Curvature | Sahithyan's Notes

Let $C$ be a graph of a $C^1$ vector-valued function $\boldsymbol{r}$ , parametrized by arc length $s$ , with $T'(s)$ existing. The curvature of $C$ at $s$ is

\kappa(s) = \left\lVert \frac{\text{d}T}{\text{d}s} \right\rVert

Since $T(s) = \boldsymbol{r}'(s)$ for an arc-length parametrization, if $\boldsymbol{r}''(s)$ exists,

\kappa(s) = \lVert \boldsymbol{r}''(s) \rVert

If $\boldsymbol{r}(t)$ is twice differentiable and $\boldsymbol{r}'(t) \ne \vec{0}$ , the curvature of $C$ is

\kappa(t) = \frac{\lVert T'(t) \rVert}{\lVert \boldsymbol{r}'(t) \rVert}

Equivalently,

\kappa(t) = \frac{\lVert \boldsymbol{r}'(t) \times \boldsymbol{r}''(t) \rVert}{\lVert \boldsymbol{r}'(t) \rVert^3}

Examples:

The curvature of a straight line is $0$ .
The curvature of a circle with radius $a > 0$ is $\frac{1}{a}$ .

Osculating Circle

If a curve $C$ in two-dimensional space has non-zero curvature $\kappa$ at a point $P$ , the circle of radius

\rho = \frac{1}{\kappa}

that shares a common tangent with $C$ at $P$ and is centered on the concave side of the curve at $P$ is the osculating circle, or circle of curvature, at $P$ .

$\rho$ : radius of curvature at $P$

Center of Curvature

The center of curvature of $C$ at a point $P = \boldsymbol{r}(s)$ with $\kappa(s) \ne 0$ is the center of the osculating circle at $P$ :

\boldsymbol{r}(s) + \rho\, N(s)

Since $N(s)$ points toward the concave side of $C$ , the center of curvature lies at distance $\rho$ from $P$ on that side.