Iso-Optic Plane of the Regular Tetrahedron

This Demonstration shows the points in space that subtend an angle from a regular tetrahedron with vertices .

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DETAILS

In the plane, it is easy to show those points from which a segment subtends an angle because they form a circle. However, for a segment in space, the points subtended form a torus, where (i.e., the torus intersects itself). The long derivation for a tetrahedron is not shown; only the result is used.
You can select between and .
In the first snapshot you can see that if is small, the figure looks like a sphere; the second snapshot shows the general case; and in the third snapshot spheres appear because is .
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