### Differential geometry notes 6

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.