Extrinsic Euler characteristic of a non-immersed hypersurface (focal surfaces, pedal images, and extrinsic Morse indices)

We use elementary descriptive methods to extend the Boy–Izumiya–Marar (extrinsic) Euler characteristic formula for the image of a compact 2-manifold mapped into a 3-manifold subject to regularity conditions at the singular points of the mapping. In addition to Boy’s “terminating double points” and triple points we allow stable conical singularities which occur generically in focal surfaces (i.e., Lagrangian critical images) and in pedal surfaces in Euclidean 3-space. As a consequence we derive a Morse theoretic formula for this extrinsic Euler characteristic in terms of a height function, and its extrinsic Morse indices.