Complex evolutoids of holomorphic plane curves

Abstract

We study the local geometry, singularity classification, and wavefront dynamics of complex evolutoids for regular holomorphic curves in ${\mathbb{C}}^2$ equipped with the standard complex bilinear metric. By integrating the generating family of rotated complex affine lines, we derive an explicit parametrization for the 1-parameter family of complex wavefronts ${\Gamma }^{\theta ,w}$ and establish that their singular cusp locus sweeps out the complex evolutoid $E^{\theta }$. Using a square-root-free formulation, we classify the local behavior of $E^{\theta }$, proving that evolutoids remain regular and tangent to the curve at isotropic points ($q=0$) while escaping to infinity at complex inflections ($\kappa =0$). Finally, we analyze the three-dimensional complex discriminant surface ${\mathcal{D}}_F\subset {\mathbb{C}}^3$ formed by the evolving family $E^{\theta }$. We prove that $A_2$ cusps are versally unfolded by spatial parameters, whereas $A_3$ swallowtail singularities are holomorphically versally unfolded by $(x_1,x_2,\theta )$ if and only if the complex non-degeneracy condition ${\kappa }^4+{\kappa }_s^2\ne 0$ holds. Crucially, we demonstrate that simple isotropic points drive the breakdown of this versality condition, inducing degenerate, non-transverse sections of the complex swallowtail surface.