Lengths of Parallel Curves

Let's start with something simple: Given a circle with radius $R$, draw a concentric circle with radius 1 larger. How much longer is the circumference of the bigger circle?

A simple calculation yields that the difference in circumferences is $2\pi (R + 1) - 2\pi R = 2\pi$.

(An interesting, slightly mind-bending application is that if we build a rope-fence that is 1 meter off the ground that goes all the way around the equator, we only need 6.3 meters more rope than when we wrap a rope around the earth exactly on the equator.)

The follow up question: does this extend to other kinds of curves?

The answer turns out to depend on the winding number of the curve we start from.

Winding number

Let $\gamma: \mathbb{R} \to \mathbb{R}^2$ be a periodic regular map, so its image is a closed smooth parametrized curve. We do not assume that $\gamma$ is a simple curve, and explicitly allow self-intersections. Since $\gamma$ is regular, we can without loss of generality assume that $\gamma$ is parametrized by arc length. In this case the period $\tau$ of the mapping is the total length of the curve $\gamma$.

The Gauss map for the curve $\gamma$ is a $\tau$-periodic mapping $\eta : \mathbb{R} \to \mathbb{S}^1 \subset \mathbb{R}^2$ into the unit circle. Let $J$ denote the linear mapping $\mathbb{R}^2 \to \mathbb{R}^2$ given by $(x,y) \mapsto (y,-x)$, we can write

\[ \eta = J \gamma' \]

Note that $J$ is the clockwise rotation by $\pi/2$ radians, and hence $\eta$ is a normal vector to $\gamma$.

Now let $\Theta$ denote the unit one-form along $\mathbb{S}^1$. The integral $\int_{\mathbb{S}^1} \Theta = 2\pi$. We can pull-back $\Theta$ to a one-form along $\gamma$ and consider

\[ \int_{\gamma} \eta^\star \Theta \]

One sees easily that the integral evaluates to a integer multiple of $2\pi$, the integer being the topological degree of the map $\eta$. (Alternatively, consider the image of $[0,\tau]$ along $\eta$: this is a curve along $\mathbb{S}^1$ that returns to its starting point. The integer mentioned before counts how many times this curve goes around the circle (in the counterclockwise direction) between $0$ and $\tau$.)

Parallel curves

Given $\gamma$ (parametrized by arc length), and a fixed number $\rho$, we can consider the smoothly parametrized curve

\[ \gamma + \rho \eta : \mathbb{R} \to \mathbb{R}^2 \]

Observe that since $\eta \perp \gamma'$ and $|\eta| = 1$, we have that $\eta' \parallel \gamma'$. And hence for the same parameter, the tangents to $\gamma$ and $\gamma + \rho\eta$ are parallel.

Let us consider the quantity

\[ \int_0^\tau \gamma' \cdot (\gamma' + \rho \eta') ~\mathrm{d}s. \]

When the integrand is signed, this is (up to sign) the arc length of the curve $\gamma + \rho \eta$. (However when $|\rho|$ is between the minimum and maximum radii of curvature of $\gamma$, this quantity may change signs and is no longer directly related to the arc length of $\gamma + \rho \eta$.)

This quantity can be computed fairly simply to equal

\[ \int_0^\tau |\gamma'|^2 + \rho \eta' \cdot \gamma' ~\mathrm{d}s. \]

The first term integrates to $\tau$ using that $\gamma$ was parametrized by arc length so $|\gamma'| = 1$. To evaluate the second term we can observe that

\[ \int_0^\tau \eta' \cdot \gamma' ~\mathrm{d}s = \int_{\gamma} \eta^\star \Theta \]

exactly. And hence if $w$ is the winding number of $\gamma$, we have that the integral evaluates to $\tau + 2 \pi \rho w$.

Willie WY Wong
Associate Professor

My research interests include partial differential equations, geometric analysis, fluid dynamics, and general relativity.