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- W2219150070 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 $$X varepsilonmathbb{R}^3$$ . 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 $$t varepsilonmathbb{R}$$ describes the same unit circle as the one mentioned above.2" @default.
- W2219150070 created "2016-06-24" @default.
- W2219150070 creator A5012775344 @default.
- W2219150070 date "2009-01-01" @default.
- W2219150070 modified "2023-10-16" @default.
- W2219150070 title "Geometry of Curves and Surfaces" @default.
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- W2219150070 doi "https://doi.org/10.1007/978-0-8176-4803-9_5" @default.
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