Complex evolutoids of holomorphic plane curves
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 $Γ^{θ, w}$ and establish that their singular cusp locus sweeps out the complex evolutoid $E^θ$. Using a square-root-free formulation, we classify the local behavior of $E^θ$, proving that evolutoids remain regular and tangent to the curve at isotropic points ($q=0$) while escaping to infinity at complex inflections ($κ=0$). Finally, we analyze the three-dimensional complex discriminant surface $\mathcal{D}_F \subset \mathbb{C}^3$ formed by the evolving family $E^θ$. We prove that $A_2$ cusps are versally unfolded by spatial parameters, whereas $A_3$ swallowtail singularities are holomorphically versally unfolded by $(x_1, x_2, θ)$ if and only if the complex non-degeneracy condition $κ^4 + κ_s^2 \neq 0$ 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.