Notes on Differential Geometry
نویسنده
چکیده
Part 1. Geometry of Curves We assume that we are given a parametric space curve of the form (1) x(u) = x 1 (u) x 2 (u) x 3 (u) u 0 ≤ u ≤ u 1 and that the following derivatives exist and are continuous (2) x (u) = dx du x (u) = d 2 x du 2 1. Arc Length The total arc length of the curve from its starting point x(u 0) to some point x(u) on the curve is defined to be (3) s(u) = u u 0 √ x ·x du It is also common to express this equation in a differential form: (4) ds 2 = dx·dx The differential ds is referred to as the element of arc of the curve. Because we know that ds/du = 0, it is always permissible to reparame-terize the curve x(u) in terms of its arc length x(s). This reparameterized curve has derivatives: (5) ˙ x(s) = dx ds¨x(s) = d 2 x ds 2 Such a parameterization of the curve is often called a unit-speed parameter-ization because ˙ x = 1.
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