# Geometry of Curves and Surfaces

Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

## Abstract

This chapter consists of a variety of topics in geometry. The approach to geometry that is taken in this chapter and throughout this book is one in which the objects of interest are described as being embedded1 in Euclidean space. There are two natural ways to describe such embedded objects: (1) parametrically and (2) implicitly. The vector-valued functions x = x(t) and x = x(u, v) are respectively parametric descriptions of curves and surfaces when $$X \varepsilon\mathbb{R}^3$$. For example, $$x(\Psi ) = [\cos \Psi , \sin \Psi , 0]^T$$ for ψ ∈ [0, 2π) is a parametric description of a unit circle in $$\mathbb{R}^3$$, and $$x(\phi ,\theta ) = [\cos \phi \sin {\rm \theta },\sin \phi \sin \theta {\rm ,}\cos {\rm \theta }]^T$$ for φ ∈ [0, 2π) and θ ∈ [0, π] is a parametric description of a unit sphere in $$\mathbb{R}^3$$. Parametric descriptions are not unique. For example, $$x(t) = [{\rm 2t/(1 + t}^{\rm 2} {\rm ), (1 } - {\rm t}^{\rm 2} {\rm )/(1 + t}^{\rm 2} {\rm ), 0]}^{\rm T}$$ for $$t \varepsilon\mathbb{R}$$ describes the same unit circle as the one mentioned above.2

## Keywords

Fundamental Form Gaussian Curvature Euler Characteristic Inverse Kinematic Total Curvature
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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