Geometry of Curves and Surfaces

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