| Sphere, {{math>a {{=}} b {{=}} c {{=}} 4}}, top; | c {{=}} 3}}, bottom left; | Tri-axial ellipsoid, {{math>a {{=}} 4.5}}, {{math | b {{=}} 6}}; {{math>c {{=}} 3}}, bottom right}}An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.An ellipsoid is a quadric surface;  that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is bounded, which means that it may be enclosed in a sufficiently large sphere.An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, or simply axes of the ellipsoid. If the three axes have different lengths, the figure is a triaxial ellipsoid (rarely scalene ellipsoid), and the axes are uniquely defined.If two of the axes have the same length, then the ellipsoid is an ellipsoid of revolution, also called a spheroid. In this case, the ellipsoid is invariant under a rotation around the third axis, and there are thus infinitely many ways of choosing the two perpendicular axes of the same length. If the third axis is shorter, the ellipsoid is an oblate spheroid; if it is longer, it is a prolate spheroid. If the three axes have the same length, the ellipsoid is a sphere.Standard equationThe general ellipsoid, also known as triaxial ellipsoid, is a quadratic surface which is defined in Cartesian coordinates as:
frac{x^2}{a^2} + frac{y^2}{b^2} + frac{z^2}{c^2} = 1,
where a, b and c are the length of the semi-axes. The points (a, 0, 0), (0, b, 0) and (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, because {{math|a, b, c}} are half the length of the principal axes. They correspond to the semi-major axis and semi-minor axis of an ellipse.In spherical coordinate system for which (x,y,z)=(rsinthetacosvarphi, rsinthetasinvarphi,rcostheta), the general ellipsoid is defined as:
{r^2sin^2thetacos^2varphiover a^2}+{r^2sin^2thetasin^2varphi over b^2}+{r^2cos^2theta over c^2}=1,
where theta is the polar angle and varphi is the azimuthal angle.When a=b=c, the ellipsoid is a sphere.When a=bneq c, the ellipsoid is a spheroid or ellipsoid of revolution. In particular, if a = b > c, it is an oblate spheroid; if a = b | < c, it is a prolate spheroid. Parameterization
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
begin{align}
x &= asin(theta)cos(varphi),
y &= bsin(theta)sin(varphi),
z &= ccos(theta),
end{align},!where
0 le theta le pi,qquad
0 le varphi < 2pi.
These parameters may be interpreted as spherical coordinates, where {{mvar|θ}} is the polar angle and {{mvar|φ}} is the azimuth angle of the point {{math|(x, y, z)}} of the ellipsoid.{{harvtxt|Kreyszig|1972|pp=455–456}}Measuring from the equator rather than a pole,
begin{align}
x &= acos(theta)cos(lambda),
y &= bcos(theta)sin(lambda),
z &= csin(theta),
end{align},!where
-tfrac{pi}2 le theta le tfrac{pi}2,qquad
0 le lambda < 2pi,
{{mvar|θ}} is the reduced latitude, parametric latitude, or eccentric anomaly and {{mvar|λ}} is azimuth or longitude.Measuring angles directly to the surface of the ellipsoid, not to the circumscribed sphere,
begin{bmatrix}
x y z
end{bmatrix} =
R begin{bmatrix}
cos(gamma)cos(lambda)
cos(gamma)sin(lambda)
sin(gamma)
end{bmatrix}
,!where
begin{align}
R ={} &frac{abc}{sqrt{c^2 left(b^2cos^2lambda + a^2sin^2lambdaright) cos^2gamma
+ a^2 b^2sin^2gamma}}, [3pt]
&-tfrac{pi}2 le gamma le tfrac{pi}2,qquad
0 le lambda < 2pi.
end{align}{{mvar|γ}} would be geocentric latitude on the Earth, and {{mvar|λ}} is longitude. These are true spherical coordinates with the origin at the center of the ellipsoid.{{citation needed|date=April 2020}}In geodesy, the geodetic 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, see ellipsoidal latitude.Volume
The volume bounded by the ellipsoid is
V = tfrac{4}{3}pi abc.
In terms of the principal diameters {{math|A, B, C}} (where {{math|A {{=}} 2a}}, {{math|B {{=}} 2b}}, {{math|C {{=}} 2c}}), the volume is
V = tfrac16 pi ABC.
This equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of an oblate or prolate spheroid when two of them are equal.The volume of an ellipsoid is {{sfrac|2|3}} the volume of a circumscribed elliptic cylinder, and {{sfrac|{{math|π}}|6}} the volume of the circumscribed box. The volumes of the inscribed and circumscribed boxes are respectively:
V_text{inscribed} = frac{8}{3sqrt{3}} abc,qquad
V_text{circumscribed} = 8abc.
Surface area
{{see also|Area of a geodesic polygon}}The surface area of a general (triaxial) ellipsoid isF.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 WEB,weblink Triaxial Ellipsoids, 2012-01-08,
S = 2pi c^2 + frac{2pi ab}{sin(varphi)}left(E(varphi, k),sin^2(varphi) + F(varphi, k),cos^2(varphi)right),
where
cos(varphi) = frac{c}{a},qquad
k^2 = frac{a^2left(b^2 - c^2right)}{b^2left(a^2 - c^2right)},qquad
a ge b ge c,
and where {{math|F(φ, k)}} and {{math|E(φ, k)}} are incomplete elliptic integrals of the first and second kind respectively.WEB,weblink DLMF: 19.2 Definitions, The surface area of this general ellipsoid can also be expressed using the {{math|RG}}, one of Carlson symmetric forms of the elliptic integrals by simply substituting the above formula to the respective definition:
S = 3VR_{G}left(a^{-2},b^{-2},c^{-2}right),
where {{math|V}} is the volume of the ellipsoid.Unlike the expression with {{math|F(φ, k)}} and {{math|E(φ, k)}}, this equation is valid for arbirary ordering of {{math|a}}, {{math|b}}, and {{math|c}}. The surface area of an ellipsoid of revolution (or spheroid) may be expressed in terms of elementary functions:
S_text{oblate} = 2pi a^2left(1 + frac{c^2}{ea^2} operatorname{artanh}eright),
qquadtext{where }e^2 = 1 - frac{c^2}{a^2}text{ and }(c < a),
or
S_text{oblate} = 2pi a^2left(1 + frac{1 - e^2}{e} operatorname{artanh}eright)
or
S_text{oblate} = 2pi a^2 + frac{pi c^2}{e}lnfrac{1+e}{1-e}and
S_text{prolate} = 2pi a^2left(1 + frac{c}{ae} arcsin eright)
qquadtext{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 for {{math|Soblate}} can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases {{mvar|e}} may again be identified as the eccentricity of the ellipse formed by the cross section through the symmetry axis. (See ellipse). Derivations of these results may be found in standard sources, for example Mathworld.WEB,weblink Prolate Spheroid, Weisstein, Eric, W., mathworld.wolfram.com, 25 March 2018, live,weblink" title="web.archive.org/web/20170803085757weblink">weblink 3 August 2017, Approximate formula
S approx 4pi sqrt[p]{frac{a^p b^p + a^p c^p + b^p c^p}{3}}.,!
Here {{math|p ≈ 1.6075}} yields a relative error of at most 1.061%;Final answers {{webarchive |url=https://web.archive.org/web/20110930084035weblink |date=2011-09-30}} by Gerard P. Michon (2004-05-13). See Thomsen's formulas and Cantrell's comments. a value of {{math|1=p = {{sfrac|8|5}} = 1.6}} is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.In the "flat" limit of {{mvar|c}} much smaller than {{mvar|a}} and {{mvar|b}}, the area is approximately {{math|2πab}}, equivalent to {{math|1=p = log23 ≈ 1.5849625007}}. Plane sections
{{see also|Earth section}}(File:Ellipsoid-ebener-Schnitt.svg|300px|thumb|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.{{citation|first=Abraham Adrian|last=Albert|title=Solid Analytic Geometry|year=2016|orig-year=1949|publisher=Dover|isbn=978-0-486-81026-3|page=117}} Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see Circular section). Determining the ellipse of a plane section
(File:Ellipso-eb-beisp.svg|thumb|Plane section of an ellipsoid (see example))Given: Ellipsoid {{math|{{sfrac|x2|a2}} + {{sfrac|y2|b2}} + {{sfrac|z2|c2}} {{=}} 1}} and the plane with equation {{math|nxx + nyy + nzz {{=}} d}}, which have an ellipse in common.Wanted: Three vectors {{math|f0}} (center) and {{math|f1}}, {{math|f2}} (conjugate vectors), such that the ellipse can be represented by the parametric equation
mathbf x = mathbf f_0 + mathbf f_1cos t + mathbf f_2sin t
(see ellipse).(File:Ellipso-eb-ku.svg|300px|thumb|Plane section of the unit sphere (see example))Solution: The scaling {{math|1=u = {{sfrac|x|a}}, v = {{sfrac|y|b}}, w = {{sfrac|z|c}}}} transforms the ellipsoid onto the unit sphere {{math|u2 + v2 + w2 {{=}} 1}} and the given plane onto the plane with equation
n_x au + n_y bv + n_z cw = d.
Let {{math|muu + mvv + mww {{=}} δ}} be the Hesse normal form of the new plane and
;mathbf m = begin{bmatrix} m_u m_v m_w end{bmatrix};
mathbf e_0 = delta mathbf m ;
is the center of the intersection circle and
;rho = sqrt{1 - delta^2};
mathbf e_1 = begin{bmatrix} rho 0 0 end{bmatrix},qquad mathbf e_2 = begin{bmatrix} 0 rho 0 end{bmatrix}.
Where {{math|mw ≠±1}}, let
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 vectors {{math|e1, e2}} are orthogonal, parallel to the intersection plane and have length {{mvar|ρ}} (radius of the circle). Hence the intersection circle can be described by the parametric equation
;mathbf u = mathbf e_0 + mathbf e_1cos t + mathbf e_2sin t;.
Pins-and-string construction
(File:Ellipse-gaertner-k.svg|upright=1|thumb|Pins-and-string construction of an ellipse:{{math|{{abs|S1 S2}}}}, length of the string (red))(File:Fokalks-ellipsoid.svg|thumb|upright=1.2|Pins-and-string construction of an ellipsoid, blue: focal conics)(File:Fokalks-ellipsoid-xyz.svg|thumb|upright=1.2|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 two pins and a string (see diagram).A pins-and-string construction of an ellipsoid of revolution is given by the pins-and-string construction of the rotated ellipse. The construction of points of a triaxial ellipsoid is more complicated. First ideas are due to the Scottish physicist J. C. Maxwell (1868). W. Böhm: Die FadenKonstruktion der Flächen zweiter Ordnung, Mathemat. Nachrichten 13, 1955, S. 151 Main investigations and the extension to quadrics was done by the German mathematician O. Staude in 1882, 1886 and 1898.Staude, O.: Ueber Fadenconstructionen des Ellipsoides. Math. Ann. 20, 147–184 (1882) Staude, O.: Ueber neue Focaleigenschaften der Flächen 2. Grades. Math. Ann. 27, 253–271 (1886). Staude, O.: Die algebraischen Grundlagen der Focaleigenschaften der Flächen 2. Ordnung Math. Ann. 50, 398 - 428 (1898). The description of the pins-and-string construction of ellipsoids and hyperboloids is contained in the book Geometry and the imagination written by D. Hilbert & S. Vossen,D. Hilbert & S Cohn-Vossen: Geometry and the imagination, Chelsea New York, 1952, {{ISBN|0-8284-1087-9}}, p. 20 . too. Steps of the construction
- Choose an ellipse {{mvar|E}} and a hyperbola {{mvar|H}}, which are a pair of focal conics: begin{align}
E(varphi) &= (acosvarphi, bsinvarphi, 0) H(psi) &= (ccoshpsi, 0, bsinhpsi),quad c^2 = a^2 - b^2end{align} with the vertices and foci of the ellipse S_1 = (a, 0, 0),quad F_1 = (c, 0, 0),quad F_2 = (-c, 0, 0),quad S_2 = (-a, 0, 0) and a string (in diagram red) of length {{mvar|l}}.
- Pin one end of the string to vertex {{math|S1}} and the other to focus {{math|F2}}. The string is kept tight at a point {{mvar|P}} with positive {{mvar|y}}- and {{mvar|z}}-coordinates, such that the string runs from {{math|S1}} to {{mvar|P}} behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string from {{mvar|P}} to {{math|F2}} runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance {{math|{{abs|S1 P}}}} 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.
- Then: {{mvar|P}} is a point of the ellipsoid with equation begin{align}
&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^2_x - c^2}}, quad
r_z = {textstyle sqrt{r^2_x - a^2}}.
end{align}
- The remaining points of the ellipsoid can be constructed by suitable changes of the string at the focal conics.
Semi-axes
Equations for the semi-axes of the generated ellipsoid can be derived by special choices for point {{mvar|P}}:
Y = (0, r_y, 0),quad Z = (0, 0, r_z).
The lower part of the diagram shows that {{math|F1}} and {{math|F2}} are the foci of the ellipse in the {{mvar|xy}}-plane, too. Hence, it is confocal to the given ellipse and the length of the string is {{math|l {{=}} 2rx + (a − c)}}. Solving for {{mvar|rx}} yields {{math|rx {{=}} {{sfrac|1|2}}(l − a + c)}}; furthermore {{math|r{{su|p=2|b=y}} {{=}} r{{su|p=2|b=x}} − c2}}.From the upper diagram we see that {{math|S1}} and {{math|S2}} are the foci of the ellipse section of the ellipsoid in the {{mvar|xz}}-plane and that {{math|r{{su|p=2|b=z}} {{=}} r{{su|p=2|b=x}} − a2}}. Converse
If, conversely, a triaxial ellipsoid is given by its equation, then from the equations in step 3 one can derive the parameters {{mvar|a}}, {{mvar|b}}, {{mvar|l}} for a pins-and-string construction. Confocal ellipsoids
If {{overline|{{mathcal|E}}}} is an ellipsoid confocal to {{mathcal|E}} with the squares of its semi-axes
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 of {{mathcal|E}}
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-axes {{math|a, b, c}} as ellipsoid {{mathcal|E}}. 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 the focal curves of the ellipsoid.O. Hesse: Analytische Geometrie des Raumes, Teubner, Leipzig 1861, p. 287The converse statement is true, too: if one chooses a second string of length {{math|{{overline|l}}}} and defines
lambda = r^2_x - overline r^2_x
then the equations
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
In case of {{math|a {{=}} c}} (a spheroid) one gets {{math|S1 {{=}} F1}} and {{math|S2 {{=}} F2}}, which means that the focal ellipse degenerates to a line segment and the focal hyperbola collapses to two infinite line segments on the {{mvar|x}}-axis. The ellipsoid is rotationally symmetric around the {{mvar|x}}-axis and
r_x = tfrac12l,quad r_y = r_z = {textstyle sqrt{r^2_x - c^2}}.
Properties of the focal hyperbola
(File:Ellipsoid-pk-zk.svg|thumb|upright=1.5|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 point {{mvar|V}} of its focal hyperbola, than it seems to be a sphere, that is its apparent shape is a circle. Equivalently, the tangents of the ellipsoid containing point {{mvar|V}} are the lines of a circular cone, whose axis of rotation is the tangent line of the hyperbola at {{mvar|V}}.D. Hilbert & S Cohn-Vossen: Geometry and the Imagination, p. 24O. Hesse: Analytische Geometrie des Raumes, p. 301 If one allows the center {{mvar|V}} to disappear into infinity, one gets an orthogonal parallel projection with the corresponding asymptote of the focal hyperbola as its direction. The true curve of shape (tangent points) on the ellipsoid is not a circle.{{paragraph}} 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 center {{mvar|V}} and main point {{mvar|H}} on the tangent of the hyperbola at point {{mvar|V}}. ({{mvar|H}} is the foot of the perpendicular from {{mvar|V}} onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the origin {{mvar|O}} is the circle's center; in the central case main point {{mvar|H}} is the center.
- Umbilical points
- The focal hyperbola intersects the ellipsoid at its four umbilical points.W. Blaschke: Analytische Geometrie, p. 125
Property of the focal ellipse
The focal ellipse together with its inner part can be considered as the limit surface (an infinitely thin ellipsoid) of the pencil of confocal ellipsoids determined by {{math|a, b}} for {{math|rz → 0}}. For the limit case one gets
r_x = a,quad r_y = b,quad l = 3a - c.
Ellipsoids in higher dimensions and general position
Standard equation
A hyperellipsoid, or ellipsoid of dimension n - 1 in a Euclidean space of dimension n, is a quadric hypersurface defined by a polynomial of degree two that has a 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
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 an {{mvar|n}}-dimensional hyperellipsoid can be obtained by replacing {{mvar|Rn}} by the product of the semi-axes {{math|a1a2...an}} in the formula for the volume of a hypersphere:
V = frac{pi^frac{n}{2}}{Gammaleft(frac{n}{2} + 1right)}a_1a_2cdots a_n approx frac{1}{sqrt{pi n}} cdot left(frac{2 e pi}{n}right)^{n/2} a_1a_2cdots a_n
(where {{math|Γ}} 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 R^n, then the set of points {{math|x}} that satisfy the equation
(mathbf{x}-mathbf{v})^mathsf{T}! boldsymbol{A}, (mathbf{x}-mathbf{v}) = 1
is an n-dimensional ellipsoid centered at {{mvar|v}}. The expression (mathbf{x}-mathbf{v})^mathsf{T}! boldsymbol{A}, (mathbf{x}-mathbf{v}) 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.{{Rp|page=67|location=}}The eigenvectors of {{mvar|A}} are the principal axes of the ellipsoid, and the eigenvalues of {{mvar|A}} are the reciprocals of the squares of the semi-axes (in three dimensions these are {{math|a−2}}, {{math|b−2}} and {{math|c−2}}).WEB,weblink Archived copy, 2013-10-12, live,weblink" title="web.archive.org/web/20130626233838weblink">weblink 2013-06-26, pp. 17–18. In particular:
- The diameter of the ellipsoid is twice the longest semi-axis, which is twice the square-root 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 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 3 × 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 boldsymbol{A}, there exists a unique positive definite matrix denoted {{math|A1/2}}, such that 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" of boldsymbol{A}. The ellipsoid defined by (mathbf{x}-mathbf{v})^mathsf{T}! boldsymbol{A}, (mathbf{x}-mathbf{v}) = 1 can also be presented asGEOMETRIC ALGORITHMS AND COMBINATORIAL OPTIMIZATION, {{Rp|page=67|location=}}A^{-1/2}cdot S(mathbf{0},1) + mathbf{v}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|f0}} and a regular 3 × 3 matrix {{math|A}}:
mathbf x mapsto mathbf f_0 + boldsymbol A mathbf x = mathbf f_0 + xmathbf f_1 + ymathbf f_2 + zmathbf f_3
where {{math|f1, f2, f3}} 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:
mathbf x(theta, varphi) = mathbf f_0 + mathbf f_1 costheta cosvarphi + mathbf f_2 costheta sinvarphi + mathbf f_3 sintheta, qquad -tfrac{pi}{2} < theta < tfrac{pi}{2},quad 0 le varphi < 2pi.
If the vectors {{math|f1, f2, f3}} form an orthogonal system, the six points with vectors {{math|f0 ± f1,2,3}} are the vertices of the ellipsoid and {{math|{{abs|f1}}, {{abs|f2}}, {{abs|f3}}}} are the semi-principal axes.A surface normal vector at point {{math|x(θ, φ)}} is
mathbf n(theta, varphi) = mathbf f_2 times mathbf f_3costhetacosvarphi + mathbf f_3 times mathbf f_1costhetasinvarphi + mathbf f_1 times mathbf f_2sintheta.
For any ellipsoid there exists an implicit representation {{math|F(x, y, z) {{=}} 0}}. If for simplicity the center of the ellipsoid is the origin, {{math|f0 {{=}} 0}}, the following equation describes the ellipsoid above:Computerunterstützte Darstellende und Konstruktive Geometrie. {{webarchive |url=https://web.archive.org/web/20131110190049weblink |date=2013-11-10}} Uni Darmstadt (PDF; 3,4 MB), S. 88.
F(x, y, z) = operatorname{det}left(mathbf x, mathbf f_2, mathbf f_3right)^2 + operatorname{det}left(mathbf f_1,mathbf x, mathbf f_3right)^2 + operatorname{det}left(mathbf f_1, mathbf f_2, mathbf xright)^2 - operatorname{det}left(mathbf f_1, mathbf f_2, mathbf f_3right)^2 = 0
Applications
The ellipsoidal shape finds many practical applications:
- Geodesy
- Mechanics
- Crystallography
Computer science
- Lighting
- Medicine
- Measurements obtained from MRI imaging of the prostate can be used to determine the volume of the gland using the approximation {{math|L × W × H × 0.52}} (where 0.52 is an approximation for {{sfrac|{{math|π}}|6}})JOURNAL, Bezinque, Adam, etal, Determination of Prostate Volume: A Comparison of Contemporary Methods, Academic Radiology, 25, 12, 1582–1587, 10.1016/j.acra.2018.03.014, 29609953, 2018, 4621745,
Dynamical properties
The mass of an ellipsoid of uniform density {{mvar|Ï}} is
m = V rho = tfrac{4}{3} pi abc rho.
The moments of inertia of an ellipsoid of uniform density are
begin{align}
I_mathrm{xx} &= tfrac{1}{5}mleft(b^2 + c^2right), &
I_mathrm{yy} &= tfrac{1}{5}mleft(c^2 + a^2right), &
I_mathrm{zz} &= tfrac{1}{5}mleft(a^2 + b^2right), [3pt]
I_mathrm{xy} &= I_mathrm{yz} = I_mathrm{zx} = 0.
end{align}For {{math|1=a = b = c}} these moments of inertia reduce to those for a sphere of uniform density.File:2003EL61art.jpg|right|thumb|Artist's conception of {{dp|Haumea}}, a Jacobi-ellipsoid dwarf planetdwarf planetEllipsoids and cuboids 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.Goldstein, H G (1980). Classical Mechanics, (2nd edition) Chapter 5.One practical effect of this is that scalene astronomical bodies such as {{dp|Haumea}} generally rotate along their minor axes (as does Earth, which is merely oblate); in addition, because of tidal locking, moons in synchronous orbit such as Mimas orbit with their major axis aligned radially to their planet.A spinning body of homogeneous self-gravitating fluid will assume the form of either a Maclaurin spheroid (oblate spheroid) or Jacobi ellipsoid (scalene ellipsoid) when in hydrostatic equilibrium, and for moderate rates of rotation. At faster rotations, non-ellipsoidal (:wikt:pyriform|piriform) or oviform shapes can be expected, but these are not stable.Fluid dynamics
The ellipsoid is the most general shape for which it has been possible to calculate the creeping 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 of microorganisms.Dusenbery, David B. (2009).Living at Micro Scale, Harvard University Press, Cambridge, Massachusetts {{isbn|978-0-674-03116-6}}.In probability and statistics
The elliptical distributions, which generalize the multivariate normal distribution and are used in finance, can be defined in terms of their density functions. When they exist, the density functions {{mvar|f}} have the structure:
f(x) = k cdot gleft((mathbf x - boldsymbolmu)boldsymbolSigma^{-1}(mathbf x - boldsymbolmu)^mathsf{T}right)
where {{mvar|k}} is a scale factor, {{math|x}} is an {{mvar|n}}-dimensional random row vector with median vector {{math|μ}} (which is also the mean vector if the latter exists), {{math|Σ}} is a positive definite matrix which is proportional to the covariance matrix if the latter exists, and {{mvar|g}} is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statistics & Probability Letters, 63(3), 275–286. The multivariate normal distribution is the special case in which {{math|g(z) {{=}} exp(−{{sfrac|z|2}})}} for quadratic form {{mvar|z}}.Thus the density function is a scalar-to-scalar transformation of a quadric expression. Moreover, the equation for any iso-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
Notes
References
- {{citation |last1=Kreyszig |first1=Erwin |author-link=Erwin Kreyszig |title=Advanced Engineering Mathematics |publisher=Wiley |location=New York |edition=3rd |year=1972 |isbn=0-471-50728-8 |url-access=registration |url=https://archive.org/details/advancedengineer00krey}}
External links
{{Commons category|Ellipsoids}}
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