Ellipsoid in three-dimensional Euclidean space

Definition

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.

Implicit and parametric descriptions

Degree of generality	Implicit description	What the parameters mean	Parametric description	What the additional parameters mean	Comment
Arbitrary	Fill this in later
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, $q=b/a,r=c/a$