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

2. Marble Texture

3. 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

4. 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

5. 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

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) *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. 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))

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

12. 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’

13. Bump Mapping

14. 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

15. 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)

16. 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

17. 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

18. 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

19. 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

20. Bump Mapping A Bump Map

21. Bump Mapping Resulting Image

22. Bump Mapping - Another Example

23. Bump Mapping Another Example

24. Bump Mapping Procedurally Defined Bump Map

25. 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?

26. 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

27. 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

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

29. 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

30. Jim Blinn
Both bump mapping and environment mapping concepts are due to Jim Blinn
Pioneer figure in computer graphics
keynote/slides.html

31. 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