Visualization of Complex Projective Curves

Abstract

We introduce a nonlinear map $\alpha :{\mathbb{C}}^2\to {\mathbb{R}}^3$ with the purpose of visualizing curves. Basic properties of $\alpha$ are proved, including preservation of orthogonality, recovery of the magnitudes of vectors in the preimage, and continuous extension of $\alpha$ to $\tilde{\alpha }:{\mathbb{P}}_{\mathbb{C}}^2\to {\mathbb{R}}^3$. For plane curves $Z\subset {\mathbb{P}}_{\mathbb{C}}^2$, it is proved that $\tilde{\alpha }(Z)$ is the union of boundaries of star-shaped domains. Methods are established to descend finite-order automorphisms of smooth projective curves to rotations of their images in ${\mathbb{R}}^3$. Efficient techniques for creating meshes and ray-traced images of $\tilde{\alpha }(Z)$ are developed.

arXiv comment18 pages, 9 figures