椭球面积计算程序

Nuclear Instruments and Methods in Physics Research

物理研究中的核仪器与方法

We present a geometric method for computing an ellipse that subtends the same solid-angle domain as an arbitrarily positioned ellipsoid. With this method we can extend existing analytical solid-angle calculations of ellipses to ellipsoids. Our idea consists of applying a linear transformation on the ellipsoid such that it is transformed into a sphere from which a disk that covers the same solid-angle domain can be computed. We demonstrate that by applying the inverse linear transformation on this disk we obtain an ellipse that subtends the same solid-angle domain as the ellipsoid. We provide a MATLAB implementation of our algorithm and we validate it numerically.

我们提出了一种计算椭圆的几何方法，该椭圆包含与任意放置的椭球相同的立体角域。 使用这种方法，我们可以将椭圆的现有分析立体角计算扩展到椭圆体。 我们的想法包括在椭球上应用线性变换，以便将其变换为一个球体，从该球体可以计算出覆盖相同立体角域的圆盘。 我们证明了通过在此磁盘上应用逆线性变换，我们得到一个椭圆，该椭圆与椭圆体对着相同的立体角域。 我们提供了算法的MATLAB实现，并进行了数值验证。

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动机 (Motivation)

Ellipsoids are often used in rendering applications, for instance as proxies for occluders or as area lights. In these applications, we need to compute the solid angle covered by the ellipsoid in respect to the shading point. However, no closed-form expression was available for this and approximations were used. We were able to solve this problem thanks to our recent research on linear transformations and spherical distributions.

椭球体通常用于渲染应用程序，例如作为遮挡物的代理或区域照明。 在这些应用中，我们需要计算相对于阴影点的椭球所覆盖的立体角。 但是，没有封闭形式的表达式可用于此，并且使用了近似值。 由于我们对线性变换和球面分布的最新研究 ， 我们得以解决此问题。

We published this result in the physics journal Nuclear Instruments and Methods in Physics Research because we thought that this result might be interesting for the physics community as well. Indeed, solid angles are used for computing the efficiency of detectors or antennas and many previous results on solid-angle computations were actually published by the physics community. It is somehow amusing to see that video-game research can also contribute to advance other fields!

我们将此结果发表在物理学杂志《物理研究中的核仪器与方法》中，因为我们认为该结果对物理学界也可能很有趣。 确实，将立体角用于计算探测器或天线的效率，并且有关立体角计算的许多先前结果实际上是由物理学界发布的。 有趣的是，看到视频游戏研究也可以促进其他领域的发展！

翻译自: https://blogs.unity3d.com/2017/02/08/analytical-calculation-of-the-solid-angle-subtended-by-an-arbitrarily-positioned-ellipsoid-to-a-point-source/

椭球面积计算程序