Ray Tracing. Last Time? Reading for Today. Reading for Today

Transcription

1 Last Time? Ray Tracing Keyframing Procedural Animation Physically-Based Animation Forward and Inverse Kinematics Motion Capture Two solutions Reading for Today Artist-Directed Dynamics for 2D Animation, Bai, Kaufman, Liu, & Popović, SIGGRAPH 2016 Reading for Today Articulated Swimming Creatures Jie Tan, Yuting Gu, Greg Turk, and C. Karen Liu, SIGGRAPH

2 Reading for Today Today Ray Casting Flexible Muscle-Based Locomotion for Bipedal Creatures, Geijtenbeek, van de Panne, van der Stappen, SIGGRAPH Asia 2013 Ray-Plane Intersection Ray-Sphere Intersection Point in Polygon Ray Tracing Recursive Ray Tracing Distribution Ray Tracing Durer s Ray Casting Machine Ray Casting Albrecht Durer, 16th century For every pixel Finding the intersection and normal is the central part of ray casting For every object in the scene Construct a ray from the eye Find intersection with the ray Keep if closest Shade depending on light and normal vector

3 A Note on Local Shading Ray Representation? Surface/Scene Characteristics: Two vectors: N surface normal direction to light viewpoint L Origin Direction (normalized is better) V Parametric line (explicit representation) Material Properties P(t) = origin + t * direction color/texture diffuse (matte) specular (shiny) P(t) direction More later! Diffuse sphere Specular spheres 3D Plane Representation? Plane defined by Po = (x,y,z) n = (A,B,C) H(p) = d > 0 P 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! origin Explicit vs. Implicit? Ray equation is explicit normal Po H P' H(p) = d < 0 P(t) = Ro + t * Rd Parametric Generates points Harder 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

4 Ray-Plane Intersection Additional Housekeeping Intersection means both are satisfied So, insert explicit equation of ray into implicit equation of plane & solve for t Verify that intersection is closer than previous P(t) < tcurrent Verify that it is not out of range (behind eye) P(t) = Ro + t * Rd P(t) > tmin H(P) = n P + D = 0 n (Ro + t * Rd) + D = 0 t = -(D + n Ro) / n Rd P(t) P(t) Normal at Surface Intersection Ray-Triangle Intersection Needed for shading Intersect with the plane Then use barycentric coordinates: diffuse: dot product between light and normal Normal of a plane is constant! normal P(α, β, γ) = αa + βb + γc with α + β + γ =1 If 0 < α < 1 & 0 < β < 1 & 0 < γ < 1 then the point is inside the triangle! c P Rd a b Ro

5 How Do We Compute α, β, γ? Using Cramer s Rule Ratio of opposite sub-triangle area to total area Used to solve for one variable at a time in system of equations α = Aa/A β = Ab/A γ = Ac/A Use signed areas for points outside the triangle c c Aa Aa A A P P a α>0 a b α<0 denotes the determinant Can be copied mechanically into code b But how do I know if the point is outside the triangle? That s what I was trying to determine! Sphere Representation? Ray-Sphere Intersection Implicit sphere equation Insert explicit equation of ray into implicit equation of sphere & solve for t Assume centered at origin (easy to translate) H(P) = P P - r2 = 0 P(t) = Ro + t*rd H(P) = P P - r2 = 0 (Ro + trd) (Ro + trd) - r2 = 0 Rd Rdt2 + 2Rd Rot + Ro Ro - r2 = 0 Rd Ro Rd Ro

6 Ray-Sphere Intersection Questions? Quadratic: at2 + bt + c = 0 a = 1 (remember, Rd = 1) b = 2Rd Ro c = Ro Ro r2 with discriminant and solutions What does it mean if there are no solutions, 1 solution, or 2 solutions? depth Reading for Next Friday Today "An improved illumination model for shaded display" Turner Whitted, Ray Casting Ray Tracing normals Shadows Reflection Refraction Recursive Ray Tracing Distribution Ray Tracing local shading

7 How Can We Add Shadows? Mirror Reflection Find the point to be shaded For every light, Construct ray from point to light For every object find intersection of ray with object If no objects between point and light Add contribution from light Cast ray symmetric with respect to the normal Multiply by reflection coefficient (color) Reflection Transparency Reflection angle = view angle R = V 2 (V N) N Cast ray in refracted direction Multiply by transparency coefficient (color) N V R θv θr V N N V N N V

8 Qualitative Refraction Refraction I M sin Өi ηi ηt I N M = (N cos Өi I) / sin Өi Өi N cos Өi T = N cos ӨT + M sin ӨT M = N cos ӨT + (N cos Өi I) sin ӨT / sin Өi = N cos ӨT + (N cos Өi I) ηr ӨT -N = N cos Өi M sin Өi Note: The math works the same tracing the ray either forwards or backwards, but it s really easy to get confused and have get a sign error in the direction. T = [ ηr cos Өi cos ӨT ] N ηr I = [ ηr cos Өi (1 sin2 ӨT )] N ηr I = [ ηr cos Өi (1 ηr2 sin2 Өi )] N ηr I = [ ηr cos Өi (1 ηr2 (1 cos2 Өi ) )] N ηr I = [ ηr (N I) (1 ηr2 (1 (N I)2 ))] N ηr I Total internal reflection when the square root is imaginary Don t forget to normalize! From Color and Light in Nature by Lynch and Livingston Refraction & the Sidedness of Objects Total Internal Reflection Make sure you know whether you re entering or leaving the transmissive material: N I I ηi = material index ηi=1 ηt = material index ηt= 1 T T N What about intersecting transparent objects? From Color and Light in Nature by Lynch and Livingston

9 Questions? Today Ray Tracing trace ray Intersect all objects color = ambient term For every light cast shadow ray Stopping criteria: color += local shading term Recursion depth If mirror color += colorrefl * Stop after a number trace reflected ray of bounces If transparent Ray contribution color += colortrans * Stop if reflected / trace transmitted ray transmitted contribution Does it ever end? becomes too small Ray Casting Ray Tracing Recursive Ray Tracing Distribution Ray Tracing The Ray Tree Eye T3 R2 N2 T1 R1 L2 R3 L1 N3 N1 L1 R1 L3 T1 L2 Eye Ni surface normal Ri reflected ray Li shadow ray Ti transmitted (refracted) ray R2 L3 R3 Complexity? T3

10 Ray Debugging Other Reading for Next Friday Visualize the ray tree for single image pixel "Distributed Ray Tracing", Cook, Porter, & Carpenter, SIGGRAPH incoming reflected ray shadow ray transmitted (refracted) ray Today Shadows one shadow ray per intersection per point light source Ray Casting Ray Tracing Recursive Ray Tracing Distribution Ray Tracing no shadow rays Soft shadows Antialiasing (getting rid of jaggies) Glossy reflection Motion blur Depth of field (focus) one shadow ray

11 Shadows & Light Sources Soft Shadows multiple shadow rays to sample area light source one shadow ray clear bulb frosted bulb lots of shadow rays Antialiasing Supersampling multiple rays per pixel jaggies w/ antialiasing Reflection one reflection ray per intersection point light area light θ θ perfect mirror

12 Glossy Reflection Motion Blur multiple reflection rays Sample objects temporally Justin Legakis θ θ polished surface Rob Cook Depth of Field Ray Tracing Algorithm Analysis multiple rays per pixel film focal length Justin Legakis Ray casting Lots of primitives Recursive Distributed Ray Tracing Effects Soft shadows Anti-aliasing Glossy reflection Motion blur Depth of field cost height * width * num primitives * intersection cost * size of recursive ray tree * num shadow rays * num supersamples * num glossy rays * num temporal samples * num focal samples *... can we reduce this? these can serve double duty