DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 2
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 2
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2009
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对于a < t < B。注释的整个1.3节可以立即重新表述为空间中的曲线：定义1.3.2（闭区间上曲线的长度），定义1.3.3和定理1.3.4（关于曲线的重新参数化），定义1.3.4（正则曲线），定理1.3.6和命题1.3.7（关于弧长参数化）。至于1.4节（即曲率和曲线基本定理），情况就不同了。首先，一个纯粹形式上的观察是，定义1.4.1. (of曲率）将不令人满意：“显然”，仅x（t）和y（t）（及其导数）不足以确定曲率，还需要将z（t）带入该游戏。但是没有什么能阻止我们要求定理1.4.3.，第一个方程：
for a < t < b. The entire Section 1.3 of the notes can be immediately reformulated for curves in the space: Definition 1.3.2 (of the length of a curve over a closed interval), Definition 1.3.3 and Theorem 1.3.4 (concerning reparametrization of curves), Definition 1.3.4 (of a regular curve), Theorem 1.3.6 and Proposition 1.3.7 (concerning parametrization by arc length). As about Section 1.4 (that is, the curvature and the fundamental theorem of curves), things are different. First of all, a purely formal observation is that Definition 1.4.1. (of the curvature) will not be satisfactory: it is “obvious” that only x(t) and y(t) (and their derivatives) are not enough to determine the curvature, one needs to bring also z(t) into this game. But nothing prevents us from requesting Theorem 1.4.3., the first equation: