Images in axially symmetric gravitational lenses from elliptical sources: the elimination method

This work focuses on developing an analytical method for constructing images in gravitational lenses. Building on our previously proposed elimination method, which utilized algebraic geometry for constructing images from circular sources, this study emphasizes images from elliptical sources using axially symmetric gravitational lenses. We detail the rational parameterization of an arbitrarily located source and use our algorithm to derive an expression that determines all images for an arbitrary axially symmetric gravitational lens. To validate this expression, we considered several limiting cases leading to well-known results. Specifically, we examined two simple models: a single-point lens and a homogeneous disk lens. By placing an elliptical source at the origin of the source plane and reducing it to a circle, we reproduced all previously known images, such as the Einstein ring for a single-point lens, and a double Einstein ring for a disk lens. Additionally, we demonstrated the Construction of images in other arbitrary lens-source configurations.