# Ellipsoid in three-dimensional Euclidean space

This surface type is not unique up to isometry or even up to similarity transformations, but rather, depends upon multiple nonzero parameters $a,b,c$. If we're considering the surface up to rigid isometries, the parameters are unique up to permutations. If we're considering the surface up to similarity transformations, the parameters are unique up to the action of permutations and projective equivalence.
Up to rotations $\frac{(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} + \frac{z - z_0)^2}{c^2} = 1$
Up to rigid motions (rotations, translations, reflections) $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
Up to similarity transformations $x^2 + \frac{y^2}{q^2} + \frac{z^2}{r^2} = 1$ Here, $q = b/a, r = c/a$