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 , and 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 and be continuous functions of a real variable defined in an interval . Then there exists one and only one arc of a curve, determined up to a direct congruent transformation, with arc length whose curvature and torsion are given by the functions and .
Involutes and evolutes: Tangents to a curve generate a surface called the tangent surface. The tangent surface can be represented as:
Involutes of a curve are curves on the tangent surface that are orthogonal to the generating tangents. Thus the involutes can be represented as:
where is chosen such that the orthogonality condition is satisfied, i.e. . Because:
it follows that or where is constant. Thus, the involutes can be expressed as:
If is an involute of , then is called an evolute of . It can be shown that the evolutes of a curve can be represented in the form:
where and is a constant.