Lens::getXi

This function calculates the apparent position $\xi$ in the lens plane, given the actual position $\chi\eta$. There are three approaches.

• The SIS lens has constant deflection, and hence the raytrace equation can be used.

  • $\xi = \chi\eta + \nabla\psi(\chi\eta)$

  • The same formula applies to SIE, because the direction from the lens is known, and the deflection does not depend on the distance from the centre.

• The point mass lens gives a quadratic equation whose solution is

  • $\chi\eta/2 + \sqrt{\frac14+R_E^2}\cdot\frac{\chi\eta}{||\chi\eta||}$

• The universal solution is to use fix-point iteration to find the solution numerically