# How does displacement mapping generalize bump mapping?

This PDF discusses bump mapping, a special case of displacement mapping. A surface is represented as a function $$\\overrightarrow{O}\left(u,\,v\right):\:\Bbb R^2\mapsto\Bbb R^3\$$. In bump mapping, we shift this surface by $$\\delta\overrightarrow{O}=B(u,\,v)\overrightarrow{N}\$$, with $$\\overrightarrow{N}=\overrightarrow{O}_u\times\overrightarrow{O}_v\$$ the surface normal. What happens in more general displacement mapping? Do we, for example, replace $$\B\$$ with a matrix, or include multiples of $$\\overrightarrow{O}_u,\,\overrightarrow{O}_v\$$ in $$\\delta\overrightarrow{O}\$$? If there's too broad a variety of displacement maps to describe in one answer, I'll settle for one or more mathematically detailed references.

• I can't give you a mathematical reference. But displacement mapping uses 3 channels of a texture to displace the vertices of a mesh by using 1 channel for each x y z axis. I think there are no other forms of displacement mapping. – Bartimaeus Sep 24 '19 at 17:16
• @Bartimaeus That would be something like $\delta O_i=\sum_j B_{ij}(u,\,v)N_j$, i.e. $B$ is replaced with a matrix, which was one of my guesses. Thanks! – J.G. Sep 24 '19 at 17:28