Editing Ellipsoid (section)

== In higher dimensions and general position {{anchor|In higher dimensions|Ellipsoids in higher dimensions and general position}} ==


A '''hyperellipsoid''', or ellipsoid of dimension <math>n - 1</math> in a [[Euclidean space]] of dimension <math>n</math>, is a [[quadric hypersurface]] defined by a polynomial of degree two that has a [[homogeneous polynomial|homogeneous part]] of degree two which is a [[positive definite quadratic form]].

One can also define a hyperellipsoid as the image of a sphere under an invertible [[affine  transformation]]. The spectral theorem can again be used to obtain a standard equation of the form 
:<math>\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}+\cdots + \frac{x_n^2}{a_n^2}=1.</math>

The volume of an {{mvar|n}}-dimensional ''hyperellipsoid'' can be obtained by replacing {{mvar|R<sup>n</sup>}} by the product of the semi-axes {{math|''a''<sub>1</sub>''a''<sub>2</sub>...''a<sub>n</sub>''}} in the formula for the [[Volume of an n-ball#The volume|volume of a hypersphere]]:
:<math>V = \frac{\pi^\frac{n}{2}}{\Gamma{\left(\frac{n}{2} + 1\right)}} a_1a_2\cdots a_n \approx  \frac{1}{\sqrt{\pi n}} \cdot \left(\frac{2 e \pi}{n}\right)^{n/2} a_1a_2\cdots a_n </math>
(where {{math|&Gamma;}} is the [[gamma function]]).

=== As a quadric ===
If {{mvar|'''A'''}} is a real, symmetric, {{mvar|n}}-by-{{mvar|n}} [[positive-definite matrix]], and {{mvar|'''v'''}} is a vector in <math>\R^n,</math> then the set of points {{math|'''x'''}} that satisfy the equation
:<math>(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) = 1</math>
is an ''n''-dimensional ellipsoid centered at {{mvar|'''v'''}}. The expression <math>(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) </math> is also called the '''ellipsoidal norm''' of {{math|'''x''' − '''v'''}}. For every ellipsoid, there are unique {{mvar|'''A'''}} and {{math|'''v'''}} that satisfy the above equation.<ref name=":0" />{{Rp|page=67|location=}}

The [[eigenvector]]s of {{mvar|'''A'''}} are the principal axes of the ellipsoid, and the [[eigenvalue]]s of {{mvar|'''A'''}} are the reciprocals of the squares of the semi-axes (in three dimensions these are {{math|''a''<sup>−2</sup>}}, {{math|''b''<sup>−2</sup>}} and {{math|''c''<sup>−2</sup>}}).<ref>{{cite web |url=http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf |title=Lecture 15 – Symmetric matrices, quadratic forms, matrix norm, and SVD |access-date=2013-10-12 |url-status=live |archive-url=https://web.archive.org/web/20130626233838/http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf |archive-date=2013-06-26}} pp. 17–18.</ref> In particular:

* The [[diameter]] of the ellipsoid is twice the longest semi-axis, which is twice the square-root of the reciprocal of the largest eigenvalue of {{mvar|'''A'''}}.
* The [[width]] of the ellipsoid is twice the shortest semi-axis, which is twice the square-root of the reciprocal of the smallest eigenvalue of {{mvar|'''A'''}}.

An invertible [[linear transformation]] applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable [[rotation]], a consequence of the [[polar decomposition]] (also, see [[spectral theorem]]). If the linear transformation is represented by a [[symmetric matrix|symmetric 3&nbsp;×&nbsp;3 matrix]], then the eigenvectors of the matrix are orthogonal (due to the [[spectral theorem]]) and represent the directions of the axes of the ellipsoid; the lengths of the semi-axes are computed from the eigenvalues. The [[singular value decomposition]] and [[polar decomposition]] are matrix decompositions closely related to these geometric observations.

For every positive definite matrix <math>\boldsymbol{A}</math>, there exists a unique positive definite matrix denoted {{math|'''''A'''''<sup>1/2</sup>}}, such that  <math>\boldsymbol{A} = \boldsymbol{A}^{1/ 2}\boldsymbol{A}^{1/ 2}; </math> this notation  is motivated by the fact that this matrix can be seen as the "positive square root" of <math>\boldsymbol{A}.</math> The ellipsoid defined by <math>(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) = 1</math> can also be presented as<ref name=":0">{{Cite Geometric Algorithms and Combinatorial Optimization}}</ref>{{Rp|page=67|location=}}<blockquote><math>A^{-1/2}\cdot S(\mathbf{0},1) + \mathbf{v}</math></blockquote>where S('''0''',1) is the [[unit sphere]] around the origin.

=== Parametric representation ===
[[File:Ellipsoid-affin.svg|300px|thumb|ellipsoid as an affine image of the unit sphere]]
The key to a parametric representation of an ellipsoid in general position is the alternative definition:
: ''An ellipsoid is an affine image of the unit sphere.''

An [[affine transformation]] can be represented by a translation with a vector {{math|'''f'''<sub>0</sub>}} and a regular 3&nbsp;×&nbsp;3 matrix {{math|'''''A'''''}}:
: <math>\mathbf x \mapsto \mathbf f_0 + \boldsymbol A \mathbf x = \mathbf f_0 + x\mathbf f_1 + y\mathbf f_2 + z\mathbf f_3</math>

where {{math|'''f'''<sub>1</sub>, '''f'''<sub>2</sub>, '''f'''<sub>3</sub>}} are the column vectors of matrix {{math|'''''A'''''}}.

A parametric representation of an ellipsoid in general position can be obtained by the parametric representation of a unit sphere (see above) and an affine transformation:
: <math>\mathbf x(\theta, \varphi) = \mathbf f_0 + \mathbf f_1 \cos\theta \cos\varphi + \mathbf f_2 \cos\theta \sin\varphi + \mathbf f_3 \sin\theta, \qquad -\tfrac{\pi}{2} < \theta < \tfrac{\pi}{2},\quad 0 \le \varphi < 2\pi</math>.

If the vectors {{math|'''f'''<sub>1</sub>, '''f'''<sub>2</sub>, '''f'''<sub>3</sub>}} form an orthogonal system, the six points with vectors {{math|'''f'''<sub>0</sub> ± '''f'''<sub>1,2,3</sub>}} are the vertices of the ellipsoid and {{math|{{abs|'''f'''<sub>1</sub>}}, {{abs|'''f'''<sub>2</sub>}}, {{abs|'''f'''<sub>3</sub>}}}} are the semi-principal axes.

A surface normal vector at point {{math|'''x'''(''&theta;'', ''&phi;'')}} is
: <math>\mathbf n(\theta, \varphi) = \mathbf f_2 \times \mathbf f_3\cos\theta\cos\varphi + \mathbf f_3 \times \mathbf f_1\cos\theta\sin\varphi + \mathbf f_1 \times \mathbf f_2\sin\theta.</math>

For any ellipsoid there exists an [[Implicit surface|implicit representation]] {{math|''F''(''x'', ''y'', ''z'') {{=}} 0}}. If for simplicity the center of the ellipsoid is the origin, {{math|'''f'''<sub>0</sub> {{=}} '''0'''}}, the following equation describes the ellipsoid above:<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf ''Computerunterstützte Darstellende und Konstruktive Geometrie.''] {{webarchive |url=https://web.archive.org/web/20131110190049/http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf |date=2013-11-10}} Uni Darmstadt (PDF; 3,4&nbsp;MB), S.&nbsp;88.</ref>
: <math>F(x, y, z) = \operatorname{det}\left(\mathbf x, \mathbf f_2, \mathbf f_3\right)^2 + \operatorname{det}\left(\mathbf f_1,\mathbf x, \mathbf f_3\right)^2 + \operatorname{det}\left(\mathbf f_1, \mathbf f_2, \mathbf x\right)^2 - \operatorname{det}\left(\mathbf f_1, \mathbf f_2, \mathbf f_3\right)^2 = 0</math>