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x2/a2 + y2/b2 + z2/a2 - 1 = 0 (equation 1)

where a and b are the semi-major and semi-minor axes, respectively. If you have any two of x,y,z-values, you can solve for the third, simply by rearranging the above equation:

x = +/- sqrt(1 - y2/b2 - z2/a2)*a

y = +/- sqrt(1 - x2/a2 - z2/a2)*b

z = +/- sqrt(1 - x2/a2 - y2/b2)*a (equations 2 thru 4)

Once you accumulate a large number of (x,y,z) points covering the spheroid surface, you can rotate and translate them in cartesian space. Each point moves to a new position, defined generally by:

x'' = M1*x + M2*y + M3*z + M4

y'' = M5*x + M6*y + M7*z + M8

z'' = M9*x + M10*y + M11*z + M12 (equations 5 thru 7)

My question is, if I want to find the implicit equation of the spheroid after it has been rotated & translated, do I substitute equations 2-4 into equations 5-7, and then substitute 5-7 into equation 1 ? Or do I just substitute equations 5-7 into equation 1 ? Or am I totally lost and using the wrong approach?