WindingNumber

Some Math Concept:

Solid Angle $\Omega(\bold{p})$

定义： $$ \Omega(\bold{p}) = \iint_\mathcal{S}\sin(\phi)\mathrm{d}\theta\mathrm{d}\phi $$ 可直观理解为将 $\mathcal{S}$ 投影到以 $\bold{p}$ 为圆心的单位圆上的圆上的投影面积，$\Omega(\bold{p})\in[-4\pi, 4\pi]$。

对于三角形面的数值计算：

Let a = vi − p, b = vj − p, c = vk − p and a = ‖a‖, b = ‖b‖, c = ‖c‖ $$ \tan\left(\dfrac{\Omega(\bold{p})}{2}\right) = \frac{\det([\bold{a}, \bold{b}, \bold{c}])}{abc + (\bold{a}\cdot \bold{b})c + (\bold{b}\cdot \bold{c})a+(\bold{c}\cdot \bold{a})b} $$ Some Related Work:

Jacobson, Alec, Ladislav Kavan, and Olga Sorkine-Hornung. “Robust Inside-Outside Segmentation Using Generalized Winding Numbers.” ACM Transactions on Graphics 32, no. 4 (July 21, 2013): 1–12. https://doi.org/10.1145/2461912.2461916.

Jacobson, Gavin Barill, Neil Dickson, Ryan Schmidt, David I. W. Levin, Alec. “Fast Winding Numbers for Soups and Clouds.” Accessed September 27, 2022. https://www.dgp.toronto.edu/projects/fast-winding-numbers/.

Some Learning Notes:

Basic Eigen

实际处理时的特殊情况：

当 $[\bold{a}, \bold{b}, \bold{c}]$ 共线时，$\tan\left(\dfrac{\Omega(\bold{p})}{2}\right) = 0$

由于 C 语言中 atan2 函数的返回定义域为 $[-\pi, \pi]$，具体如下

• If y is ±0 and x is negative or -0, ±π is returned
• If y is ±0 and x is positive or +0, ±0 is returned