1. So Far We have assumed that we know: The point The surface normal
The viewer location (or direction)
The light location (or direction)
But commonly, normal vectors are only given at the vertices and it is expensive to compute lighting for every point
Objects rendered using Phong reflection model and Gouraud or Phong interpolated shading often appear rather ‘plastic’ and ‘floating in air’
Breaking the scene into smaller and smaller polygonal objects increases the detail BUT it is very hard to model and very time-consuming to render

2. Texture Mapping
Texture effects can be added to give more realistic looking surface appearance
Texture mapping associates the color of a point with the color in a texture image - a 2D image is ‘painted’ onto the object

3. + = Parameterization geometry image texture map
Q: How do we decide where on the geometry each color from the image should go?

4. Option: Varieties of projections
[Paul Bourke]

5. Option: unfold the surface
[Piponi2000]

6. How to map object to texture?
To each vertex (x,y,z in object coordinates), must associate 2D texture coordinates (s,t)
So texture fits “nicely” over object

7. Idea: Use Map Shape Map shapes correspond to various projections
Planar, Cylindrical, Spherical
First, map (square) texture to basic map shape
Then, map basic map shape to object
Or vice versa: Object to map shape, map shape to square
Usually, this is straightforward
Maps from square to cylinder, plane, sphere well defined
Maps from object to these are simply spherical, cylindrical, cartesian coordinate systems

8. Planar mapping Like projections, drop z coord (s,t) = (x,y)
Problems: what happens near z = 0?

9. Cylindrical Mapping Cylinder: r, θ, z with (s,t) = (θ/(2π),z)
Note seams when wrapping around (θ = 0 or 2π)

10. Spherical Mapping Convert to spherical coordinates: use latitude/long.
Singularities at north and south poles

11. Cube Mapping

12. Cube Mapping

13. Texture Mapping Main Issues How do we map between 3D
How does the texture modify the shading of pixels?
How do we sample the texture for a fragment?

14. Texture Mapping v u Interpolation Texture Mapping (painting)
We need to define a mapping between the 3D world space (x,y,z) and 2D image space (s,t) [0,1]2
Texture coordinates defined at vertices serve this purpose
Specify (s,t) at each vertex: u v V2 V1 V0

15. Texture Mapping Interpolate in the interior: v u
Mapping the object on the left to the texture on the right

16. Texture Mapping v (50, 50) Interpolate in the interior: (100, 15)
(3, 3)
V1(20, 3)
(100, 15)
V0(0,0)
(0, 0)
Consider the y value of 3 in the interpolation across the triangle.
The rendering process determines that the range of object coordinates are from (3, 3) to (20, 3).
The first co-ordinate comes from the interpolation along the left edge of the triangle at a parametric value t = 0.3.

17. Texture Mapping v (50, 50) Interpolate in the interior: (15, 15)
(3, 3)
(15, 15)
V1(20, 3)
(100, 15)
V0(0,0)
(0, 0)
The texture location for this left edge location is (15, 15). Why?
Object locations are interpolated from (3, 3) to (20, 3)
Texture locations are interpolated from (15, 15) to (100, 15)

18. Texture Mapping v (50, 50) Interpolate in the interior: (15, 15)
(3, 3)
(15, 15)
V1(20, 3)
(100, 15)
V0(0,0)
(0, 0)
The texture gives a value that will influence the shading process. Specifically, the value from the texture replaces the diffuse constant in the calculation of the diffusion component in the illumination equation. By doing so, the texture values change the object appearance, but shape cues that come from the specular highlights will stay the same.

19. Texture Mapping v (50, 50) Interpolate in the interior: y (u, v)
(x, y)
(3, 3)
(15, 15)
V1(20, 3)
(100, 15)
V0(0,0)
(0, 0) x

20. Texture Mapping v (50, 50) Interpolate in the interior: y (u, v)
(x, y)
(3, 3)
(15, 15)
V1(20, 3)
(100, 15)
V0(0,0)
(0, 0) x
A texture location can be calculated from object location directly. In general, mapping an object with x coordinates in the range from xmin to xmax and y coordinates in the range ymin to ymax into the texture coordinate range can be computed as:
umin , umax , vmin , vmax are determined by the range of texture values assigned to the object vertices.

21. Texture Mapping Interpolate in the interior: - Painted texture Simple
But can have problems – especially for highly irregular surfaces

22. Artifacts
McMillan’s demo of this is at
Another example
What artifacts do you see?
Why?
Why not in standard Gouraud shading?
Hint: problem is in interpolating parameters

23. Interpolating Parameters
The problem turns out to be fundamental to interpolating parameters in screen-space
Uniform steps in screen space  uniform steps in world space

24. Texture Mapping Linear interpolation of texture coordinates
Correct interpolation
with perspective divide
Hill Figure 8.42

25. Interpolating Parameters
Perspective foreshortening is not getting applied to our interpolated parameters
Parameters should be compressed with distance
Linearly interpolating them in screen-space doesn’t do this

26. Perspective-Correct Interpolation
Skipping a bit of math to make a long story short…
Rather than interpolating u and v directly, interpolate u/z and v/z
These do interpolate correctly in screen space
Also need to interpolate z and multiply per-pixel
Problem: we don’t know z anymore
Solution: we do know w  1/z
So…interpolate uw and vw and w, and compute u = uw/w and v = vw/w for each pixel
This unfortunately involves a divide per pixel

27. Texture Map Filtering Naive texture mapping aliases badly
Moire pattern

28. Mapping to Curved Surfaces
A pixel
A curved
image
Parametric surface
x=x(u,v)
y=y(u,v)
z=z(u,v)
The task is to map texture
to surface, or to find a mapping
u=as+bt+c
v=ds+et+f

29. 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, to give the illusion of surface ‘bumps’

30. 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

31. Bump Mapping Example
Bump map Result

32. Bump Mapping Example Texture = change in surface normal!
Sphere w/ diffuse texture and swirly bump map
Sphere w/ diffuse texture
Swirly bump map

33. Illumination Maps Texture map: Texture map + light map:
Quake introduced illumination maps or light maps to capture lighting effects in video games
Texture map:
Light map
Texture map + light map:

34. Environment Maps Images from Illumination and Reflection Maps:
Simulated Objects in Simulated and Real Environments
Gene Miller and C. Robert Hoffman
SIGGRAPH 1984 “Advanced Computer Graphics Animation” Course Notes