5.3 Serret-Frenet Equations

The natural triad ?, ?, ? can serve as coordinate axes of the space; this is used when studying local properties of a curve. Frenet established a system of ordinary differential equations which governs the triad along the curve. We have already derived two of these three equations: they are formulas (5.5) and (5.8). The former will be written as ? ?=k ₁ ?. Let us derive the third formula of the Serret-Frenet system. We have ?= ? ?. By this,

Let us collect the Serret-Frenet equations together:

(5.10)

We recall that these equations are written out when the curve has the natural parameterization.

Let us note that if the curvatures k ₁ (s) and k ₂ (s) are given functions of s, the system (5.10) becomes a linear system of ordinary differential equations (when written in component form it becomes a system of nine equations in nine unknowns). Fixing some point of the curve in space, by this system we can define ?(s), ?(s), and ?(s) uniquely; then, by the equation r' (s)= ?(s), we define the curve r= r (s) uniquely as well. Thus k ₁ (s) and k ₂ (s) define the curve up to a motion in space. That is why the pair of equations for k ₁ (s), k ₂ (s) is called the set of natural equations of...