AI 中文总结
该研究引入非线性映射α将复射影曲线映射到三维欧氏空间，证明其性质并建立自同构降维方法，开发可视化技术，实现复射影曲线的有效可视化。
AI 中文摘要
我们引入非线性映射α:ℂ²→ℝ³以实现曲线可视化。证明了α的基本性质，包括正交性保持、原像中向量模长的恢复，以及α可连续延拓为α̃:ℙ²_ℂ→ℝ³。对于平面曲线Z⊂ℙ²_ℂ，证明了α̃(Z)是星形域边界的并集。建立了将光滑射影曲线的有限阶自同构降为其在ℝ³中图像旋转的方法，开发了生成α̃(Z)的网格和光线追踪图像的高效技术。
英文摘要
We introduce a nonlinear map $α:\mathbb{C}^2\rightarrow\mathbb{R}^3$ with the purpose of visualizing curves. Basic properties of $α$ are proved, including preservation of orthogonality, recovery of the magnitudes of vectors in the preimage, and continuous extension of $α$ to $\widetildeα:\mathbb{P}^2_\mathbb{C}\rightarrow\mathbb{R}^3$. For plane curves $Z\subset\mathbb{P}^2_\mathbb{C}$, it is proved that $\widetildeα(Z)$ 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 $\widetildeα(Z)$ are developed.
Comments18 pages, 9 figures
