Same volume. Different geometry.

Two ways to interpolate covariance ellipses.

Log-Euclidean interpolation averages matrix logarithms; Kubo–Ando interpolation averages after whitening one endpoint. Each outline is a unit-Mahalanobis ellipse: its radii are square roots of covariance eigenvalues.

Interpolation t = 0.50

0 · matrix Amatrix B · 1

Log-Euclidean

exp((1 − t) log A + t log B)

E_t(A, B)

Kubo–Ando

A¹ᐟ² (A⁻¹ᐟ² B A⁻¹ᐟ²)ᵗ A¹ᐟ²

A #_t B

A B current t t = 0.1, …, 0.9 fixed time-average Equal axes · shared, time-independent scale

Show logarithmic mean (fixed time-average) in the Kubo–Ando panel

Log-Euclidean: average logs, then exponentiate. Logarithmic mean: average all Kubo–Ando times, not tied to the slider t.

Relative orientation

Relative rotation

Rotate B while A’s axes stay fixed.

Shared shape

Anisotropy (axis ratio)

Change both ellipses’ axis ratio; 1 means circles.

Relative path gap at t

Common path determinant

Time-average vs log midpoint · ‖ · ‖_F

Normalized commutator

Residual in the predicted log determinant: log-Euclidean ; Kubo–Ando . Residuals show numerical rounding, not a rigorous error certificate.