1. GR2Advanced Computer GraphicsAGR Lecture 12 Solid Textures Bump Mapping Environment Mapping

2. Marble Texture

3. Y object space V texture space U X W Z Solid Texture • A difficulty with 2D textures is the mapping from the object surface to the texture image • ie constructing fu(x,y,z) and fv(x,y,z) • This is avoided in 3D, or solid, texturing • texture now occupies a volume • can imagine object being carved out of the texture volume Mapping functions trivial: u = x; v = y; w = z

4. Defining the Texture • The texture volume itself is usually defined procedurally • ie as a function that can be evaluated, such as: texture (u, v, w) = sin (u) sin (v) sin (w) • this is because of the vast amount of storage required if it were defined by data values

5. V texture space U W Example: Wood Texture • Wood grain texture can be modelled by a set of concentric cylinders • cylinders coloured dark, gaps between adjacent cylinders coloured light radius r = sqrt(u*u + w*w) if radius r = r1, r2, r3, then texture (u,v,w) = dark else texture (u,v,w) = light looking down: cross section view

6. Example: Wood Texture • It is a bit more interesting to apply a sinusoidal perturbation • radius:= radius + 2 * sin( 20*) , with 0<<2 • .. and a twist along the axis of the cylinder • radius:= radius + 2 * sin( 20* + v/150 ) • This gives a realistic wood texture effect

7. Wood Texture

8. How to do Marble? • First create noise function (in 1D): • noise [i] = random numbers on lattice of points • Next create turbulence: • turbulence (x) = noise(x) + 0.5*noise(2x) + 0.25*noise(4x) + … • Marble created by: • basic pattern: • marble (x) = marble_colour (sin (x) ) • with turbulence: • marble (x) = marble_colour (sin (x + turbulence (x) ) )

9. Marble Texture

10. Bump Mapping • This is another texturing technique • Aims to simulate a dimpled or wrinkled surface • for example, surface of an orange • Like Gouraud and Phong shading, it is a trick • surface stays the same • but the true normal is perturbed, or jittered, to give the illusion of surface ‘bumps’

11. Bump Mapping

12. How Does It Work? • Looking at it in 1D: original surface P(u) bump map b(u) add b(u) to P(u) in surface normal direction, N(u) new surface normal N’(u) for reflection model

13. How It Works - The Maths! • Any 3D surface can be described in terms of 2 parameters • eg cylinder of fixed radius r is defined by parameters (s,t) x=rcos(s); y=rsin(s); z=t • Thus a point P on surface can be written P(s,t) where s,t are the parameters • The vectors: Ps = dP(s,t)/ds and Pt = dP(s,t)/dt are tangential to the surface at (s,t)

14. How it Works - The Maths • Thus the normal at (s,t) is: N = Ps x Pt • Now add a bump map to surface in direction of N: P’(s,t) = P(s,t) + b(s,t)N • To get the new normal we need to calculate P’s and P’t P’s = Ps + bsN + bNs approx P’s = Ps + bsN - because b small • P’t similar • P’t = Pt + btN

15. How it Works - The Maths • Thus the perturbed surface normal is: N’ = P’s x P’t or N’ = Ps x Pt + bt(Ps x N) + bs(N x Pt) + bsbt(N x N) • But since • Ps x Pt = N and N x N = 0, this simplifies to: N’ = N + D • where D = bt(Ps x N) + bs(N x Pt) = bs(N x Pt) - bt(N x Ps ) = A - B

16. Worked Example for a Cylinder • P has co-ordinates: • Thus: • and then x (s,t) = r cos (s) y (s,t) = r sin (s) z (s,t) = t Ps : xs (s,t) = -r sin (s) ys (s,t) = r cos (s) zs (s,t) = 0 Pt : xt (s,t) = 0 yt (s,t) = 0 zt (s,t) = 1 N = Ps x Pt : Nx = r cos (s) Ny = r sin (s) Nz = 0

17. Worked Example for a Cylinder • Then: D = bt(Ps x N) + bs(N x Pt) becomes: • and perturbed normal N’ = N + D is: D : bt *0 + bs*r sin (s) = bs*r sin (s) bt *0 - bs*r cos (s) = - bs*r cos (s) bt*(-r2) + bs*0 = - bt*(r2) N’ : r cos (s) + bs*r sin (s) r sin (s) - bs*r cos (s) -bt*r2

18. Bump MappingA Bump Map

19. Bump MappingResulting Image

20. Bump Mapping - Another Example

21. Bump MappingAnother Example

22. Bump MappingProcedurally Defined Bump Map

23. Environment Mapping • This is another famous piece of trickery in computer graphics • Look at a highly reflective surface • what do you see? • does the Phong reflection model predict this? • Phong reflection is a local illumination model • does not convey inter-object reflection • global illumination methods such as ray tracing and radiosity provide this • .. but can we cheat?

24. Environment Mapping - Recipe • Place a large cube around the scene with a camera at the centre • Project six camera views onto faces of cube - known as an environment map projection of scene on face of cube - environment map camera

25. Environment Mapping - Rendering • When rendering a shiny object, calculate the reflected viewing direction (called R earlier) • This points to a colour on the surrounding cube which we can use as a texture when rendering environment map eye point

26. Environment Mapping - Limitations • Obviously this gives far from perfect results - but it is much quicker than the true global illumination methods (ray tracing and radiosity) • It can be improved by multiple environment maps (why?) - one per key object • Also known as reflection mapping • Can use sphere rather than cube

27. Environment Mapping

28. Environment Mapping

29. Jim Blinn • Both bump mapping and environment mapping concepts are due to Jim Blinn • Pioneer figure in computer graphics www.research.microsoft.com/~blinn www.siggraph.org/s98/conference/ keynote/slides.html