The ellipticity manifold

The space of ellipticities is the hyperbolic plane, which is a Riemannian manifold. More specifically, \(\epsilon\)-ellipticities are points in the Poincaré disk model of the hyperbolic plane.

Distance

cosmicshear.distance(x, y=None)

Compute the intrinsic distance between ellipticities.

Isometry

The hyperbolic plane has an isometry \(T_{\epsilon_0}\) that maps the point \(\epsilon_0\) to the origin,

\[T_{\epsilon_0}(\epsilon) = \frac{\epsilon - \epsilon_0}{1 - \epsilon_0^* \epsilon} \;.\]

In weak lensing, the action of a reduced shear \(g\) is precisely \(T_{-g}\).

cosmicshear.isometry(epsilon, origin)

An isometry of the ellipticity epsilon that maps origin to the origin.

Exponential map

cosmicshear.exponential_map(vec, origin=None)

Compute the exponential map of vector vec in origin origin (default 0).

cosmicshear.normal_coordinates(epsilon, origin=None)

Compute normal coordinates of ellipticity epsilon in origin (default 0).

Intrinsic mean

cosmicshear.mean(epsilon, weight=None, *, axis=None, initial=None, maxiter=100, tol=1e-06)

Compute the Fréchet mean of a sample of ellipticities epsilon. If weight is given, the mean is weighted.