2.4 Frenet-Serret formulae

From (2.20) and (2.44), we found that

From these equations we deduce

In matrix form we can express the differential equations as

Thus, , , are completely determined by the curvature and torsion of the curve as a function of parameter . The equations , are called intrinsic equations of the curve. The formulae (2.56) are known as the Frenet-Serret formulae and describe the motion of a moving trihedron ( ) along the curve. From these , , the shape of the curve can be determined apart from a translation and rotation. For arbitrary speed curve the Frenet-Serret formulae are given by

where is the parametric speed.

Example 2.4.1 As shown in Example 2.3.1 the intrinsic equations of circular helix are given by , , where . In this example we derive the parametric equations of circular helix from these intrinsic equations. Substituting the intrinsic equations into the Frenet-Serret equations we obtain

We first differentiate the first equation twice and the second equation once with respect to , which yield

where the third equation is used to replace . Eliminating , , and recognizing that , we obtain the fourth order differential equation

The general solution to this differential equation is given by

where , , and are the vector constants determined by the initial conditions. In this case we assume the following initial conditions

which yield

thus, we have .