Hyperspherical Geodesic Alignment

Rotate the gold points R Ŝ_M until they land where the black points Ŝ_N are dense. The sphere is colored by the Gaussian kernel density of Ŝ_N, measured in geodesic distance.

Ŝ_N fixed

R Ŝ_M rotating

J_σ(R)

0.0000

σ = 0.40 rad

low 0.00

0.00 high

Rotate Ŝ_M

About the x-axis0.0°

About the y-axis0.0°

About the z-axis0.0°

Include a reflection (det R = −1)

Kernel width

σ0.40 rad

Drag the sphere to change your viewing angle. This does not change R.

Current R

Objective

J_σ(R) = (1/mn) Σ_i,j exp( −arccos²⟨R p̂_i, q̂_j⟩ / 2σ² )

The sphere shows f(x) = (1/n) Σ_j exp( −d(x, q̂_j)² / 2σ² ), so J_σ(R) is the average of f over the gold points.

Demo data: n = 36 and m = 28 points drawn from the same four clusters, with the second sample rotated by a hidden rotation. The color scale runs from the smallest to the largest value of f on the sphere.