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 ℂ2 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 Γθ, w and establish that their singular cusp locus sweeps out the complex evolutoid Eθ. Using a square-root-free formulation, we classify the local behavior of Eθ, 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 𝒟F ⊂ ℂ3 formed by the evolving family Eθ. We prove that A2 cusps are versally unfolded by spatial parameters, whereas A3 swallowtail singularities are holomorphically versally unfolded by (x1, x2, θ) if and only if the complex non-degeneracy condition κ4 + κs2 ≠ 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.
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BibTeX
@misc{deolindosilva2026complex,
title = {{Complex evolutoids of holomorphic plane curves}},
author = {Jorge Luiz Deolindo-Silva and Mostafa Salarinoghabi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38546},
eprint = {2609.38546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38546},
}
RIS
TY - UNPB
TI - Complex evolutoids of holomorphic plane curves
AU - Jorge Luiz Deolindo-Silva
AU - Mostafa Salarinoghabi
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38546
UR - https://arxiv.org/abs/2609.38546
AB - We study the local geometry, singularity classification, and wavefront dynamics of complex evolutoids for regular holomorphic curves in 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 𝒟_F ⊂ 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 ≠ 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.
ER -