优化顶点时如何避免多边形变得复杂？
假设有一组 2D 点来表示初始简单多边形。现在我想根据一些成本函数优化每个点的位置。但这可能会使多边形变得复杂，即多边形与自身相交。我怎样才能避免这种情况？谢谢！
Suppose there's a set of 2D points to represent an initial simple polygon. Now I want to optimize the positions of each point according to some cost function. But this could make the polygon complex, i.e. the polygon intersects with itself. How can I avoid this? Thanks!
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如果可以假定多边形是凸多边形，那么它就很简单。只需计算每条边与下一条边之间的角度即可。对于凸多边形，每个角度必须介于 0 到 180 度之间。对于具有 N 条边的闭合多边形来说，这些角度的总和是众所周知的。这将导致简单的约束优化。 （实际上，您可以以比用三角函数计算角度更“简单”的形式编写这些约束。叉积就足够了。）
如果多边形不需要是凸的，那么您需要担心边交叉或其他简并性。
If the polygon could be presumed to be convex, then it is simple. Simply compute the angles between each side and the next side. Each angle must be between 0 and 180 degrees for a convex polygon. The sum of those angles is well known for a closed polygon with N sides. This will result in a simple constrained optimization. (Actually, you can write those constraints in a "simpler" form than computing the angles with a trigonometric function. Cross products will suffice.)
If the polygon need not be convex, then you need to worry about edges crossing, or other degeneracies.