Differential Geometry of Curves and Surfaces

Thomas F. Banchoff, Stephen T. Lovett

• Verlag: Chapman and Hall/CRC
• Jahr: 2016
• ISBN:
  9781482247473
• Auflage: 2
• Thema: Mathématiques
• Thema: Mathématiques générales

Differential Geometry of Curves and Surfaces, Second Edition takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces. Requiring only multivariable calculus and linear algebra, it develops students' geometric intuition through interactive computer graphics applets suppor

CHAPTER 1

Plane Curves: Local Properties

Just as calculus courses introduce real functions of one variable before tackling multivariable calculus, so it is natural to study curves before addressing surfaces and higher-dimensional objects. This first chapter presents local properties of plane curves, where by local property we mean properties that are defined in a neighborhood of a point on the curve. For the sake of comparison with calculus, the derivative f′(a) of a function f at a point a is a local property of the function since we only need knowledge of f(x) for x in (a − ε, a + ε), where ε is any positive real number, to define f′(a). In contrast, the definite integral of a function over an interval is a global property since we need knowledge of the function over the whole interval to calculate the integral. In contrast to this present chapter, Chapter 2 introduces global properties of plane curves.

1.1 Parametrizations

Borrowing from a physical understanding of motion in the plane, we can think about plane curves by specifying the coordinates x and y as functions of a time variable t, which give the position of a point traveling along the curve. Thus we need two functions x(t) and y(t). Using vector notation to locate a point on the curve, we often write X→(t)=(x(t),y(t)) for this pair of coordinate functions and call X→(t) a vector function into ℝ². From a mathematical standpoint, t does not have to refer to time and is simply called the parameter of the ...