# Polytopes and Manifolds

• Gregory S. Chirikjian
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

## Abstract

This chapter extends the review of geometrical ideas from the previous chapters to include geometrical objects in higher dimensions. These include hyper-surfaces and “ghyper-polyhedra” (or polytopes) in ℝn. A parametric description of an m-dimensional embedded manifold1 in an n-dimensional Euclidean space is of the form x = x(q) where x ε ℝn and q ε ℝm with m ≤ n. If m = n-1, then this is called a hyper-surface. An implicit description of an m-dimensional embedded manifold in ℝn is a system of constraint equations of the form φi(x) = 0 for i = 1,..., n-m. In the context of engineering applications, the two most important differences between the study of two-dimensional surfaces in ℝ3 and m-dimensional embedded manifolds in ℝn are: (1) there is no crossproduct operation for ℝn; and (2) if m ≪ n, it can be more convenient to leave behind Rn and describe the manifold intrinsically. For these reasons, modern mathematical concepts such as differential forms and coordinate-free differential geometry can be quite powerful.

## Keywords

Riemannian Manifold Tangent Space Convex Body Klein Bottle Coordinate Chart
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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