Determine formula to calculate the radii of a unique ellipsoid from coordinates of non-coplanar locii on its surface, and without knowing its center or rotation angles.

Determining the radius of a unique 2D inscribed circle from three points in space is possible, as is calculating the circumradius of a sphere from a four points (tetrahedral pyramid) without knowing its center. Similarly:

a. What is the minimum number of non-coplanar locii required to calculate all three radii of a unique prolate/oblate ellipsoid, not knowing the rotation angles or the center? Please justify your answer.

b. Please provide a formula to calculate all three radii of a generic ellipsoid from coordinates of non-coplanar locii on its surface radius of a generic ellipsoid, as well as formulas to determine its center and rotational angles. Along with a sketch of the solution, please provide instructions on how to do these computations on a computer.

Note: An example of its application maybe to view earth from a satellite, measuring the exactly relative locations a few cities in USA, and then calculating the three radii of earth as an oblate ellipsoid, and determine its center and rotational angles.
Algebraic Geometry Geometry Discrete Mathematics
Poincare Poincare
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