Curvature of curves

Given a curve $ \alpha(s)$ parameterized by arc length, we want to describe the bending and twisting of the curve at a point. Starting with the unit tangent vector $ t(s) = \alpha'(s)$ , we can examine the vector $ t'(s) = \alpha''(s) =
k(s)n(s)$ . This is a vector which we break into two parts: a scalar curvature $ k(s)$ and a vector normal. Hence the curvature is defined as $ k(s) = \vert\alpha''(s)\vert$ and the normal is uniquely defined if $ k(s)
\neq 0$ . The curvature describes how the curve pulls away from the tangent. The inverse of the curvature $ R = 1/k$ is called the radius of curvature and describes the size of a circle with the same curvature. The plane that contains $ t(s)$ and $ n(s)$ 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 $ k(s)
\neq 0$ so that the normal is defined everywhere.

The unit binormal vector $ b(s) = t(s) \times n(s)$ is the second direction in which the curve can bend and is perpendicular to the osculating plane by definition2.1. The rate of change of $ b(s)$ describes how the osculating planes of neighbouring points differ, or put another way, how the curve pulls out of the osculating plane.

Figure 2.1: Curve with tangent $ t$ , normal $ n$ and binormal $ b$ .
Image Curve

It is useful to know how the three vectors change with changes of parameter. By definition, we have $ t'(s) = k(s) n(s)$ . For the binormal, we have

$\displaystyle b'(s)$ $\displaystyle = t'(s) \times n(s) + t(s) \times n'(s)$ (2.5)
  $\displaystyle = k(s) n(s) \times n(s) + t(s) \times n'(s)$ (2.6)
  $\displaystyle = t(s) \times n'(s),$ (2.7)

and it follows that $ b'(s)$ is parallel to the normal.(TODO) Hence we can write $ b'(s) = \tau(s) n(s)$ , where $ \tau(s)$ is the scalar torsion2.2. Finally, the derivative of the normal can be expressed in terms of the other quantities:

$\displaystyle n'(s)$ $\displaystyle = b'(s) \times t(s) + b(s) \times t'(s)$ (2.8)
  $\displaystyle = \tau(s) n(s) \times t(s) + b(s) \times k(s) n(s)$ (2.9)
  $\displaystyle = -\tau(s) b(s) - k(s) t(s)$ (2.10)

To summarize, the behaviour of a arc length parameterized curve can be characterized by three orthonormal vectors, namely the tangent $ t(s)$ , the normal $ n(s)$ and the binormal $ b(s)$ , and two scalars, namely the curvature $ k(s)$ and the torsion $ \tau(s)$ . The frame defined by the three vectors and the point $ \alpha(s)$ is called the Frenet frame.

The fundamental theorem of the local theory of curves says that all curves with identical $ k(s)$ and $ \tau(s)$ are identical up to rigid transformations. That is, the curvature and torsion uniquely characterize the local behaviour of a curve.



Footnotes

... definition2.1
We denote the cross product of two vectors $ u,v \in \ensuremath{\mathbb{R}^3}$ as $ u \times v$ .
...torsion2.2
Some define the torsion as $ b'(s) = -\tau(s) n(s)$ .
Copyright © 2005 Adrian Secord. All rights reserved.