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Diffgeom β

Ellipsoid in three-dimensional Euclidean space

Contents

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 \ge b \ge 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 \ge b \ge c, we would get 0 < q,r \le 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 \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,\R) and the orientation-preserving isometry group is the special orthogonal group SO(3,\R).

Verification of theorems

Gauss-Bonnet theorem

Fill this in later