I have an oblate spheroid (E): (x/a)^2+(y/a)^2+(z/b)^2 = 1 and a plane (P): ux+vy+wz+d=0 / d=0 (passing by origin)
I have plugged z = - (ux+vy)/w from (P) into (E), then I became this equation:
[(wb)^2+(au)^2]x^2 + [(wb)^2+(av)^2]y^2 + (2*uva^2)xy -(wab)^2 = 0
That is the equation of conics (here in this case, an ellipse, bcz intersection of an plane passing by origin with an oblate spheroid is always an ellipse except when it is parallel to plane xy is a circle)
General equation of conics:
Ax^2 + Bxy + Cy^2 + F = 0 [Dx and Ey vanish bcz the ellipse centered always at O(0,0,0)]
I have rotated and everything done, the I got this form:
A'x^2 + C'y^2 - F' = 0 with A' = a^2*[u^2 + v^2] + (bw)^2 C' = (bw)^2 F' = (abw)^2
When I continue to find the semi-axis major a' and semi-axis minor b' of this ellipse, I get:
a' = a and b' = (abw)/sqrt[a^2*(u^2 + v^2) + (b*w)^2]
My problem, is: the semi-axis major a' shouldn't equal the semi-major of the oblate spheroid a at each case!!! What is wrong?!!
Thanks for any help, regards!