SI31 Advanced Computer Graphics AGR Lecture 10 Solid Textures Bump Mapping Environment Mapping

Marble Texture

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 X Y Z object space U V W texture space Mapping functions trivial: u = x; v = y; w = z

texture (u, v, w) = sin (u) sin (v) sin (w) 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

Example: Wood Texture Wood grain texture can be modelled by a set of concentric cylinders cylinders coloured dark, gaps between adjacent cylinders coloured light U V W texture space 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

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

Wood Texture

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) ) )

Marble Texture

Using Turbulence for Flame Simulation Flame in 2D region [-b,b] x [0,h] can be modelled as: flame(x,y) = (1-y/h) * flame_col(abs(x/b)) flame_col has max intensity at 0, min at 1 Adding turbulence factor to flame_col gives more realistic effect: flame(x,y) = (1-y/h) * flame_col(abs(x/b)+turb(x))

Animating the Turbulence The noise function, and hence the turbulence function, can be made time-dependent

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’

Bump Mapping

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

x=rcos(s); y=rsin(s); z=t Ps = dP(s,t)/ds and Pt = dP(s,t)/dt 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)

approx P’s = Ps + bsN - because b small 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

N’ = Ps x Pt + bt(Ps x N) + bs(N x Pt) + bsbt(N x N) 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

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

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

Bump Mapping A Bump Map

Bump Mapping Resulting Image

Bump Mapping - Another Example

Bump Mapping Another Example

Bump Mapping Procedurally Defined Bump Map

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?

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

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

Environment Mapping Example Environment Mapped Teapot The six views from the teapot

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

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

Acknowledgements Thanks again to Alan Watt for many of the images Flame simulation movie from Josef Pelikan, Charles University Prague Environment mapping examples from Mizutani and Reindel, Japan