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Ellipsoid

From Wikipedia, the free encyclopedia
Quadric surface that looks like a deformed sphere
Examples of ellipsoids with equation⁠x2/a2⁠ +⁠y2/b2⁠ +⁠z2/c2⁠ = 1:
  • Sphere,a =b =c = 4,top;
  • Spheroid,a =b = 5,c = 3,bottom left;
  • Tri-axial ellipsoid,a = 4.5,b = 6;c = 3,bottom right

Anellipsoid is a surface that can be obtained from asphere by deforming it by means of directionalscalings, or more generally, of anaffine transformation.

An ellipsoid is aquadric surface;  that is, asurface that may be defined as thezero set of apolynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planarcross section is either anellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It isbounded, which means that it may be enclosed in a sufficiently large sphere.

An ellipsoid has three pairwiseperpendicularaxes of symmetry which intersect at acenter of symmetry, called the center of the ellipsoid. Theline segments that are delimited on the axes of symmetry by the ellipsoid are called theprincipal axes, or simply axes of the ellipsoid. If the three axes have different lengths, the figure is atriaxial ellipsoid (rarelyscalene ellipsoid), and the axes are uniquely defined.

If two of the axes have the same length, then the ellipsoid is anellipsoid ofrevolution, also called aspheroid. In this case, the ellipsoid is invariant under arotation around the third axis, and there are thus infinitely many ways of choosing the two perpendicular axes of the same length. In the case of two axes being the same length:

  • If the third axis is shorter, the ellipsoid is a sphere that has been flattened (called anoblate spheroid).
  • If the third axis is longer, it is a sphere that has been lengthened (called aprolate spheroid).

If the three axes have the same length, the ellipsoid is a sphere.

Standard equation

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The general ellipsoid, also known as triaxial ellipsoid, is a quadratic surface which is defined inCartesian coordinates as:

x2a2+y2b2+z2c2=1,{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1,}

wherea{\displaystyle a},b{\displaystyle b} andc{\displaystyle c} are the length of the semi-axes.

The points(a,0,0){\displaystyle (a,0,0)},(0,b,0){\displaystyle (0,b,0)} and(0,0,c){\displaystyle (0,0,c)} lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, becausea,b,c are half the length of the principal axes. They correspond to thesemi-major axis andsemi-minor axis of anellipse.

Inspherical coordinate system for which(x,y,z)=(rsin⁡θcos⁡φ,rsin⁡θsin⁡φ,rcos⁡θ){\displaystyle (x,y,z)=(r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi ,r\cos \theta )}, the general ellipsoid is defined as:

r2sin2⁡θcos2⁡φa2+r2sin2⁡θsin2⁡φb2+r2cos2⁡θc2=1,{\displaystyle {r^{2}\sin ^{2}\theta \cos ^{2}\varphi \over a^{2}}+{r^{2}\sin ^{2}\theta \sin ^{2}\varphi \over b^{2}}+{r^{2}\cos ^{2}\theta \over c^{2}}=1,}

whereθ{\displaystyle \theta } is the polar angle andφ{\displaystyle \varphi } is the azimuthal angle.

Whena=b=c{\displaystyle a=b=c}, the ellipsoid is a sphere.

Whena=b≠c{\displaystyle a=b\neq c}, the ellipsoid is a spheroid or ellipsoid of revolution. In particular, ifa=b>c{\displaystyle a=b>c}, it is anoblate spheroid; ifa=b<c{\displaystyle a=b<c}, it is aprolate spheroid.

Parameterization

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The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is

x=asin⁡θcos⁡φ,y=bsin⁡θsin⁡φ,z=ccos⁡θ,{\displaystyle {\begin{aligned}x&=a\sin \theta \cos \varphi ,\\y&=b\sin \theta \sin \varphi ,\\z&=c\cos \theta ,\end{aligned}}\,\!}

where

0≤θ≤π,0≤φ<2π.{\displaystyle 0\leq \theta \leq \pi ,\qquad 0\leq \varphi <2\pi .}

These parameters may be interpreted asspherical coordinates, whereθ is the polar angle andφ is the azimuth angle of the point(x,y,z) of the ellipsoid.[1]

Measuring from the equator rather than a pole,

x=acos⁡θcos⁡λ,y=bcos⁡θsin⁡λ,z=csin⁡θ,{\displaystyle {\begin{aligned}x&=a\cos \theta \cos \lambda ,\\y&=b\cos \theta \sin \lambda ,\\z&=c\sin \theta ,\end{aligned}}\,\!}

where

−π2≤θ≤π2,0≤λ<2π,{\displaystyle -{\tfrac {\pi }{2}}\leq \theta \leq {\tfrac {\pi }{2}},\qquad 0\leq \lambda <2\pi ,}

θ is thereduced latitude,parametric latitude, oreccentric anomaly andλ is azimuth or longitude.

Measuring angles directly to the surface of the ellipsoid, not to the circumscribed sphere,

[xyz]=R[cos⁡γcos⁡λcos⁡γsin⁡λsin⁡γ]{\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}=R{\begin{bmatrix}\cos \gamma \cos \lambda \\\cos \gamma \sin \lambda \\\sin \gamma \end{bmatrix}}\,\!}

where

R=abcc2(b2cos2⁡λ+a2sin2⁡λ)cos2⁡γ+a2b2sin2⁡γ,−π2≤γ≤π2,0≤λ<2π.{\displaystyle {\begin{aligned}R={}&{\frac {abc}{\sqrt {c^{2}\left(b^{2}\cos ^{2}\lambda +a^{2}\sin ^{2}\lambda \right)\cos ^{2}\gamma +a^{2}b^{2}\sin ^{2}\gamma }}},\\[3pt]&-{\tfrac {\pi }{2}}\leq \gamma \leq {\tfrac {\pi }{2}},\qquad 0\leq \lambda <2\pi .\end{aligned}}}

γ would begeocentric latitude on the Earth, andλ is longitude. These are true spherical coordinates with the origin at the center of the ellipsoid.[citation needed]

Ingeodesy, thegeodetic latitude is most commonly used, as the angle between the vertical and the equatorial plane, defined for a biaxial ellipsoid. For a more general triaxial ellipsoid, seeellipsoidal latitude.

Volume

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Thevolume bounded by the ellipsoid is

V=43πabc.{\displaystyle V={\tfrac {4}{3}}\pi abc.}

In terms of the principaldiametersA,B,C (whereA = 2a,B = 2b,C = 2c), the volume is

V=16πABC{\displaystyle V={\tfrac {1}{6}}\pi ABC}.

This equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of anoblate orprolate spheroid when two of them are equal.

Thevolume of an ellipsoid is⁠2/3⁠ the volume of acircumscribedelliptic cylinder, and⁠π/6⁠ the volume of the circumscribed box. Thevolumes of theinscribed and circumscribedboxes are respectively:

Vinscribed=833abc,Vcircumscribed=8abc.{\displaystyle V_{\text{inscribed}}={\frac {8}{3{\sqrt {3}}}}abc,\qquad V_{\text{circumscribed}}=8abc.}

Surface area

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See also:Area of a geodesic polygon

Thesurface area of a general (triaxial) ellipsoid is[2]

S=2πc2+2πabsin⁡(φ)(E(φ,k)sin2⁡(φ)+F(φ,k)cos2⁡(φ)),{\displaystyle S=2\pi c^{2}+{\frac {2\pi ab}{\sin(\varphi )}}\left(E(\varphi ,k)\,\sin ^{2}(\varphi )+F(\varphi ,k)\,\cos ^{2}(\varphi )\right),}

where

cos⁡(φ)=ca,k2=a2(b2−c2)b2(a2−c2),a≥b≥c,{\displaystyle \cos(\varphi )={\frac {c}{a}},\qquad k^{2}={\frac {a^{2}\left(b^{2}-c^{2}\right)}{b^{2}\left(a^{2}-c^{2}\right)}},\qquad a\geq b\geq c,}

and whereF(φ,k) andE(φ,k) are incompleteelliptic integrals of the first and second kind respectively.[3]

The surface area of this general ellipsoid can also be expressed in terms of⁠RG{\displaystyle R_{G}}⁠, one of theCarlson symmetric forms of elliptic integrals:[4]

S=4πbcRG(a2b2,a2c2,1).{\displaystyle S=4\pi bcR_{G}\left({\frac {a^{2}}{b^{2}}},{\frac {a^{2}}{c^{2}}},1\right).}

Simplifying the above formula using properties ofRG,[5] this can also be expressed in terms of the volume of the ellipsoidV:

S=3VRG(a−2,b−2,c−2).{\displaystyle S=3VR_{G}\left(a^{-2},b^{-2},c^{-2}\right).}

Unlike the expression withF(φ,k) andE(φ,k), the equations in terms ofRG do not depend on the choice of an order ona,b, andc.

The surface area of an ellipsoid of revolution (or spheroid) may be expressed in terms ofelementary functions:

Soblate=2πa2(1+c2ea2artanh⁡e),where e2=1−c2a2 and (c<a),{\displaystyle S_{\text{oblate}}=2\pi a^{2}\left(1+{\frac {c^{2}}{ea^{2}}}\operatorname {artanh} e\right),\qquad {\text{where }}e^{2}=1-{\frac {c^{2}}{a^{2}}}{\text{ and }}(c<a),}

or

Soblate=2πa2(1+1−e2eartanh⁡e){\displaystyle S_{\text{oblate}}=2\pi a^{2}\left(1+{\frac {1-e^{2}}{e}}\operatorname {artanh} e\right)}

or

Soblate=2πa2 +πc2eln⁡1+e1−e{\displaystyle S_{\text{oblate}}=2\pi a^{2}\ +{\frac {\pi c^{2}}{e}}\ln {\frac {1+e}{1-e}}}

and

Sprolate=2πa2(1+caearcsin⁡e)where e2=1−a2c2 and (c>a),{\displaystyle S_{\text{prolate}}=2\pi a^{2}\left(1+{\frac {c}{ae}}\arcsin e\right)\qquad {\text{where }}e^{2}=1-{\frac {a^{2}}{c^{2}}}{\text{ and }}(c>a),}

which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula forSoblate can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both casese may again be identified as theeccentricity of the ellipse formed by the cross section through the symmetry axis. (Seeellipse). Derivations of these results may be found in standard sources, for exampleMathworld.[6]

Approximate formula

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S≈4πapbp+apcp+bpcp3p.{\displaystyle S\approx 4\pi {\sqrt[{p}]{\frac {a^{p}b^{p}+a^{p}c^{p}+b^{p}c^{p}}{3}}}.\,\!}

Herep ≈ 1.6075 yields a relative error of at most 1.061%;[7] a value ofp =⁠8/5⁠ = 1.6 is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.

In the "flat" limit ofc much smaller thana andb, the area is approximately2πab, equivalent top = log23 ≈ 1.5849625007.

Plane sections

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See also:Earth section
Plane section of an ellipsoid

The intersection of a plane and a sphere is a circle (or is reduced to a single point, or is empty). Any ellipsoid is the image of the unit sphere under some affine transformation, and any plane is the image of some other plane under the same transformation. So, because affine transformations map circles to ellipses, the intersection of a plane with an ellipsoid is an ellipse or a single point, or is empty.[8] Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (seeCircular section).

Determining the ellipse of a plane section

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Plane section of an ellipsoid (see example)

Given: Ellipsoid⁠x2/a2⁠ +⁠y2/b2⁠ +⁠z2/c2⁠ = 1 and the plane with equationnxx +nyy +nzz =d, which have an ellipse in common.

Wanted: Three vectorsf0 (center) andf1,f2 (conjugate vectors), such that the ellipse can be represented by the parametric equation

x=f0+f1cos⁡t+f2sin⁡t{\displaystyle \mathbf {x} =\mathbf {f} _{0}+\mathbf {f} _{1}\cos t+\mathbf {f} _{2}\sin t}

(seeellipse).

Plane section of the unit sphere (see example)

Solution: The scalingu =⁠x/a⁠,v =⁠y/b⁠,w =⁠z/c⁠ transforms the ellipsoid onto the unit sphereu2 +v2 +w2 = 1 and the given plane onto the plane with equation

 nxau+nybv+nzcw=d.{\displaystyle \ n_{x}au+n_{y}bv+n_{z}cw=d.}

Letmuu +mvv +mww =δ be theHesse normal form of the new plane and

m=[mumvmw]{\displaystyle \;\mathbf {m} ={\begin{bmatrix}m_{u}\\m_{v}\\m_{w}\end{bmatrix}}\;}

its unit normal vector. Hence

e0=δm{\displaystyle \mathbf {e} _{0}=\delta \mathbf {m} \;}

is thecenter of the intersection circle and

ρ=1−δ2{\displaystyle \;\rho ={\sqrt {1-\delta ^{2}}}\;}

its radius (see diagram).

Wheremw = ±1 (i.e. the plane is horizontal), let

 e1=[ρ00],e2=[0ρ0].{\displaystyle \ \mathbf {e} _{1}={\begin{bmatrix}\rho \\0\\0\end{bmatrix}},\qquad \mathbf {e} _{2}={\begin{bmatrix}0\\\rho \\0\end{bmatrix}}.}

Wheremw ≠ ±1, let

e1=ρmu2+mv2[mv−mu0],e2=m×e1 .{\displaystyle \mathbf {e} _{1}={\frac {\rho }{\sqrt {m_{u}^{2}+m_{v}^{2}}}}\,{\begin{bmatrix}m_{v}\\-m_{u}\\0\end{bmatrix}}\,,\qquad \mathbf {e} _{2}=\mathbf {m} \times \mathbf {e} _{1}\ .}

In any case, the vectorse1,e2 are orthogonal, parallel to the intersection plane and have lengthρ (radius of the circle). Hence the intersection circle can be described by the parametric equation

u=e0+e1cos⁡t+e2sin⁡t.{\displaystyle \;\mathbf {u} =\mathbf {e} _{0}+\mathbf {e} _{1}\cos t+\mathbf {e} _{2}\sin t\;.}

The reverse scaling (see above) transforms the unit sphere back to the ellipsoid and the vectorse0,e1,e2 are mapped onto vectorsf0,f1,f2, which were wanted for the parametric representation of the intersection ellipse.

How to find the vertices and semi-axes of the ellipse is described inellipse.

Example: The diagrams show an ellipsoid with the semi-axesa = 4,b = 5,c = 3 which is cut by the planex +y +z = 5.

Pins-and-string construction

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Pins-and-string construction of an ellipse:
|S1S2|, length of the string (red)
Pins-and-string construction of an ellipsoid, blue: focal conics
Determination of the semi axis of the ellipsoid

The pins-and-string construction of an ellipsoid is a transfer of the idea constructing an ellipse using twopins and a string (see diagram).

A pins-and-string construction of anellipsoid of revolution is given by the pins-and-string construction of the rotated ellipse.

The construction of points of atriaxial ellipsoid is more complicated. First ideas are due to the Scottish physicistJ. C. Maxwell (1868).[9] Main investigations and the extension to quadrics was done by the German mathematician O. Staude in 1882, 1886 and 1898.[10][11][12] A description of the pins-and-string construction of ellipsoids and hyperboloids is contained in the bookGeometry and the Imagination byHilbert &Cohn-Vossen.[13]

Steps of the construction

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  1. Choose anellipseE and ahyperbolaH, which are a pair offocal conics:E(φ)=(acos⁡φ,bsin⁡φ,0)H(ψ)=(ccosh⁡ψ,0,bsinh⁡ψ),c2=a2−b2{\displaystyle {\begin{aligned}E(\varphi )&=(a\cos \varphi ,b\sin \varphi ,0)\\H(\psi )&=(c\cosh \psi ,0,b\sinh \psi ),\quad c^{2}=a^{2}-b^{2}\end{aligned}}} with the vertices and foci of the ellipseS1=(a,0,0),F1=(c,0,0),F2=(−c,0,0),S2=(−a,0,0){\displaystyle S_{1}=(a,0,0),\quad F_{1}=(c,0,0),\quad F_{2}=(-c,0,0),\quad S_{2}=(-a,0,0)} and astring (in diagram red) of lengthl.
  2. Pin one end of the string tovertexS1 and the other to focusF2. The string is kept tight at a pointP with positivey- andz-coordinates, such that the string runs fromS1 toP behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string fromP toF2 runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance|S1P| over any hyperbola point is at a minimum. The analogous statement on the second part of the string and the ellipse has to be true, too.
  3. Then:P is a point of the ellipsoid with equationx2rx2+y2ry2+z2rz2=1rx=12(l−a+c),ry=rx2−c2,rz=rx2−a2.{\displaystyle {\begin{aligned}&{\frac {x^{2}}{r_{x}^{2}}}+{\frac {y^{2}}{r_{y}^{2}}}+{\frac {z^{2}}{r_{z}^{2}}}=1\\&r_{x}={\tfrac {1}{2}}(l-a+c),\quad r_{y}={\textstyle {\sqrt {r_{x}^{2}-c^{2}}}},\quad r_{z}={\textstyle {\sqrt {r_{x}^{2}-a^{2}}}}.\end{aligned}}}
  4. The remaining points of the ellipsoid can be constructed by suitable changes of the string at the focal conics.

Semi-axes

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Equations for the semi-axes of the generated ellipsoid can be derived by special choices for pointP:

Y=(0,ry,0),Z=(0,0,rz).{\displaystyle Y=(0,r_{y},0),\quad Z=(0,0,r_{z}).}

The lower part of the diagram shows thatF1 andF2 are the foci of the ellipse in thexy-plane, too. Hence, it isconfocal to the given ellipse and the length of the string isl = 2rx + (a −c). Solving forrx yieldsrx =⁠1/2⁠(l −a +c); furthermorer2
y =r2
x −c2.

From the upper diagram we see thatS1 andS2 are the foci of the ellipse section of the ellipsoid in thexz-plane and thatr2
z =r2
x −a2.

Converse

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If, conversely, a triaxial ellipsoid is given by its equation, then from the equations in step 3 one can derive the parametersa,b,l for a pins-and-string construction.

Confocal ellipsoids

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IfE is an ellipsoidconfocal toE with the squares of its semi-axes

r¯x2=rx2−λ,r¯y2=ry2−λ,r¯z2=rz2−λ{\displaystyle {\overline {r}}_{x}^{2}=r_{x}^{2}-\lambda ,\quad {\overline {r}}_{y}^{2}=r_{y}^{2}-\lambda ,\quad {\overline {r}}_{z}^{2}=r_{z}^{2}-\lambda }

then from the equations ofE

rx2−ry2=c2,rx2−rz2=a2,ry2−rz2=a2−c2=b2{\displaystyle r_{x}^{2}-r_{y}^{2}=c^{2},\quad r_{x}^{2}-r_{z}^{2}=a^{2},\quad r_{y}^{2}-r_{z}^{2}=a^{2}-c^{2}=b^{2}}

one finds, that the corresponding focal conics used for the pins-and-string construction have the same semi-axesa,b,c as ellipsoidE. Therefore (analogously to the foci of an ellipse) one considers the focal conics of a triaxial ellipsoid as the (infinite many) foci and calls them thefocal curves of the ellipsoid.[14]

The converse statement is true, too: if one chooses a second string of lengthl and defines

λ=rx2−r¯x2{\displaystyle \lambda =r_{x}^{2}-{\overline {r}}_{x}^{2}}

then the equations

r¯y2=ry2−λ,r¯z2=rz2−λ{\displaystyle {\overline {r}}_{y}^{2}=r_{y}^{2}-\lambda ,\quad {\overline {r}}_{z}^{2}=r_{z}^{2}-\lambda }

are valid, which means the two ellipsoids are confocal.

Limit case, ellipsoid of revolution

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In case ofa =c (aspheroid) one getsS1 =F1 andS2 =F2, which means that the focal ellipse degenerates to a line segment and the focal hyperbola collapses to two infinite line segments on thex-axis. The ellipsoid isrotationally symmetric around thex-axis and

rx=12l,ry=rz=rx2−c2{\displaystyle r_{x}={\tfrac {1}{2}}l,\quad r_{y}=r_{z}={\textstyle {\sqrt {r_{x}^{2}-c^{2}}}}}.

Properties of the focal hyperbola

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Top: 3-axial Ellipsoid with its focal hyperbola.
Bottom: parallel and central projection of the ellipsoid such that it looks like a sphere, i.e. its apparent shape is a circle
True curve
If one views an ellipsoid from an external pointV of its focal hyperbola, then it seems to be a sphere, that is its apparent shape is a circle. Equivalently, the tangents of the ellipsoid containing pointV are the lines of a circularcone, whose axis of rotation is thetangent line of the hyperbola atV.[15][16] If one allows the centerV to disappear into infinity, one gets anorthogonalparallel projection with the correspondingasymptote of the focal hyperbola as its direction. Thetrue curve of shape (tangent points) on the ellipsoid is not a circle.
The lower part of the diagram shows on the left a parallel projection of an ellipsoid (with semi-axes 60, 40, 30) along an asymptote and on the right a central projection with centerV and main pointH on the tangent of the hyperbola at pointV. (H is the foot of the perpendicular fromV onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the originO is the circle's center; in the central case main pointH is the center.
Umbilical points
The focal hyperbola intersects the ellipsoid at its fourumbilical points.[17]

Property of the focal ellipse

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The focal ellipse together with its inner part can be considered as the limit surface (an infinitely thin ellipsoid) of thepencil of confocal ellipsoids determined bya,b forrz → 0. For the limit case one gets

rx=a,ry=b,l=3a−c.{\displaystyle r_{x}=a,\quad r_{y}=b,\quad l=3a-c.}

In higher dimensions and general position

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Ahyperellipsoid, or ellipsoid of dimensionn−1{\displaystyle n-1} in aEuclidean space of dimensionn{\displaystyle n}, is aquadric hypersurface defined by a polynomial of degree two that has ahomogeneous part of degree two which is apositive definite quadratic form.

One can also define a hyperellipsoid as the image of a sphere under an invertibleaffine transformation. The spectral theorem can again be used to obtain a standard equation of the form

x12a12+x22a22+⋯+xn2an2=1.{\displaystyle {\frac {x_{1}^{2}}{a_{1}^{2}}}+{\frac {x_{2}^{2}}{a_{2}^{2}}}+\cdots +{\frac {x_{n}^{2}}{a_{n}^{2}}}=1.}

The volume of ann-dimensionalhyperellipsoid can be obtained by replacingRn by the product of the semi-axesa1a2...an in the formula for thevolume of a hypersphere:

V=πn2Γ(n2+1)a1a2⋯an≈1πn⋅(2eπn)n/2a1a2⋯an{\displaystyle V={\frac {\pi ^{\frac {n}{2}}}{\Gamma {\left({\frac {n}{2}}+1\right)}}}a_{1}a_{2}\cdots a_{n}\approx {\frac {1}{\sqrt {\pi n}}}\cdot \left({\frac {2e\pi }{n}}\right)^{n/2}a_{1}a_{2}\cdots a_{n}}

(whereΓ is thegamma function).

As a quadric

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IfA is a real, symmetric,n-by-npositive-definite matrix, andv is a vector inRn,{\displaystyle \mathbb {R} ^{n},} then the set of pointsx that satisfy the equation

(x−v)TA(x−v)=1{\displaystyle (\mathbf {x} -\mathbf {v} )^{\mathsf {T}}\!{\boldsymbol {A}}\,(\mathbf {x} -\mathbf {v} )=1}

is ann-dimensional ellipsoid centered atv. The expression(x−v)TA(x−v){\displaystyle (\mathbf {x} -\mathbf {v} )^{\mathsf {T}}\!{\boldsymbol {A}}\,(\mathbf {x} -\mathbf {v} )} is also called theellipsoidal norm ofx −v. For every ellipsoid, there are uniqueA andv that satisfy the above equation.[18]: 67 

Theeigenvectors ofA are the principal axes of the ellipsoid, and theeigenvalues ofA are the reciprocals of the squares of the semi-axes (in three dimensions these area−2,b−2 andc−2).[19] In particular:

  • Thediameter of the ellipsoid is twice the longest semi-axis, which is twice the square-root of the reciprocal of the largest eigenvalue ofA.
  • Thewidth of the ellipsoid is twice the shortest semi-axis, which is twice the square-root of the reciprocal of the smallest eigenvalue ofA.

An invertiblelinear transformation applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitablerotation, a consequence of thepolar decomposition (also, seespectral theorem). If the linear transformation is represented by asymmetric 3 × 3 matrix, then the eigenvectors of the matrix are orthogonal (due to thespectral theorem) and represent the directions of the axes of the ellipsoid; the lengths of the semi-axes are computed from the eigenvalues. Thesingular value decomposition andpolar decomposition are matrix decompositions closely related to these geometric observations.

For every positive definite matrixA{\displaystyle {\boldsymbol {A}}}, there exists a unique positive definite matrix denotedA1/2, such thatA=A1/2A1/2;{\displaystyle {\boldsymbol {A}}={\boldsymbol {A}}^{1/2}{\boldsymbol {A}}^{1/2};} this notation is motivated by the fact that this matrix can be seen as the "positive square root" ofA.{\displaystyle {\boldsymbol {A}}.} The ellipsoid defined by(x−v)TA(x−v)=1{\displaystyle (\mathbf {x} -\mathbf {v} )^{\mathsf {T}}\!{\boldsymbol {A}}\,(\mathbf {x} -\mathbf {v} )=1} can also be presented as[18]: 67 

A−1/2⋅S(0,1)+v{\displaystyle A^{-1/2}\cdot S(\mathbf {0} ,1)+\mathbf {v} }

where S(0,1) is theunit sphere around the origin.

Parametric representation

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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.

Anaffine transformation can be represented by a translation with a vectorf0 and a regular 3 × 3 matrixA:

x↦f0+Ax=f0+xf1+yf2+zf3{\displaystyle \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}}

wheref1,f2,f3 are the column vectors of matrixA.

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:

x(θ,φ)=f0+f1cos⁡θcos⁡φ+f2cos⁡θsin⁡φ+f3sin⁡θ,−π2<θ<π2,0≤φ<2π{\displaystyle \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\leq \varphi <2\pi }.

If the vectorsf1,f2,f3 form an orthogonal system, the six points with vectorsf0 ±f1,2,3 are the vertices of the ellipsoid and|f1|, |f2|, |f3| are the semi-principal axes.

A surface normal vector at pointx(θ,φ) is

n(θ,φ)=f2×f3cos⁡θcos⁡φ+f3×f1cos⁡θsin⁡φ+f1×f2sin⁡θ.{\displaystyle \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 .}

For any ellipsoid there exists animplicit representationF(x,y,z) = 0. If for simplicity the center of the ellipsoid is the origin,f0 =0, the following equation describes the ellipsoid above:[20]

F(x,y,z)=det⁡(x,f2,f3)2+det⁡(f1,x,f3)2+det⁡(f1,f2,x)2−det⁡(f1,f2,f3)2=0{\displaystyle 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}

Applications

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The ellipsoidal shape finds many practical applications:

Geodesy
Mechanics
Crystallography

Computer science

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Lighting
Medicine
  • Measurements obtained fromMRI imaging of theprostate can be used to determine the volume of the gland using the approximationL ×W ×H × 0.52 (where 0.52 is an approximation for⁠π/6⁠)[21]

Dynamical properties

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Themass of an ellipsoid of uniformdensityρ is

m=Vρ=43πabcρ.{\displaystyle m=V\rho ={\tfrac {4}{3}}\pi abc\rho .}

Themoments of inertia of an ellipsoid of uniform density are

Ixx=15m(b2+c2),Iyy=15m(c2+a2),Izz=15m(a2+b2),Ixy=Iyz=Izx=0.{\displaystyle {\begin{aligned}I_{\mathrm {xx} }&={\tfrac {1}{5}}m\left(b^{2}+c^{2}\right),&I_{\mathrm {yy} }&={\tfrac {1}{5}}m\left(c^{2}+a^{2}\right),&I_{\mathrm {zz} }&={\tfrac {1}{5}}m\left(a^{2}+b^{2}\right),\\[3pt]I_{\mathrm {xy} }&=I_{\mathrm {yz} }=I_{\mathrm {zx} }=0.\end{aligned}}}

Fora =b =c these moments of inertia reduce to those for a sphere of uniform density.

Artist's conception ofHaumea, a Jacobi-ellipsoiddwarf planet, with its two moons

Ellipsoids andcuboids rotate stably along their major or minor axes, but not along their median axis. This can be seen experimentally by throwing an eraser with some spin. In addition,moment of inertia considerations mean that rotation along the major axis is more easily perturbed than rotation along the minor axis.[22]

One practical effect of this is that scalene astronomical bodies such asHaumea generally rotate along their minor axes (as does Earth, which is merelyoblate); in addition, because oftidal locking, moons insynchronous orbit such asMimas orbit with their major axis aligned radially to their planet.

A spinning body of homogeneous self-gravitating fluid will assume the form of either aMaclaurin spheroid (oblate spheroid) orJacobi ellipsoid (scalene ellipsoid) when inhydrostatic equilibrium, and for moderate rates of rotation. At faster rotations, non-ellipsoidalpiriform oroviform shapes can be expected, but these are not stable.

Fluid dynamics

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The ellipsoid is the most general shape for which it has been possible to calculate thecreeping flow of fluid around the solid shape. The calculations include the force required to translate through a fluid and to rotate within it. Applications include determining the size and shape of large molecules, the sinking rate of small particles, and the swimming abilities ofmicroorganisms.[23]

In probability and statistics

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Theelliptical distributions, which generalize themultivariate normal distribution and are used infinance, can be defined in terms of theirdensity functions. When they exist, the density functionsf have the structure:

f(x)=k⋅g((x−μ)Σ−1(x−μ)T){\displaystyle f(x)=k\cdot g\left((\mathbf {x} -{\boldsymbol {\mu }}){\boldsymbol {\Sigma }}^{-1}(\mathbf {x} -{\boldsymbol {\mu }})^{\mathsf {T}}\right)}

wherek is a scale factor,x is ann-dimensionalrandom row vector with median vectorμ (which is also the mean vector if the latter exists),Σ is apositive definite matrix which is proportional to thecovariance matrix if the latter exists, andg is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.[24] The multivariate normal distribution is the special case in whichg(z) = exp(−⁠z/2⁠) for quadratic formz.

Thus the density function is a scalar-to-scalar transformation of a quadric expression. Moreover, the equation for anyiso-density surface states that the quadric expression equals some constant specific to that value of the density, and the iso-density surface is an ellipsoid.

See also

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Notes

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  1. ^Kreyszig (1972, pp. 455–456).
  2. ^F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark, editors, 2010,NIST Handbook of Mathematical Functions (Cambridge University Press), Section 19.33"Triaxial Ellipsoids". Retrieved2012-01-08.
  3. ^"DLMF: 19.2 Definitions".
  4. ^"Surface Area of an Ellipsoid".analyticphysics.com. Retrieved2024-07-23.
  5. ^"DLMF: §19.20 Special Cases ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals".dlmf.nist.gov. Retrieved2024-07-23.
  6. ^Weisstein., Eric."Prolate Spheroid".Wolfram MathWorld (Wolfram Research).Archived from the original on 3 August 2017. Retrieved25 March 2018.
  7. ^Final answersArchived 2011-09-30 at theWayback Machine by Gerard P. Michon (2004-05-13). See Thomsen's formulas and Cantrell's comments.
  8. ^Albert, Abraham Adrian (2016) [1949],Solid Analytic Geometry, Dover, p. 117,ISBN 978-0-486-81026-3
  9. ^ W. Böhm:Die FadenKonstruktion der Flächen zweiter Ordnung, Mathemat. Nachrichten 13, 1955, S. 151
  10. ^Staude, O.:Ueber Fadenconstructionen des Ellipsoides. Math. Ann. 20, 147–184 (1882)
  11. ^ Staude, O.:Ueber neue Focaleigenschaften der Flächen 2. Grades. Math. Ann. 27, 253–271 (1886).
  12. ^ Staude, O.:Die algebraischen Grundlagen der Focaleigenschaften der Flächen 2. Ordnung Math. Ann. 50, 398 - 428 (1898).
  13. ^D. Hilbert & S Cohn-Vossen:Geometry and the imagination, Chelsea New York, 1952,ISBN 0-8284-1087-9, p. 20
  14. ^O. Hesse:Analytische Geometrie des Raumes, Teubner, Leipzig 1861, p. 287
  15. ^D. Hilbert & S Cohn-Vossen:Geometry and the Imagination, p. 24
  16. ^O. Hesse:Analytische Geometrie des Raumes, p. 301
  17. ^W. Blaschke:Analytische Geometrie, p. 125
  18. ^abGrötschel, Martin;Lovász, László;Schrijver, Alexander (1993),Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics, vol. 2 (2nd ed.), Springer-Verlag, Berlin,doi:10.1007/978-3-642-78240-4,ISBN 978-3-642-78242-8,MR 1261419
  19. ^"Lecture 15 – Symmetric matrices, quadratic forms, matrix norm, and SVD"(PDF).Archived(PDF) from the original on 2013-06-26. Retrieved2013-10-12. pp. 17–18.
  20. ^Computerunterstützte Darstellende und Konstruktive Geometrie.Archived 2013-11-10 at theWayback Machine Uni Darmstadt (PDF; 3,4 MB), S. 88.
  21. ^Bezinque, Adam; et al. (2018). "Determination of Prostate Volume: A Comparison of Contemporary Methods".Academic Radiology.25 (12):1582–1587.doi:10.1016/j.acra.2018.03.014.PMID 29609953.S2CID 4621745.
  22. ^Goldstein, H G (1980).Classical Mechanics, (2nd edition) Chapter 5.
  23. ^Dusenbery, David B. (2009).Living at Micro Scale, Harvard University Press, Cambridge, MassachusettsISBN 978-0-674-03116-6.
  24. ^Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.

References

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External links

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