Solving the Alhazen–Ptolemy Problem: Determining Specular Points on Spherical Surfaces for Radiative Transfer of Titan’s Seas
Solving the Alhazen–Ptolemy Problem: Determining Specular Points on Spherical Surfaces for Radiative Transfer of Titan’s Seas
复制标题
解决阿尔哈森-托勒密问题：确定泰坦海洋辐射传输的球面上的镜面点
DOI：
10.3847/psj/abe4dd
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发表时间：
2020
影响因子：
--
通讯作者：
S. MacKenzie
中科院分区：
文献类型：
--
作者：
W. Miller;J. Barnes;S. MacKenzie
给定一个光源、一个球面反射器和一个观察者，光在球体表面的什么地方直接反射到观察者，即镜面反射点在哪里？这被称为Alhazen-Ptolemy问题，找到球面反射器的这个镜面反射点在从计算机渲染到大气建模到GPS通信的各种应用中都很有用。现有的解决方案依赖于寻找四次方程的根，并用数值计算哪个根提供真实的镜面反射点。我们提供了一个公式及其两个解，其正确的根是预先确定的，从而允许构造我们所给出的完全解析解。由于计算速度更快，我们的解决方案在需要重复计算镜面反射点的情况下应该被证明是有用的，例如蒙特卡罗辐射转移，包括土卫六碳氢化合物海的反射。
Given a light source, a spherical reflector, and an observer, where on the surface of the sphere will the light be directly reflected to the observer, i.e., where is the specular point? This is known as the Alhazen–Ptolemy problem and finding this specular point for spherical reflectors is useful in applications ranging from computer rendering to atmospheric modeling to GPS communications. Existing solutions rely upon finding the roots of a quartic equation and evaluating numerically which root provides the real specular point. We offer a formulation, and two solutions thereof, for which the correct root is predeterminable, thereby allowing the construction of the fully analytical solutions we present. Being faster to compute, our solutions should prove useful in cases that require repeated calculation of the specular point, such as Monte Carlo radiative transfer, including reflections off of Titan’s hydrocarbon seas.