit equation of sphere & solve for t P(t) = R o + t*R d H(P) = P·P - r 2 = 0 (R o + tR d ) · (R o + tR d ) - r 2 = 0 R d ·R d t 2 + 2R d ·R o t + R o ·R o - r 2 = 0 RdRd RoRo

32 Ray-Sphere Intersection Quadratic: at 2 + bt + c = 0 –a = 1 (remember, ||R d || = 1) –b = 2R d ·R o –c = R o ·R o – r 2 with discriminant and solutions

33 Ray-Sphere Intersection 3 cases, depending on the sign of b 2 – 4ac What do these cases correspond to? Which root (t+ or t-) should you choose? –Closest positive! (usually t-)

MIT EECS 6.837, Durand 34 Ray-Sphere Intersection It's so easy that all ray-tracing images have spheres!

MIT EECS 6.837, Durand 35 Geometric Ray-Sphere Intersection Shortcut / easy reject What geometric information is important? –Ray origin inside/outside sphere? –Closest point to ray from sphere origin? –Ray direction: pointing away from sphere?

MIT EECS 6.837, Durand 36 Geometric Ray-Sphere Intersection Is ray origin inside/outside/on sphere? –(R o ·R o r 2 / R o ·R o = r 2 ) –If origin on sphere, be careful about degeneracies… RoRo r O RdRd

37 Geometric Ray-Sphere Intersection Is ray origin inside/outside/on sphere? Find closest point to sphere center, t P = - R o · R d –If origin outside & t P < 0 → no hit RoRo O RdRd tPtP r

38 Geometric Ray-Sphere Intersection Is ray origin inside/outside/on sphere? Find closest point to sphere center, t P = - R o · R d. Find squared distance, d 2 = R o ·R o - t P 2 –If d 2 > r 2 → no hit RoRo O RdRd d tPtP r

MIT EECS 6.837, Durand 39 Geometric Ray-Sphere Intersection RoRo O RdRd d t t’t’ Is ray origin inside/outside/on sphere? Find closest point to sphere center, t P = - R o · R d. Find squared distance: d 2 = R o ·R o - t P 2 Find distance (t ’ ) from closest point (t P ) to correct intersection: t ’ 2 = r 2 - d 2 –If origin outside sphere → t = t P - t ’ –If origin inside sphere → t = t P + t ’ tPtP r

40 Geometric vs. Algebraic Algebraic is simple & generic Geometric is more efficient –Timely tests –In particular for rays outside and pointing away

41 Sphere Normal Simply Q/||Q|| –Q = P(t), intersection point –(for spheres centered at origin) Q normal RoRo O RdRd

42 Questions?

43 Overview of Today Ray-Sphere Intersection Ray-Triangle Intersection Implementing CSG

Ray-Triangle Intersection First, intersect ray with plane Then, check if point is inside triangle P P0P0 V

Ray-Triangle Intersection Check if point is inside triangle parametrically P P0P0 Compute  P =  (T 2 -T 1 ) +  (T 3 -T 1 ) Check if point inside triangle. 0   1 and 0   1  V   T1T1 T2T2 T3T3

46 Ray-Triangle Intersection Use general ray-polygon Or try to be smarter –Use barycentric coordinates RoRo RdRd c a b P

47 Barycentric Definition of a Plane P(  ) =  a +  b +  c with  =1 Is it explicit or implicit? [Möbius, 1827] c a b P P is the barycenter: the single point upon which the plane would balance if weights of size  are placed on points a, b, & c.

48 Barycentric Definition of a Triangle P(  ) =  a +  b +  c with  =1 AND 0 <  < 1 & 0 <  < 1 & 0 <  < 1 c a b P

49 How Do We Compute  Ratio of opposite sub-triangle area to total area –  = A a /A  = A b /A  = A c /A Use signed areas for points outside the triangle  c a b P AaAa A

50 Intuition Behind Area Formula P is barycenter of a and Q A a is the interpolation coefficient on aQ All points on lines parallel to bc have the same   All such triangles have same height/area) Q c a b P AaAa A

51 Simplify Since  =1, we can write  P(  ) =  a +  b +  c P(  ) =  a +  b +  c = a +  b-a) +  c-a) c a b P Non-orthogonal coordinate system of the plane rewrite

52 Intersection with Barycentric Triangle Set ray equation equal to barycentric equation P(t) = P(  ) R o + t * R d = a +  b-a) +  c-a) Intersection if  0 (and t > t min … ) RoRo RdRd c a b P

53 Intersection with Barycentric Triangle R o + t * R d = a +  b-a) +  c-a) R ox + tR dx = a x +  b x -a x ) +  c x -a x ) R oy + tR dy = a y +  b y -a y ) +  c y -a y ) R oz + tR dz = a z +  b z -a z ) +  c z -a z ) Regroup & write in matrix form: 3 equations, 3 unknowns      ozz oyy oxx dzzzzz dyyyyy dxxxxx Ra Ra Ra Rcaba Rcaba Rcaba                                t  

54 Cramer’s Rule A RRaba RRaba RRaba dzozzzz dyoyyyy dxoxxxx     A Racaba Racaba Racaba t ozzzzzz oyyyyyy oxxxxxx     A RcaRa RcaRa RcaRa dzzzozz dyyyoyy dxxxoxx     | | denotes the determinant Can be copied mechanically into code Used to solve for one variable at a time in system of equations

55 Advantages of Barycentric Intersection Efficient Stores no plane equation Get the barycentric coordinates for free –Useful for interpolation, texture mapping RoRo RdRd c a b P

56 Overview of Today Ray-Sphere Intersection Ray-Triangle Intersection Implementing CSG

MIT EECS 6.837, Durand 57 For example:

MIT EECS 6.837, Durand 58 How can we implement CSG? Union Intersection Subtraction Points on A, Outside of B Points on B, Outside of A Points on B, Inside of A Points on A, Inside of B

MIT EECS 6.837, Durand 59 Collect all the intersections Union Intersection Subtraction

MIT EECS 6.837, Durand 60 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

61 Precision What happens when –Origin is on an object? –Grazing rays? Problem with floating-point approximation

62 The evil  In ray tracing, do NOT report intersection for rays starting at the surface (no false positive) –Because secondary rays –Requires epsilons reflection refraction shadow

63 The evil  a hint of nightmare Edges in triangle meshes –Must report intersection (otherwise not watertight) –No false negative