Expected chord length on a sphere, Geometric Probability

Problem 61.1

Two points are chosen independently and uniformly on the unit sphere $S^2 \subset \mathbb{R}^3$ . What is the expected Euclidean distance (chord length) between them?

Solution. By rotational symmetry of the sphere, condition on the first point at the north pole $\mathbf{u} = (0, 0, 1)$ . The second point $\mathbf{v}$ is uniform on $S^2$ , and by Archimedes’s hat-box theorem (see Chapter 28), its $z$ -coordinate $z = \cos \Theta$ is uniform on $[-1, 1]$ . Here $\Theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$ .

The chord length is $|\mathbf{u} - \mathbf{v}| = \sqrt{2 - 2 \cos \Theta} = 2 \sin(\Theta / 2).$ Hence $\mathbb{E}[|\mathbf{u} - \mathbf{v}|] = \mathbb{E}[2 \sin(\Theta/2)].$ With $z = \cos \Theta$ uniform on $[-1, 1]$ , substitute $z = \cos \Theta$ , $dz = -\sin \Theta \, d\Theta$ . The density of $\Theta$ on $[0, \pi]$ is therefore $\tfrac12 \sin \Theta$ . Then $\mathbb{E}[2 \sin(\Theta/2)] = \int_0^{\pi} 2 \sin(\Theta/2) \cdot \tfrac12 \sin \Theta \, d\Theta = \int_0^{\pi} \sin(\Theta/2) \sin \Theta \, d\Theta.$ Use $\sin \Theta = 2 \sin(\Theta/2) \cos(\Theta/2)$ and substitute $u = \sin(\Theta/2), du = \tfrac12 \cos(\Theta/2) d\Theta$ : $= 2 \int_0^{\pi} \sin^2(\Theta/2) \cos(\Theta/2) \, d\Theta = 4 \int_0^1 u^2 \, du = \frac{4}{3}.$ Therefore $\boxed{\mathbb{E}[|\mathbf{u} - \mathbf{v}|] = \frac{4}{3}.} \qedhere$

Compare:

Unit circle ( $S^1$ , Chapter 13): $\mathbb{E}[\text{chord}] = \tfrac{4}{\pi} \approx 1.27$ .
Unit sphere ( $S^2$ , here): $\mathbb{E}[\text{chord}] = \tfrac{4}{3} \approx 1.33$ .
Unit disc (interior, Chapter 35): $\tfrac{128}{45\pi} \approx 0.91$ .
Unit ball (interior, Chapter 59): $\tfrac{36}{35} \approx 1.03$ .

The surface versions are consistently larger than the interior versions, because points pushed to the boundary are farther from one another on average.

import numpy as np
u = np.random.randn(3, 2, 10**7); u /= np.linalg.norm(u, axis=0)
d = np.linalg.norm(u[:,0] - u[:,1], axis=0)
print(f"sim: {d.mean():.5f}   exact: {4/3:.5f}")
# sim: 1.33313   exact: 1.33333