为平截头体寻找最小边界球 [英] Finding a minimum bounding sphere for a frustum

问题描述

我有一个截锥体(截棱锥)，我需要为这个截锥体计算一个尽可能小的边界球.我可以选择中心正好在截锥体的中心，半径是到远"角之一的距离，但这通常会在截锥体的窄端周围留下很多松弛

I have a frustum (truncated pyramid) and I need to compute a bounding sphere for this frustum that's as small as possible. I can choose the centre to be right in the centre of the frustum and the radius be the distance to one of the "far" corners, but that usually leaves quite a lot of slack around the narrow end of the frustum

这看起来像简单的几何图形，但我似乎无法弄清楚.有什么想法吗?

This seems like simple geometry, but I can't seem to figure it out. Any ideas?

推荐答案

好吧，有 http://www.cgafaq.info/wiki/Minimal_enclosure_sphere 当然(通过 Google).

Well, there's http://www.cgafaq.info/wiki/Minimal_enclosing_sphere of course (via Google).

我认为有两种可能性.一个(如果平截头体非常平坦)是底部的相对点成为球体上的相对点.另一个(如果截锥体非常高)是截锥体的相对点将在球体上，您可以从这四个点(底部上的一个点，底部上的第一个点对面，一个在较高方格上的第一个对面，一个与较高方格上的第一个相邻).

I'd think there are two possibilities. One (if the frustum is very flat) would be that the opposite points of the base become opposite points on the sphere. The other (if the frustum is very tall) would be that opposite points of the frustum would be on the sphere and you'd figure out the sphere from those four points (one point on the base, one opposite the first on the base, one opposite the first on the higher square, one adjacent the first on the higher square).

找出第一个球体.如果平截头体适合它，那就是你的答案.否则，第二个球体就是你的答案.

Figure out the first sphere. If the frustum fits in it, that's your answer. Otherwise, the second sphere would be your answer.

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