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Next: Forward Differencing Up: 10.6 Bump Mapping with Previous: Approximating Bump Mapping   Contents Tangent Space

Recall that the bump map normal $\vec{N'}$ is formed by $\vec{Pu} \times \vec{Pv}$. Assume that the surface $P$ is coincident with the $XY$ plane and that changes in $u$ and $v$ correspond to changes in $X$ and $Y$, respectively. Then $F$ can be substituted for $P'$, resulting in the following expression for the vector $N'$:

\begin{displaymath}N' = \pmatrix{
-\frac{\partial F}{\partial u} \cr
-\frac{\partial F}{\partial v} \cr
1 \cr

In order to evaluate the lighting equation, $N'$ must be normalized. If the displacements in the bump map are restricted to small values, however, the length of $N'$ will be so close to one as to be approximated by one. Then $N'$ itself can be substituted for $\vec{N}$ without normalization. If the diffuse intensity component $\vec{N} \cdot \vec{L}$ of the lighting equation is evaluated with the value presented above for $N'$, the result is the following:

N' \cdot L = - \frac{\partial F}{\partial u}L_x -
\frac{\partial F}{\partial v}L_y +
\end{displaymath} (1)

This expression requires the surface to lie in the $XY$ plane and that the $u$ and $v$ parameters change in $X$ and $Y$, respectively. Most surfaces, however, will have arbitrary locations and orientations in space. In order to use this simplification to perform bump mapping, the view direction $\vec{V}$, and light source direction $\vec{L}$ are transformed into tangent space.

Tangent space has 3 axes, $\vec{T}$, $\vec{B}$ and $\vec{N}$. The tangent vector, $\vec{T}$, is parallel to the direction of increasing $s$ on the surface. The normal vector, $\vec{N}$, is perpendicular to the surface. The binormal, $\vec{B}$, is perpendicular to both $\vec{N}$ and $\vec{T}$, and like $\vec{T}$, lies in the plane tangent to the surface. These vectors form a coordinate system that is attached to and varies over the surface.

% latex2html id marker 10120
\vrule width 0pt height 0....
...Figure \thefigure . Tangent Space Defined at Polygon Vertices}\\

The light source is transformed into tangent space at each vertex of the polygon. To find the tangent space vectors at a vertex, use the vertex normal for $\vec{N}$ and find the tangent axis $\vec{T}$ by finding the vector direction of increasing $s$ in the object's coordinate system. The direction of increasing $t$ may also be used. Find $\vec{B}$ by computing the cross product of $\vec{N}$ and $\vec{T}$. These unit vectors form the transformation shown below:

\pmatrix{x' \cr y' \cr z\lq  \cr w\lq  \cr} =
T_x & T_y ...
...V_z \cr
0 & 0 & 0 & 1 \cr
\pmatrix{ x \cr y \cr z \cr w \cr}
\end{displaymath} (2)

This transformation brings coordinates into tangent space, where the plane tangent to the surface lies in the $X-Y$ plane, and the normal to the surface coincides with the $Z$ axis. Note that the tangent space transformation varies for vertices representing a curved surface, and so this technique makes the approximation that curved surfaces are flat and the tangent space transformation is interpolated from vertex to vertex.

next up previous contents
Next: Forward Differencing Up: 10.6 Bump Mapping with Previous: Approximating Bump Mapping   Contents