Simple Texture Mapping and Simple RayTracing Texture Map

Simple Texture Mapping and Simple Ray-Tracing

Texture Map • Each pixel in a texture map is called a Texel • Each Texel is associated with a (s, t) 2 D texture coordinate • The range of s, t is [0. 0, 1. 0] due to normalization

Simple Mapping Texture Coordinates Projection Coordinates

Simple Mapping Texture Coordinates Projection Coordinates

Simple Mapping

Basic File Structure • Add texture coordinates v 1 x 1 y 1 z 1 u 1 v 2 x 2 y 2 z 2 u 2 v 3 x 3 y 3 z 3 u 3 v 4 x 4 y 4 z 4 u 4 v 4 f 1 1 3 2 f 1 1 4 3 f 1 1 2 4 f 1 2 3 4

Texture Filtering Get an interpolated value from near four neighbors

Bilinear Interpolation

Perspective Correction Linear Interpolation of Texture Coordinate Corrected Interpolation

Perspective Correction Let’s assume that the viewport is located 1 unit away from the projection center

Perspective Correction Compare linear interpolation in screen space

Perspective Correction

Perspective Correction • We can interpolate arbitrary parameters of a surface on the screen coordinates in the linear fashion – Use s instead t ! – (ex) Gouraud shading, Phong normal. . etc

Perspective Correction • Further reading – http: //www. whisqu. se/per/docs/graphics 16. htm – http: //p 205. ezboard. com/fyabasicprogrammingfrm 20. show. Me ssage? topic. ID=20. topic • (Perspective correction with Z-buffering) – http: //easyweb. easynet. co. uk/~mrmeanie/tmap. htm

Ray Tracing Concept eye view plane Binary Tree

Ray Tracing Procedure For each ray r from eye to pixel, color the pixel the value returned by the function: ray_cast(r) { s nearest_intersected_surface(r); p point_of_intersection(r, s); u reflect(r, s, p); v refract(r, s, p); c phong(p, s, r) + s. kreflect ray_cast(u) + s. krefract ray_cast(v); return(c); }

Ray Tracing Procedure • s nearest_intersected_surface(r); – Use geometric searching to find the nearest surface s intersected by the ray r • p point_of_intersection(r, s); – Compute p, the point of intersection of ray r with surface s • u reflect(r, s, p); v refract(r, s, p); – Compute the reflected ray u and the refracted ray v using Snell’s Laws

Ray Tracing Procedure • phong(p, s, r) – Evaluate the Phong reflection model for the ray r at point p on surface s, taking shadowing into account • s. kreflect ray_cast(u) – Multiply the contribution from the reflected ray u by the specular-reflection coefficient kreflect for surface s • s. krefract ray_cast(v) – Multiply the contribution from the refracted ray v by the specular-refraction coefficient krefract for surface s

Reflection and Refraction • Reflected and refracted rays are computed using Snell’s Law reflected ray surface normal incident ray surface refracted ray

Shadowing • If any object intersects the shadow ray between the surface and the point light source, the surface is in shadow w. r. t. that light source blocked ! u L N r

Surface Intersection • Ray equation ath p y ra

Surface Intersection • For sphere, If the discriminant is negative, the ray does not intersect the sphere. Otherwise, intersection coordinates are obtained using smaller s.