Having an oblate spheroid (S): (x/a)^2+(y/a)^2+(z/b)^2=1 eqt(1) with a>b
and a plane not passing through the origin (P): ux+vy+wz+D=0 eqt(2)
The result of this intersection is an ellipse (except special cases circle or a point). I would get the semi-axis length (major and minor) and the coordinates of the center of this ellipse.
from eqt(2) I get z = -(ux+vy+D)/w eqt(3) then I subtitute z in eqt(1) to get an equation in general conic quadratic form Ax^2+Bxy+Cy^2+Dx+Ey+F=0
A = (b*w)^2 + (a*u)^2 B = 2*u*v*a^2 C = (b*w)^2 + (a*v)^2 D = 2*u*a^2 E = 2*v*a^2 F = (a*D)^2 - (a*b*w)^2
Numerical example: (S): (x/20)^2 + (y/20)^2 + (z/16)^2 =1 and (P): 3x -2y +4z -8 =0
A = 7696 B = -710740992 C = 5696 D = 2400 E = -1600 F = -1612800
Somehow there is an error and probably by B and F
The derivation of semi-axis major and minor lengths, the coordinates of the center from the general quadratic conic form, I get from Osmund's answer 1 and answer 2 But it seems that there is a problem by him in the equation where he derives the distance center-focus, isn't?