# Derivation of surfaces

In an Euclidean linear plane, the evolute of a given curve C with support function h(t) can be regarded as a kind of derivative C' of C. Indeed, C' has support function h'(Pi/2 -t).

Is there any analogue in dimension 3 ?

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en.wikipedia.org/wiki/Evolute . It won't have the simple formula as for a curve, but in general you can use the caustic of the normal mapping. –  Charlie Frohman Feb 19 '12 at 12:54

For a surface in 3-dimensional Euclidean space, there is an an analogue of the evolute of a curve but ... in Lorentz-Minkowski space-time, not in 3-dimensional Euclidean space :

Derivation of convex surfaces of $\Bbb R^3$ in Lorentz space and study of their focals. C. R., Math., Acad. Sci. Paris 348, No. 23-24, 1307-1310 (2010).

The author does not consider the case of dimensions greater than 3.

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Consider a convex body $K \subset {\mathbb R}^{2n}$ with smooth boundary $\partial K$. Using the identification of $\mathbb R^{2n}$ and $\mathbb C^n$ we can define the normal hyperplane at a point $z \in \partial K$ as $\sqrt{-1} \ T_z \partial K$ or, equivalently, as the hyperplane that contains the normal to $\partial K$ at $z$ and the complex subspace of maximal dimension inside the tangent space $T_z \partial K$.