Title: An explanation of the number or points and symmetries of starbursts Show Chinese title

Title: 星爆的点数和对称性的解释

Abstract: Starbursts are the light intensity patterns seen when small bright sources are looked at night, typically stars. Starburst shapes are produced when the presence of the eye's wave aberrations generates caustics (light concentration) at the retina. A fascinating, but never explained fact about starbursts is that they usually present a $p$-fold symmetry pattern. We provide a theoretical explanation of the number of points and symmetries of starbursts, based on the geometric and algebraic properties of the wave aberration function expressed as a Zernike polynomial expansion. Specifically, we investigate the number and distribution of saddle cusps of Gauss of the Hessian of the wave aberration function. We also establish the connections between those points with the symmetries and the number of starburst points. We found that starbursts are likely generated by axially symmetric dominated wave aberrations with some amount of non-axially symmetric terms. For instance, whereas a wave aberration with a dominant spherical aberration (Zernike polynomial $Z_4^{0}$) plus $Z_3^{3}$ may induce a $3$ points starburst with a $3$-fold symmetry, a wave aberration combining $Z_4^{0}$ and $Z_4^{4}$ may induce a $4$-fold symmetry starburst with $4$ or $8$ points. Show Chinese abstract

Abstract: 当夜间观察小而明亮的光源时，会看到星爆状的光强模式，通常为星星。 星爆形状是当眼睛的波前像差存在时，在视网膜上产生焦散线（光集中）所致。 一个有趣但从未被解释的事实是，星爆通常呈现出$p$-fold 对称性图案。 我们提供了一种理论解释，说明星爆的点数和对称性的数量，这是基于波前像差函数作为泽尼克多项式展开的几何和代数性质。 具体而言，我们研究了波前像差函数的Hessian矩阵的高斯马鞍尖点的数量和分布。 我们还建立了这些点与对称性和星爆点数量之间的联系。 我们发现，星爆很可能是由轴对称主导的波前像差产生的，同时包含一定量的非轴对称项。 例如， whereas 一个以球面像差（Zernike多项式$Z_4^{0}$）为主并加上$Z_3^{3}$的波前像差可能会引起具有$3$点星爆且具有$3$-重对称性的现象，而一个结合$Z_4^{0}$和$Z_4^{4}$的波前像差可能会引起具有$4$-重对称性的星爆现象，且有$4$或$8$个点。