Given a curve parameterized by arc length, we want to describe the bending and twisting of the curve at a point. Starting with the unit tangent vector , we can examine the vector . This is a vector which we break into two parts: a scalar curvature and a vector normal. Hence the curvature is defined as and the normal is uniquely defined if . The curvature describes how the curve pulls away from the tangent. The inverse of the curvature is called the radius of curvature and describes the size of a circle with the same curvature. The plane that contains and is called the osculating plane. If a curve is completely contained in a plane, then the osculating plane at every point coincides with the containing plane. For this section, we will assume that so that the normal is defined everywhere.
The unit binormal vector is the second direction in which the curve can bend and is perpendicular to the osculating plane by definition^{2.1}. The rate of change of describes how the osculating planes of neighbouring points differ, or put another way, how the curve pulls out of the osculating plane.
It is useful to know how the three vectors change with changes of parameter. By definition, we have . For the binormal, we have
(2.5) | ||
(2.6) | ||
(2.7) |
(2.8) | ||
(2.9) | ||
(2.10) |
To summarize, the behaviour of a arc length parameterized curve can be characterized by three orthonormal vectors, namely the tangent , the normal and the binormal , and two scalars, namely the curvature and the torsion . The frame defined by the three vectors and the point is called the Frenet frame.
The fundamental theorem of the local theory of curves says that all curves with identical and are identical up to rigid transformations. That is, the curvature and torsion uniquely characterize the local behaviour of a curve.