Differential geometry notes 6

by Emin Orhan

Natural equations of a curve: Is it possible to characterize a curve in a manner independent of the choice of coordinates, except for the position of the curve in space, that is, to within congruent transformations? For example, we have seen that s, \kappa and \tau are quantities of this sort (i.e. independent of the choice of coordinates). The following theorem, called the fundamental theorem of the theory of curves, answers this question.

Theorem 20.1: Let \kappa(s) and \tau(s) be continuous functions of a real variable s defined in an interval I: 0\leq s \leq a. Then there exists one and only one arc \mathbf{x}(s) of a curve, determined up to a direct congruent transformation, with arc length s whose curvature and torsion are given by the functions \kappa(s) and \tau(s).

Involutes and evolutes: Tangents to a curve \mathbf{x}(s) generate a surface called the tangent surface. The tangent surface can be represented as:

\mathbf{y}(s,u) = \mathbf{x}(s) + u \mathbf{t}(s)

Involutes of a curve are curves on the tangent surface that are orthogonal to the generating tangents. Thus the involutes can be represented as:

\mathbf{z}(s) = \mathbf{x}(s) + u(s)\mathbf{t}(s)

where u(s) is chosen such that the orthogonality condition is satisfied, i.e. \mathbf{\dot{z}} \cdot \mathbf{t} = 0. Because:

\dot{\mathbf{z}} = \mathbf{t} + \frac{d}{ds}(u \mathbf{t}) = \mathbf{t} + u \kappa \mathbf{p} + \dot{u} \mathbf{t}

it follows that 1 + \dot{u}=0 or u(s) = c-s where c is constant. Thus, the involutes can be expressed as:

\mathbf{z}(s) = \mathbf{x}(s) + (c-s) \mathbf{t}(s)

If C is an involute of C^*, then C^* is called an evolute of C. It can be shown that the evolutes of a curve \mathbf{x}(s) can be represented in the form:

\mathbf{y}(s) = \mathbf{x}(s) + \rho(s) \big[ \mathbf{p}(s) + \mathbf{b}(s) \cot \alpha(s) \big]

where \alpha(s) = \int_0^s \tau(\sigma) d\sigma + k^* and k^* is a constant.