W. Boehm
Differential Geometry I
Differential geometry is based largely on the pioneering work of L. Euler (1707–1783), C. Monge (1746–1818), and C. F. Gauss (1777–1855). One of their concerns was the description of local curve and surface properties such as curvature. These concepts are also of interest in modern computer-aided geometric design. The main tool for the development of general results is the use of local coordinate systems, in terms of which geometric properties are easily described and studied. This introduction discusses local properties of curves independent of a possible embedding into a surface.
10.1 Parametric Curves and Arc Length
A curve in is given by the parametric representation
(10.1)
where its cartesian coordinates ...
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