The photorealism quest An introduction to ray tracing The basic idea Cast a ray from the viewer s eye through each pixel on the screen to see where it hits some object object view plane eye of viewer Apply illumination principles from physics to determine the intensity of the light that is reflected from that spot toward the eye Example a spherical object Mathematics problem How can we find the spot where a ray hits a sphere We apply our knowledge of vector algebra Describe sphere s surface by an equation Describe ray s trajectory by an equation Then solve this system of two equations There could be 0 1 or 2 distinct solutions Describing the sphere A sphere consists of points in space which lie at a fixed distance from a given center Let r denote this fixed distance radius Let c cx cy cz be the sphere s center If w wx wy wz be any point in space then distance w c is written w c Formula w c sqrt w c w c Sphere is a set w R3 w c w c r Describing a ray A ray is a geometrical half line in space It has a beginning point b bx by bz It has a direction vector d dx dy dz It can be described with a parameter t 0 If w wx wy wz is any point in space then w will be on the half line just in case w b td for some choice of the parameter t 0 Computing the hit time To find WHERE a ray hits a sphere we think of w as a point traveling in space and we ask WHEN will w hit the shere We can substitute the formula for a point on the half line into our formula for points that belong to the sphere s surface getting an equation that has t as its only variable It s easy to solve such an equation for t to find when w hits the sphere s surface The algebra details Sphere w c r Half line w b td t 0 Substitution b td c r Replacement Let q c b Simplification td q r Square td q td q r2 Expand t2 d d 2td q q q r2 Transpose t2 d d 2td q q q r2 0 Apply Quadratic Formula To solve t2 d d 2td q q q r2 0 Format At2 Bt C 0 Formula t B 2A sqrt B2 4AC 2A Notice there could be 0 1 or 2 hit times OK to use a simplifying assumption d d 1 Equation becomes t2 2td q q q r2 0 Application We want the earliest hit time t d q sqrt d q 2 q q r2 Interpretations half line w2 w1 b c A half line that begins inside the sphere will only hit the spgere s surface once half line Second example A planar surface Mathematical problem How can we find the spot where a ray hits a plane Again we can employ vector algebra Any plane is determined by a two entities a point chosen arbitrarily that lies in it and a vector that is perpendicular normal to it Let p be the point and n be the vector Plane is the set w R3 w p n 0 When does ray hit plane Plane w p n 0 Half line w b td t 0 Substitution b td p n 0 Replacement Let q p b Simplification td q n 0 Expand td n q n 0 Solution t q n d n Interpretations half line n p d w If a half line begins from a point outside a given plane then it can only hit that plane once and it might possibly not even hit that plane at all half line Achromatic light Achromatic light brightness but no color Light can come from point sources and light can come from ambient sources Incident light shining on the surface of an object can react in three discernable ways By being absorbed By being reflected By being transmitted into the interior Illumination Besides knowing where a ray of light will hit a surface we will also need to know whether that ray is reflected or absorbed If the ray of light is reflected by a surface does it travel mainly in one direction Or does it scatter off in several directions Various surface materials react differently Some terminology Scattering of light diffuse reradiation color may be affected by the surface Reflection of light specular reflection mirror like shininess color isn t affected Geometric ingredients Normal vector to the surface at a point Direction vector from point to viewer s eye Direction vector from point to light source
View Full Document
Unlocking...