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.

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