2.4 Frenet-Serret formulae

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 .