Given the vector points A and B on a sphere, relative to center, I want to find point C, which would be on the other side of B seen from A, equal in distance and in the same direction. I don't know any fancy math words, so it's hard to explain in text, and even harder to google for answers. Attached is a picture of what I'm trying to find, represented in 2D. I was thinking it should be possible to identify the "rotation" offset between A and B, and apply that to B to find C, but I don't know how, or if it's possible, or if there are better ways.
Let $r$ be the radius of the sphere.
Let n be the dot product $A.B$ It will be a number between $-r^2$ and $r^2$.
Let $D$ be the point $ nB/r^2$ It will be on the same diameter as $B$ but inside the sphere.
Let $C$ be the point $2D-A$. It will be on the same straight line in 3space as A and D.