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 definition^2.1. 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 $b$ .

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

$$b'(s) = t'(s) \times n(s) + t(s) \times n'(s)$$ (2.5)

$$= k(s) n(s) \times n(s) + t(s) \times n'(s)$$ (2.6)

$$= t(s) \times n'(s),$$ (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 torsion^2.2. Finally, the derivative of the normal can be expressed in terms of the other quantities:

$$n'(s) = b'(s) \times t(s) + b(s) \times t'(s)$$ (2.8)

$$= \tau(s) n(s) \times t(s) + b(s) \times k(s) n(s)$$ (2.9)

$$= -\tau(s) b(s) - k(s) t(s)$$ (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

... definition^2.1

We denote the cross product of two vectors $u,v \in \ensuremath{\mathbb{R}^3}$ as $u \times v$ .

...torsion^2.2

Some define the torsion as $b'(s) = -\tau(s) n(s)$ .