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The interpretation of gravitational microlensing events caused by planetarysystems or multiple stars is based on the n-point-mass lens model. The firstplanets detected by microlensing were well described by the two-point-massmodel of a star with one planet. By the end of 2014, four events involvingthree-point-mass lenses had been announced. Two of the lenses were stars withtwo planetary companions each; two were binary stars with a planet orbiting onecomponent. While the two-point-mass model is well understood, the same cannotbe said for lenses with three or more components. Even the range of possiblecritical-curve topologies and caustic geometries of the three-point-mass lensremains unknown. In this paper we provide new tools for mapping thecritical-curve topology and caustic cusp number in the parameter space ofn-point-mass lenses. We perform our analysis in the image plane of the lens. Weshow that all contours of the Jacobian are critical curves of re-scaledversions of the lens configuration. Utilizing this property further, weintroduce the cusp curve to identify cusp-image positions on all contourssimultaneously. In order to track cusp-number changes in caustic metamorphoses,we define the morph curve, which pinpoints the positions of metamorphosis-pointimages along the cusp curve. We demonstrate the usage of both curves on simpletwo- and three-point-mass lens examples. For the three simplest causticmetamorphoses we illustrate the local structure of the image and source planes.

Image-plane Analysis of n-point-mass Lens Critical Curves and Caustics