Points on the sphere

Displacements

The displacement of a point on the sphere is described by movement along a geodesic. In the language of ships, this is called “great-circle navigation”, while in differential geometry, it is the so-called “exponential map”.

On the sphere, we encode displacement as a complex value \(\alpha\) with spin weight \(1\). The absolute value \(|\alpha|\) is the angular distance of the displacement: If \(\hat{u}\) and \(\hat{u}'\) are two points on the sphere, and \(\alpha\) is the complex displacement between them, then \(\hat{u} \cdot \hat{u}' = \cos|\alpha|\).

The complex argument \(\arg\alpha\) (in radians) is the direction of the displacement: By convention, \(\arg\alpha = 0\) is north, \(\arg\alpha = \frac{\pi}{2}\) is east, \(\arg\alpha = \pi\) is south, and \(\arg\alpha = -\frac{\pi}{2}\) is west. With this definition, the gradients of potential fields recover their intuitive meaning, e.g. in gravitational lensing.

There are two functions for working with points and displacements: Given a set of points and their displacements, the displace() function computes the new set of points after applying the displacement. Conversely, given an initial and a final set of points, the displacement() function computes the displacement that would transform the one set into the other.

angst.displace(lon, lat, alpha)

Displace points with longitude lon and latitude lat (both in degrees) by alpha. Inputs must be array-like. The displacement alpha must be complex-valued or have a leading axis of size 2 for its real and imaginary component.

Returns a tuple newlon, newlat of longitude and latitude (both in degrees) of the displaced points.

angst.displacement(from_lon, from_lat, to_lon, to_lat)

Compute the complex displacement that transforms points with longitude from_lon and latitude from_lat into points with longitude to_lon and latitude to_lat (all in degrees). All inputs must be array-like.