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 positive 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.

The surface, however, is unique up to affine transformations, which include transformations that do not preserve the geometric structure.

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				This version need not be centered at the origin and need not be oriented parallel to the axes.
Up to rotations	${\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}+{\frac {(z-z_{0})^{2}}{c^{2}}}=1$		$x=x_{0}+a\cos u\sin v,y=y_{0}+b\sin u\sin v,z=z_{0}+c\cos v$		This version need not be centered at the origin but is oriented parallel to the axes.
Up to rigid motions (rotations, translations, reflections)	${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1$	$a$, $b$, $c$ are positive numbers that represent the lengths of the three ellipsoidal semi-axes. Note that because we allow rigid motions, we can permute and rearrange to force $a\geq b\geq c$.	$x=a\cos u\sin v,y=b\sin u\sin v,z=c\cos v$	Here, $u$ and $v$ are ellipsoidal equivalents of the angles $\theta ,\phi$ used in spherical polar coordinates.	This version is centered at the origin and oriented parallel to the axes.
Up to similarity transformations	$x^{2}+{\frac {y^{2}}{q^{2}}}+{\frac {z^{2}}{r^{2}}}=1$	Here, $q=b/a,r=c/a$. If we chose $a\geq b\geq c$, we would get $0<q,r\leq 1$.			For this version, we can normalize one of the three values $a,b,c$. In the form given here, we chose to normalize $a$.
Up to arbitrary affine transformations (not permissible if we are interested in studying geometric structure)	$x^{2}+y^{2}+z^{2}=1$				For this version, we can choose different scaling factors in each coordinate and hence scale down to the unit sphere.

Particular cases

Case	Name of surface we get	Additional comments
All parameter values $a,b,c$ are equal	2-sphere in Euclidean space	This has full rotational symmetry about every axis through its center and reflection symmetry about every plane through its center.
Two of the parameter values are equal, and they are both greater than the third	oblate spheroid	This has full rotational symmetry about the third axis, and can be obtained by taking the surface of revolution obtained from an ellipse in a plane through that axis. The ellipse is being revolved about its minor axis.
Two of the parameter values are equal, and they are both less than the third	prolate spheroid	This has full rotational symmetry about the third axis, and can be obtained by taking the surface of revolution obtained from an ellipse in a plane through that axis. The ellipse is being revolved about its major axis.

Basic topology

The ellipsoid is topologically homeomorphic, and in fact, diffeomorphic, to the 2-sphere. It is a compact regular surface and the interior region it bounds is diffeomorphic to an open unit disk in $\mathbb {R} ^{3}$.

Fundamental forms and curvatures

For the table below, we consider the parametric description that is valid for

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1$

namely the description:

$x=a\cos u\sin v,b=y\sin u\sin v,c=z\cos v$

Quantity	Meaning in general	Value
$E,F,G$ for first fundamental form using this parametrization	The Riemannian metric is given by $ds^{2}=E\,du^{2}+2F\,du\,dv+G\,dv^{2}$	Fill this in later
$e,f,g$ for second fundamental form using this parametrization	Fill this in later	Fill this in later
principal curvatures using this parametrization	eigenvalues of the shape operator	Fill this in later
mean curvature	arithmetic mean of the principal curvatures = half the trace of the shape operator	Fill this in later
Gaussian curvature	product of the principal curvatures = determinant of shape operator	Fill this in later

Sections

For the discussion here, we assume the ellipsoid is given in the form:

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1$

Below we describe the nature of sections:

Plane with which we're intersecting	Description of intersection
$x=x_{0}$	empty if $|x_{0}|>a$ a single point $(x_{0},0,0)$ if $|x_{0}|=a$. an ellipse of the form ${\frac {y^{2}}{b^{2}(1-(x_{0}/a))^{2}}}+{\frac {z^{2}}{c^{2}(1-(x_{0}/a))^{2}}}=1,x=x_{0}$ in the plane $x=x_{0}$.
$y=y_{0}$	Fill this in later
$z=z_{0}$

Automorphisms

Isometry group

For a generic ellipsoid, where the three parameters $a,b,c$ are pairwise distinct, the isometry group is an elementary abelian group of order 8 (see groupprops:elementary abelian group:E8), given by the identity automorphisms and seven other automorphisms of orer two, which includes three reflections about planes, three rotations by angles of $\pi$ about planes, and one reflection about the origin. For an ellipsoid given by the equation:

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1$

the group of isometries is given as follows:

Image of point $(x,y,z)$ on ellipsoid under isometry	Description of isometry	Orientation-preserving?
$(x,y,z)$	identity map	Yes
$(-x,y,z)$	reflection about $yz$-plane	No
$(x,-y,z)$	reflection about $xz$-plane	No
$(x,y,-z)$	reflection about $xy$-plane	No
$(-x,-y,z)$	rotation by angle of $\pi$ in $xy$-plane; equivalently, reflection about $z$-axis	Yes
$(-x,y,-z)$	rotation by angle of $\pi$ in $xz$-plane; equivalently, reflection about $y$-axis	Yes
$(x,-y,-z)$	rotation by angle of $\pi$ in $yz$-plane; equivalently, reflection about $x$-axis	Yes
$(-x,-y,-z)$	reflection about the origin	No

The subgroup of orientation-preserving isometries is elementary abelian of order four (i.e., it is a Klein four-group), comprising the ientity element and the three rotations by angles of $\pi$.

In the case that two of the three values $a,b,c$ are equal and the third is different (thus, we may have an oblate spheroid or a prolate spheroid) the isometry group is infinite and can be described as the direct product of the generalized dihedral group corresponding to the circle group and the cyclic group of order two. The orientation-preserving isometry group is simply the generalized dihedral group corresponding to the circle group.

In the case that all three values $a,b,c$ are equal, we get a 2-sphere in Euclidean space. The isometry group is the orthogonal group $O(3,\mathbb {R} )$ and the orientation-preserving isometry group is the special orthogonal group $SO(3,\mathbb {R} )$.

Verification of theorems

Gauss-Bonnet theorem

Fill this in later