Editing Ellipsoid (section)

=== 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>