8. Properties of Curves

f. Curvature and Torsion

1. Curvature

The curvature, \(\kappa\), of a parametric curve, \(\vec{r}(t)\), is \[ \kappa=\left|\dfrac{d\hat{T}}{ds}\right| \] where \(\hat{T}\) is the unit tangent vector to the curve and \(s\) is the arc length parameter on the curve.

1) We use the unit tangent vector \(\hat{T}\) instead of the velocity vector \(\vec{v}\) because \(\vec{v}\) includes information about the speed \(|\vec{v}|\) while the curvature is only supposed to measure the change in the direction of the tangent vector which is \(\hat{T}\).

2) We take the derivative with respect to arc length \(s\) instead of the parameter \(t\) because we don't want the curvature to depand on how fast we travel on the curve. So \(\dfrac{d\hat{T}}{ds}\) measures the rate of change of the direction of the tangent vector, \(\hat{T}\), with respect to arc length, \(s\).
3) We take the magnitude of \(\dfrac{d\hat{T}}{ds}\) because we want the curvature to be a single number rather than a vector.

It is often easier to calculate curvature using one of the following formulas:

The curvature can be calculated from: \[ \kappa=\dfrac{1}{|\vec{v}|}\left|\dfrac{d\hat{T}}{dt}\right| =\dfrac{|\vec{v}\times\vec{a}|}{|\vec{v}|^{3}} \] Proof

Find the curvature \(\kappa\) of the twisted cubic \(\vec{r}=\left\langle t,t^2,\dfrac{2}{3}t^{3}\right\rangle\).

On the previous page, in the example to find \(\hat{T}\), \(\hat{N}\) and \(\hat{B}\) for this twisted cubic, we found: \[\begin{aligned} &\vec{v}=\langle 1,2t,2t^2\rangle &&|\vec{v}|=1+2t^2 \\ &\vec{a}=\langle 0,2,4t\rangle &&|\vec{v}\times\vec{a}|=2(1+2t^2) \end{aligned}\] So \[ \kappa=\dfrac{|\vec{v}\times\vec{a}|}{|\vec{v}|^{3}} =\dfrac{2(1+2t^2)}{(1+2t^2)^{3}}=\dfrac{2}{(1+2t^2)^2} \]

As promised, the curvature, \(\kappa\), of a circle is the reciprocal of its radius, \(R\). Based on this fact, we define:

The radius of curvature of a curve is the reciprocal of its curvature: \[ R=\dfrac{1}{\kappa} \]

Thus, the radius of curvature at a point on the curve is the radius of the circle which best fits the curve at that point. That circle is called the osculating circle .

The osculating circle at a point on a curve is the circle which instantaneously lies in the same plane as the circle (the \(\hat T\hat N\)-plane), is tangent to the curve at that point, bends in the same direction as the curve and has radius equal to the radius of curvature of the curve at that point. It is regarded as the best fit circle at that point.

If \(R\) is the radius of curvature at a point \(\vec{r}(a)\) on a curve \(\vec{r}(t)\), then the osculating circle at the point \(\vec{r}(a)\) may be parametrized by: \[ \vec{r}_a(\phi) =\vec{r}(a)+R\vec{N}+R\cos\phi\,\hat{T}+R\sin\phi\,\hat{N} \] Derivation