Texture Mapping

David S. Ebert

September 2000

Map in object space

In texture space for quadrilaterals


P0v = P00 + v (P01 - P00)

P1v = P10 + v (P11 - P10)

Puv = P0v + u (P1v - P0v)

Puv = {P00 + v (P01 - P00)} + u{ (P10 + v (P11 - P01)) - (P00 + v (P10 - P00))}


solve for u and v.

In general case, add Pij's u,v coordinates into equation.

Before do this,

Now what do you do ?

Solution: Antialiased texture mapping

Object Space vs. World Space for Mapping

  • In world space, object will move through the texture. -- The texture is not stuck to the object.

  • Example:

Other Mapping Functions

  • Spherical Mapping Example
  • Cylindrical Mapping Examples
  • Box Mapping Examples

Bump Mapping [Blinn 78]

  • Texture map contains displacement to the surface.

Mathematical Derivation:

Decompose N into partial surface derivatives in the u and v directions: Pu Pv.

N = Pu X Pv

F(u,v) is displacement of surface,

P' = P + F(u,v) * N /||N||

Assuming F is negligibly small,

N' = P'u X P'v

P'u = Pu + Fu * N /||N||

P'v = Pv + Fv * N /||N||

N' = N + D

D = [Fu(N X Pv) - Fv(N X Pu)] / ||N||

How do we calculate Fu, Fv?

Given F(u,v) function, ep = 1 / (TextureMapResolution)

Fu = (F(u+ep, v) - F(u - ep, v)) / (2*ep)

Fv = (F(u, v+ep) - F(u, v - ep)) / (2*ep)

How do we calculate Pu, Pv?

  • Displace world space point in +u, -u direction and +v, -v direction by small amount and calculate approximated partial derivative as :
    Pu = (P(p+uepsilon) - P(p-uepsilon)) / (2 * epsilon)
    Pv = (P(p+vepsilon) - P(p-vepsilon)) / (2 * epsilon)

  • If smooth shading, need to adjust this because of interpolated normals:
    PVtemp = Pu X N
    Pu = N X PVtemp

    PUtemp = Pv X N
    Pv = N X PUtemp

Dependence on scale:

If scale object, need to adjust displacement map accordingly

D' = alpha * D * |N|/|D| ,

alpha = sqrt ( Fu2 + Fv2)

N'' = N + D'

Displacement Mapping [Cook 84]

  • Use texture map to displace actual surface geometry. (vertices, control vertices)

  • Done as a pre-process to geometry before the actual rendering

  • Requires fine resolution in geometry or will get unexpected holes.

  • Alternative is for program to add geometry when displacement variation can't smoothly be represented in geometry.

  • Example of Bump Mapping vs. Displacement Mapping

  • How do you tell the difference between the two?

Environment Mapping (Reflection Mapping) [Newell 76]

  • Used to simulate ray traced reflections.

  • Use reflection vector as index into texture map.

  • Texture map is either hand drawn or precomputed rendering of scene viewed from that location (often times 6 directional rendering ).

  • Normally use box projection or spherical.

  • Examples of box projection:

  • Notice the differences in the ray traced vs. environmental mapped images:

  • Key to tell which is which? Look for object reflecting itself.

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