1. Physics fundamentals of 3D rendering Szirmay-Kalos László Science is either physics or stamp collecting. Rutherford

2. 3D rendering Tone mapping Real worldcolor pixelVirtual world

3. Radiance = Power radiated by a unit projected area into a unit solid angle [W/sr/m 2 ] L(x, V ) =   A cos    A pixel  A surface Wanted:  pixel =  surface  pixel  surface  surface Geometry:  pixel cos  pixel  pixel =  surface cos  surface  surface Radiance: L(x, V)

4. L(x, V ) = L in (x, L)  cos  ’  f r (L,x,V) Radiance = Irradiance  BRDF x ’’ ’’ L in f r (L,x,V) = f r (V,x,L) Helmholtz law: V L f r (L,x,V) = L (x,V)I(x,L)L (x,V)I(x,L) def Light-matter interaction

5. Wavelenths are independent Relativistic mass is small: m=E/c 2 = hf/c 2 Photon energy (frequency) does not change in elastic scattering e-e- e-e-

6. F =F = cos  ’ - (n+k j ) cos  cos  ’+ (n+k j ) cos  2 cos  - (n+k j ) cos  ’ cos  + (n+k j ) cos  ’  ’’ n =n = sin  ’ sin  2 ’’ 1212 1212 + F  F0 + (1-F0)  (1-cos  ’) 5, F0 = (n -1) 2 + k 2 (n+1) 2 + k 2 L in  F L in  (1-F) L in „Smooth” = surface visible in a single pixel is a plane Smooth surfaces: Fresnel equation

7. ’’ ’’ gold silver F (,  ’) Fresnel function

8. v r = v + 2 N cos  vrvr v v + N cos   N N cos  cos  = - ( v ·N) x vec3 reflect(vec3 inDir, vec3 normal) { return inDir - normal * dot(normal, inDir) * 2.0f; }; Reflection direction

9. v t = v  n + N (cos  n -  1- (1 - cos 2  )/ n 2 ) vtvt v  v + Ncos  N NN N  sin   -Ncos  v  Ncos  sin  N =N = n =n = sin  Snellius-Descartes v t = N  sin  Ncos  v t = v  n + N (cos  n  cos  ) cos   1- sin 2   1- sin 2  / n 2 Refraction direction

10. v t = v  n + N (cos  n -  1- (1 - cos 2  )/ n 2 ) class SmoothMaterial { vec3 F0;// F0 float n;// n public: vec3 reflect(vec3 inDir, vec3 normal) { return inDir - normal * dot(normal, inDir) * 2.0f; } vec3 refract(vec3 inDir, vec3& normal) { // may flip the normal float ior = n; float cosa = -dot(normal, inDir); if (cosa < 0) { cosa = -cosa; normal = -normal; ior = 1/n; } float disc = 1 - (1 - cosa * cosa)/ior/ior; if (disc < 0) return reflect(inDir, normal); return inDir/ior + normal * (cosa/ior - sqrt(disc)); } vec3 Fresnel(vec3 inDir, vec3 normal) { float cosa = fabs(dot(normal, inDir)); return F0 + (float3(1, 1, 1) – F0) * pow(1-cosa, 5); } }; SmoothMaterial class

11. ’’ Cook-Torrance He-Torrance Surface visible in a pixel Black-box model Rough surfaces

12. l Radiance = Irradiance  BRDF is independent of the viewing direction l BRDF is independent of the viewing direction l Helmholtz: BRDF is independent of the illumination direction l BRDF is constant: l Diffuse reflection = very rough surfaces –Multiple light-matter interaction –colorful ! f r (L,x,V) = k d (x, ) ’’ L N V x Diffuse reflection

13. BRDF is constant, BUT the reflected radiance does depend on the illumination direction L ref = L in k d cos +  ’ ’’ L N cos  ’  N·L Lambert’s law

14. ’’ = diffuse + R V ’’  Halfway vector V L H N H = (L+V)/|L+V| cos  ·  L ref = L in k d cos +  ’+ L in k s (cos +  s k s ( cos +  s Specular (glossy) reflection (Phong-Blinn model)

15. Diffuse+glossy materials 5 10 20 50 shininess = diffuse glossy diffuse + glossy Single light-matter Interaction Wavelength independent for non- metals Multiple light-matter interaction ”own color”

16. RoughMaterial class class RoughMaterial { vec3 kd, ks; float shininess; public: vec3 shade( vec3 normal, vec3 viewDir, vec3 lightDir, vec3 inRad) { vec3 reflRad(0, 0, 0); float cosTheta = dot(normal, lightDir); if(cosTheta < 0) return reflRad; reflRad = inRad * kd * cosTheta; vec3 halfway = (viewDir + lightDir).normalize(); float cosDelta = dot(normal, halfway); if(cosDelta < 0) return reflRad; return reflRad + inRad * ks * pow(cosDelta,shininess); } };

17. l Geometry + emitter radiance: l Abstract light sources: –Directional: rays are parallel, intensity is constant –Positional: rays connect the light source to the shaded point, intensity decrases with the square of the distance DirectionalPositional d L e (x,V, ) V x Light sources

18. Rendering pixelVirtual world f r  LeLe l Visible surface determination l Radiance calculation l R, G, B conversion L L  L L  R, G, B

19. pixel Local illumination Recursive ray tracing Global illumination Rendering models

20. color: r, g, b 400700500600 r(r( g(g( b(b( r, g, b 645526444 nm 1 2 Color perception: Monochromatic light

21. r =   r  d g =   g  d b =   b  d  Color perception: Polychromatic light

22. model Rendering on many wavelengths R,G,B image Visible spectrum R, G, B Color matching model Rendering on 3 wavelengths R,G,B image R, G, B Spectral versus RGB rendering