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I think it would be helpful to refer to this image when trying to follow my description: http://i.imgur.com/nRXQo3W.jpg (taken from http://experilous.com/1/blog/post/procedural-planet-generation). That image shows a simple icosahedron, but I'm working with some that have potentially been subdivided many times.

What I'd like to do is find the 3D coordinates of the points labeled 2, 3, and 4 (the centroids adjacent to point 1). The information I have to work with is the coordinates of point 1, the coordinates of the three points which make up the vertices of the triangle which 1 is the center of, the coordinates of the center of the icosahedron's circumsphere, and the radius of the circumsphere.

What I've tried:

  1. Let us call the triangle which point 1 is the centroid of A.
  2. Find the midpoints of all edges of A.
  3. Find the distance between point 1 and each midpoint.
  4. Add this distance to the midpoint in the direction perpendicular to the triangle's edge.

However, this is not what I want. The new point I obtain through this method is outside of the circumsphere. It's on the line between the midpoint of an edge of A and point 1, whereas I want the point on a great circle through those two coordinates.

Is this possible to solve given the information I have? I think I might be able to solve it by finding the great circle between point 1 and a midpoint, then finding the distance along the great circle between the two points, and then finding the point that same distance away from the midpoint, which should give me 2, 3, and 4. The only problem is I've never really learned any math beyond basic vectors and have no idea where to begin regarding that. I would appreciate any help at all in solving this. Thanks!

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Reflect point 1 in the plane through the origin and two of the vertices of A.

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