非线性生态圈 [英] Non-linear Icosphere

问题描述

我需要一个所有边都相同长度的 icosphere.目前我从二十面体开始生成一个，然后线性插入每个三角形.我能找到的所有教程都使用相同的线性插值.后来有人建议只使用来自 Blender 的 icosphere，虽然也使用线性插值...

I need an icosphere with all edges the same length. Currently I generate one starting from an icosahedron and then linearly interpolate each triangle. All tutorials I could find used the same linear interpolation. Someone later suggested just using the icosphere from blender, which works, though also uses the linear interpolation...

所以有人有关于如何制作 icosphere 并确保所有边都相等的很好的教程吗?(如果你知道一个可以制作这个并将其导出为 .obj 或其他格式的程序也可以.)

So anyone has a good tutorial on how to make an icosphere and make sure all the sides are equal? (It would also be ok if you know a program which can make this and export it as a .obj or something.)

推荐答案

这不可能从等边三角形得到任意精确的球体.如果你需要等边三角形，你能得到的最好的就是二十面体(你可能已经有了).原因叫做角度缺陷":

This is not possible to have an arbitrarily precise sphere from equilateral triangles. If you require equilateral triangles, the best you can get is the icosahedron (which you probably already have). The reason is called "angle defect":

角度缺陷是一个顶点周围的顶点角度之和 360 度.任何多面体必须具有等于 720 度的总顶点缺陷:

The angle defect is 360 degrees the sum of vertex angles around a vertex. Any polyhedron must have a total vertex defect equal to 720 degrees:

• 四面体:4*180 度
• 八面体:6*120 度
• 立方体:8*90 度
• 二十面体:12*60 度
• 十二面体:20*36 度

如果五个等边三角形在一个顶点相交，它们会累积 36 度的缺陷.如果六个等边三角形在一个顶点相交，它们对缺陷没有任何贡献.

If five equilateral triangles meet at a vertex, they accumulate 36 degrees of defect. If six equilateral triangles meet at a vertex, they contribute nothing to the defect.

如果六个等边三角形在一个顶点相交，它们必须形成其中一个

If six equilateral triangles meet at a vertex, they must form either

• 一架飞机或
• 折叠(边缘)或
• 一个马鞍.

有一个叫做五角二十面体的多面体，有 32 个顶点，但如果你用等边三角形制作它，你会得到一个非凸多面体.

There is a polyhedron called pentakis icosahedron with 32 vertices but if you make it out of equilateral triangles, you'll get a non-convex polyhedron.

因此，如果您想要一个任意精确的球体，您需要允许不同长度的边，以便角度缺陷可以在整个表面上均匀分布.任意精度的等边是不可能的.

So, if you want an arbitrarily precise sphere, you need to allow different length edges so that the angle defect may distribute evenly across the surface. Equal sides with arbitrary precision are impossible.

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