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Exam Announcements related discussion Assignment 3 is due today at 11:59pm Assignment 4 is up Raycasting-based assignment Ray Casting CMSC 435 Plagiarism and citing sources 1 2 Ray Tracing Renderman Internet Ray Tracing Competition http://www.irtc.org/ http://www.irtc.org/ftp/pub/stills/2004-12-31/lb_muse.jpg Overview of Today Ray Casting Basics Camera and Ray Generation Ray-Plane Intersection Ray-Sphere Intersection 3 4 3D Raytracing Chip Shown at CeBIT 2005 http://www.saarcor.de/ Ray Tracing at ACM s site Over 700 research papers related to ray tracing http://www.acm.org/tog/resources/bib/ 1 Ray casting basics For every pixel Construct a ray from the eye For every object in the scene Find intersection with the ray Keep if closest 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 5 6 A Note on Shading Surface/Scene Characteristics: surface normal direction to light viewpoint N L V Ray Tracing Secondary rays (shadows, reflection, refraction) Material Properties Diffuse (matte) Specular (shiny) reflection refraction Much more on Wednesday! Diffuse sphere Specular spheres 7 8 2 Ray Tracing 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 9 Ray Representation? Two vectors: Origin Direction (normalized is better) Durer s Ray Casting Machine Albrecht Durer, 16th century Parametric line P(t) = origin + t * direction P(t) direction origin 11 12 3 Durer s Ray Casting Machine Albrecht Durer, 16th century Overview of Today Ray Casting Basics Camera and Ray Generation Ray-Plane Intersection Ray-Sphere Intersection 13 14 Cameras For every pixel Construct a ray from the eye For every object in the scene Find intersection with ray Keep if closest 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 15 16 4 Oldest Illustration From. R. Gemma Frisius, 1545 Simplified Pinhole Camera Eye-image pyramid (frustum) Note that the distance/size of image are arbitrary 17 18 Camera Description? Eye point e (center) Orthobasis u, v, w (horizontal, up, -direction) Field of view angle Image rectangle height, width Other Weird Cameras E.g. fish eye, omnimax, panorama 19 Image Credits: www.sparse.org 20 5 Overview of Today Ray Casting Basics Camera and Ray Generation Ray-Plane Intersection Ray-Sphere Intersection 21 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 22 Recall: Ray Representation Parametric line P(t) = Ro + t * Rd Explicit representation 3D Plane Representation? Plane defined by Po = (x,y,z) n = (A,B,C) H(p) = d > 0 P normal Po H P' H(p) = d < 0 Implicit plane equation H(P) = Ax+By+Cz+D = 0 = n P + D = 0 P(t) direction Point-Plane distance? Rd origin Ro 23 If n is normalized, distance to plane, d = H(P) d is the signed distance! 24 6 Explicit vs. Implicit? Ray equation is explicit P(t) = Ro + t * Rd Parametric Generates points Hard to verify that a point is on the ray Ray-Plane Intersection Intersection means both are satisfied So, insert explicit equation of ray into implicit equation of plane & solve for t P(t) = Ro + t * Rd H(P) = n P + D = 0 n + (Ro t * Rd) + D = 0 t = -(D + n Ro) / n Rd 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 P(t) 25 26 Additional Housekeeping Verify that intersection is closer than previous P(t) < tcurrent Normal For shading diffuse: dot product between light and normal Verify that it is not out of range (behind eye) P(t) > tmin Normal is constant normal P(t) 27 28 7 Overview of Today Ray Casting Basics Camera and Ray Generation Ray-Plane Intersection Ray-Sphere Intersection 29 Sphere Representation? Implicit sphere equation Assume centered at origin (easy to translate) H(P) = P P - r2 = 0 Rd Ro 30 Ray-Sphere Intersection Insert explicit equation of ray into implicit equation of sphere & solve for t P(t) = Ro + t*Rd H(P) = P P r2 r2 =0 (Ro + tRd) (Ro + tRd) - = 0 Rd Rdt2 + 2Rd Rot + Ro Ro - r2 = 0 Ray-Sphere Intersection Quadratic: at2 + bt + c = 0 a = 1 (remember, ||Rd|| = 1) b = 2Rd Ro c = Ro Ro r2 with discriminant and solutions Rd Ro 31 32 8 Ray-Sphere Intersection 3 cases, depending on the sign of b2 4ac What do these cases correspond to? Which root (t+ or t-) should you choose? Closest positive! (usually t-) Ray-Sphere Intersection It's so easy that all ray-tracing images have spheres! 33 34 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? Geometric Ray-Sphere Intersection Is ray origin inside/outside/on sphere? (Ro Ro < r2 / Ro Ro > r2 / Ro Ro = r2) If origin on sphere, be careful about degeneracies O r Rd 35 Ro 36 9 Geometric Ray-Sphere Intersection Is ray origin inside/outside/on sphere? Find closest point to sphere center, tP = - Ro Rd If origin outside & tP < 0 no hit Geometric Ray-Sphere Intersection Is ray origin inside/outside/on sphere? Find closest point to sphere center, tP = - Ro Rd. Find squared distance, d2 = Ro Ro - tP2 If d2 > r2 no hit O r Rd tP 37 O d Ro tP 38 r Rd Ro Geometric Ray-Sphere Intersection Is ray origin inside/outside/on sphere? Find closest point to sphere center, tP = - Ro Rd. Find squared distance: d2 = Ro Ro - tP2 Find distance (t ) from closest point (tP) to correct intersection: t 2 = r2 - d2 If origin outside sphere t = tP - t If origin inside sphere t = tP + t O r d Rd Ro tP t t 39 Geometric vs. Algebraic Algebraic is simple & generic Geometric is more efficient Timely tests In particular for rays outside and pointing away 40 10 Sphere Normal Simply Q/||Q|| Q = P(t), intersection point (for spheres centered at origin) Precision What happens when Origin is on an object? Grazing rays? Problem with floating-point approximation O Rd Ro 41 42 Q normal The evil e In ray tracing, do NOT report intersection for rays starting at the surface (no false positive) Because secondary rays Requires epsilons reflection shadow refraction The evil e: a hint of nightmare Edges in triangle meshes Must report intersection (otherwise not watertight) No false negative 43 44 11 Ray Tracing References There are many references for Ray Tracing The Shirley book has a good introduction to it. Trivia In the newly released movie, Robots they used Bezier patches Ratchet, a later model robot than Rodney, was built with sliding sheets of steel using subdivision surfaces that were converted to Bezier patches for raytracing. Article in Computer Graphics World magazine 45 Image credits: CGW 46 12

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