ABCD transfer matrices and paraxial ray tracing for elliptic and hyperbolic lenses and mirrors

A method is introduced for evaluating the ABCD transfer matrices for paraxial rays refracted/reflected by lenses/mirrors with elliptic or hyperbolic surfaces, which relies on the use of certain cusp points of the evolutes of the curves. Paraxial ray tracing techniques are developed. Applications examined include early hypothetical refracting telescopes by Descartes and Gregory