Ellipsoid - Knowledge

9501: 3977: 5223: 5187: 10116: 5215: 3998: 8490: 7140: 9208: 4233: 2067: 6019: 8892: 27: 10139: 9808: 9162: 9496:{\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} 5739: 7819: 1819: 8693: 4968: 3364: 10111:{\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),\\I_{\mathrm {xy} }&=I_{\mathrm {yz} }=I_{\mathrm {zx} }=0.\end{aligned}}} 5470: 1808: 927: 3742: 8992: 6526: 3984:

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

10367: 8648: 2806: 1539: 1281: 2622: 4781: 2062:{\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 }}},\\&-{\tfrac {\pi }{2}}\leq \gamma \leq {\tfrac {\pi }{2}},\qquad 0\leq \lambda <2\pi .\end{aligned}}} 7623: 6014:{\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}}} 10220:

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

8887:{\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 } 7574: 6724: 4802: 3194: 5290: 8396: 7979: 6945: 8063: 3466: 2426: 1623: 5084: 7128: 5252:(1868). Main investigations and the extension to quadrics was done by the German mathematician O. Staude in 1882, 1886 and 1898. The description of the pins-and-string construction of ellipsoids and hyperboloids is contained in the book 734: 2998: 7198:(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 center 5641: 4220: 3578: 3894: 1656: 742: 414: 9157:{\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 .} 3589: 6368: 4537: 10259: 8294: 8534: 2633: 1386: 1128: 3985:

circles to ellipses, the intersection of a plane with an ellipsoid is an ellipse or a single point, or is empty. Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see

2452: 4663: 1344: 7814:{\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}} 8468: 3117: 9813: 1824: 6826: 4584: 9793: 7348: 5744: 5295: 1391: 1133: 6129: 4639: 4963:{\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}\ .} 7442: 6540: 4416: 2161: 2261: 3359:{\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),} 10414:

is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve. The multivariate normal distribution is the special case in which

8324: 8319: 7907: 6837: 5475: 8218: 7893: 7997: 3375: 2341: 8687:

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:

1550: 285:

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

1040: 1108: 1072: 970: 4999: 950: 594: 556: 518: 7022: 5465:{\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}}} 2875: 1005: 10158:

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,

632: 7391: 2889: 4140: 3477: 7415: 1803:{\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}=R{\begin{bmatrix}\cos(\gamma )\cos(\lambda )\\\cos(\gamma )\sin(\lambda )\\\sin(\gamma )\end{bmatrix}}\,\!} 922:{\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,} 477: 457: 437: 3737:{\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),} 2094:

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

3783: 315: 6521:{\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 } 7163:

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 point

1122:

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

11065:

Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.

10362:{\displaystyle f(x)=k\cdot g\left((\mathbf {x} -{\boldsymbol {\mu }}){\boldsymbol {\Sigma }}^{-1}(\mathbf {x} -{\boldsymbol {\mu }})^{\mathsf {T}}\right)} 6747:. 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 8643:{\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}} 4461: 8231: 2801:{\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,} 1534:{\displaystyle {\begin{aligned}x&=a\cos(\theta )\cos(\lambda ),\\y&=b\cos(\theta )\sin(\lambda ),\\z&=c\sin(\theta ),\end{aligned}}\,\!} 1276:{\displaystyle {\begin{aligned}x&=a\sin(\theta )\cos(\varphi ),\\y&=b\sin(\theta )\sin(\varphi ),\\z&=c\cos(\theta ),\end{aligned}}\,\!} 10684: 10494:, is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid), respectively. 10972: 2617:{\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),} 223:

of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar

4776:{\displaystyle \ \mathbf {e} _{1}={\begin{bmatrix}\rho \\0\\0\end{bmatrix}},\qquad \mathbf {e} _{2}={\begin{bmatrix}0\\\rho \\0\end{bmatrix}}.} 10454:

states that the quadric expression equals some constant specific to that value of the density, and the iso-density surface is an ellipsoid.

1292: 7004:, which means that the focal ellipse degenerates to a line segment and the focal hyperbola collapses to two infinite line segments on the 10929: 10953: 5729:

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.

596:

lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, because

8404: 3026: 8187:) and represent the directions of the axes of the ellipsoid; the lengths of the semi-axes are computed from the eigenvalues. The 8153:

of the ellipsoid is twice the shortest semi-axis, which is twice the square-root of the reciprocal of the smallest eigenvalue of

6768: 10662: 8140:

of the ellipsoid is twice the longest semi-axis, which is twice the square-root of the reciprocal of the largest eigenvalue of

6328:

If, conversely, a triaxial ellipsoid is given by its equation, then from the equations in step 3 one can derive the parameters

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The focal ellipse together with its inner part can be considered as the limit surface (an infinitely thin ellipsoid) of the

4548: 9736: 7275: 7218:

onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the origin

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behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string from

10450:

Thus the density function is a scalar-to-scalar transformation of a quadric expression. Moreover, the equation for any

7569:{\displaystyle {\frac {x_{1}^{2}}{a_{1}^{2}}}+{\frac {x_{2}^{2}}{a_{2}^{2}}}+\cdots +{\frac {x_{n}^{2}}{a_{n}^{2}}}=1.} 5139: 4599: 4226: 7151:

parallel and central projection of the ellipsoid such that it looks like a sphere, i.e. its apparent shape is a circle

11098: 10801: 6719:{\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}} 5713:

runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance

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considerations mean that rotation along the major axis is more easily perturbed than rotation along the minor axis.

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This equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of an

4338: 9657: 6729:

one finds, that the corresponding focal conics used for the pins-and-string construction have the same semi-axes

11118: 6023:

The remaining points of the ellipsoid can be constructed by suitable changes of the string at the focal conics.

2116: 10989:

Bezinque, Adam; et al. (2018). "Determination of Prostate Volume: A Comparison of Contemporary Methods".

2219: 305:

The general ellipsoid, also known as triaxial ellipsoid, is a quadratic surface which is defined in

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The pins-and-string construction of an ellipsoid is a transfer of the idea constructing an ellipse using two

9565: 8391:{\displaystyle (\mathbf {x} -\mathbf {v} )^{\mathsf {T}}\!{\boldsymbol {A}}\,(\mathbf {x} -\mathbf {v} )=1} 8188: 8167:

applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable

7974:{\displaystyle (\mathbf {x} -\mathbf {v} )^{\mathsf {T}}\!{\boldsymbol {A}}\,(\mathbf {x} -\mathbf {v} )=1} 6940:{\displaystyle {\overline {r}}_{y}^{2}=r_{y}^{2}-\lambda ,\quad {\overline {r}}_{z}^{2}=r_{z}^{2}-\lambda } 8497:

The key to a parametric representation of an ellipsoid in general position is the alternative definition:

8058:{\displaystyle (\mathbf {x} -\mathbf {v} )^{\mathsf {T}}\!{\boldsymbol {A}}\,(\mathbf {x} -\mathbf {v} )} 626: 306: 8299: 3747:

which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for

10967: 10566: 8201: 7866: 3761: 3461:{\displaystyle S_{\text{oblate}}=2\pi a^{2}\left(1+{\frac {1-e^{2}}{e}}\operatorname {artanh} e\right)} 2437: 2421:{\displaystyle V_{\text{inscribed}}={\frac {8}{3{\sqrt {3}}}}abc,\qquad V_{\text{circumscribed}}=8abc.} 231:, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is 8296:

this notation is motivated by the fact that this matrix can be seen as the "positive square root" of

1618:{\displaystyle -{\tfrac {\pi }{2}}\leq \theta \leq {\tfrac {\pi }{2}},\qquad 0\leq \lambda <2\pi ,} 11148: 10503: 7614: 10403: 9579: 7854: 6032:

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

2083:

is longitude. These are true spherical coordinates with the origin at the center of the ellipsoid.

224: 4993:(radius of the circle). Hence the intersection circle can be described by the parametric equation 10473: 10468: 10234: 10197: 6157: 5238: 5089:

The reverse scaling (see above) transforms the unit sphere back to the ellipsoid and the vectors

5079:{\displaystyle \;\mathbf {u} =\mathbf {e} _{0}+\mathbf {e} _{1}\cos t+\mathbf {e} _{2}\sin t\;.} 3756:

can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases

9555: 9551: 7422: 7123:{\displaystyle r_{x}={\tfrac {1}{2}}l,\quad r_{y}=r_{z}={\textstyle {\sqrt {r_{x}^{2}-c^{2}}}}} 2880: 1013: 10881: 1081: 1045: 955: 10922: 8508: 8164: 7433: 7184: 1350: 935: 729:{\displaystyle (x,y,z)=(r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi ,r\cos \theta )} 561: 523: 485: 271: 257:, or simply axes of the ellipsoid. If the three axes have different lengths, the figure is a 212: 201: 10873: 2993:{\displaystyle S=4\pi bcR_{G}{\left({\frac {b^{2}}{a^{2}}},{\frac {c^{2}}{a^{2}}},1\right)}} 1650:

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

11090: 10909: 7418: 7246: 5636:{\displaystyle S_{1}=(a,0,0),\quad F_{1}=(c,0,0),\quad F_{2}=(-c,0,0),\quad S_{2}=(-a,0,0)} 4215:{\displaystyle \mathbf {x} =\mathbf {f} _{0}+\mathbf {f} _{1}\cos t+\mathbf {f} _{2}\sin t} 3573:{\displaystyle S_{\text{oblate}}=2\pi a^{2}\ +{\frac {\pi c^{2}}{e}}\ln {\frac {1+e}{1-e}}} 2853: 2095: 978: 4134:(conjugate vectors), such that the ellipse can be represented by the parametric equation 3184:

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

8: 10877: 10697: 9639: 9532: 8192: 8172: 7370: 7187: 7009: 5249: 3185: 2076: 1636: 242: 10869: 10216:

The ellipsoid is the most general shape for which it has been possible to calculate the

5231: 11014: 10189: 10188:

A spinning body of homogeneous self-gravitating fluid will assume the form of either a

9573: 7400: 7172: 3889:{\displaystyle S\approx 4\pi {\sqrt{\frac {a^{p}b^{p}+a^{p}c^{p}+b^{p}c^{p}}{3}}}.\,\!} 2332: 462: 442: 422: 409:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1,} 197: 9617:, ellipsoids used in crystallography to indicate the magnitudes and directions of the 11158: 11153: 11128: 11094: 11083: 11049: 11006: 10947: 10895: 10886:, Algorithms and Combinatorics, vol. 2 (2nd ed.), Springer-Verlag, Berlin, 10797: 10722: 10678: 10407: 10178: 10159: 9799: 9622: 9618: 9614: 6355: 4452: 3929:

is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.

2842: 2302: 2091: 1640: 10654: 7436:. The spectral theorem can again be used to obtain a standard equation of the form 5135:, which were wanted for the parametric representation of the intersection ellipse. 11018: 10998: 10887: 10526: 10463: 10246: 10193: 9635: 9604: 9603:, a diagram of an ellipsoid that depicts the orientation and relative magnitude of 9585: 9569: 9168: 8184: 8180: 8176: 8114:

are the reciprocals of the squares of the semi-axes (in three dimensions these are

3986: 2272: 1632: 1111: 293: 246: 10630:"DLMF: §19.20 Special Cases ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals" 7432:

One can also define a hyperellipsoid as the image of a sphere under an invertible

4532:{\displaystyle \;\mathbf {m} ={\begin{bmatrix}m_{u}\\m_{v}\\m_{w}\end{bmatrix}}\;} 10976: 10905: 10707:

by Gerard P. Michon (2004-05-13). See Thomsen's formulas and Cantrell's comments.

10704: 10170: 9600: 9593: 9522: 8289:{\displaystyle {\boldsymbol {A}}={\boldsymbol {A}}^{1/2}{\boldsymbol {A}}^{1/2};} 7394: 7233: 2268: 1075: 615: 611: 287: 208: 20: 11114: 11078: 11002: 10531: 10520: 10508: 10222: 9536: 7829: 5655: 11124: 10891: 6754:

The converse statement is true, too: if one chooses a second string of length

3768:). Derivations of these results may be found in standard sources, for example 11142: 10387: 10217: 10174: 5257: 3971: 2848:

The surface area of this general ellipsoid can also be expressed in terms of

2299: 1042:, the ellipsoid is a spheroid or ellipsoid of revolution. In particular, if 239: 10201: 10200:, and for moderate rates of rotation. At faster rotations, non-ellipsoidal 11010: 10451: 10182: 10147: 8195:

are matrix decompositions closely related to these geometric observations.

5284: 3976: 3764:

of the ellipse formed by the cross section through the symmetry axis. (See

2443: 253:

that are delimited on the axes of symmetry by the ellipsoid are called the

250: 10781:

Die algebraischen Grundlagen der Focaleigenschaften der Flächen 2. Ordnung

10135:

these moments of inertia reduce to those for a sphere of uniform density.

10561:

F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark, editors, 2010,

10484: 10165:

One practical effect of this is that scalene astronomical bodies such as

9677:

can be used to determine the volume of the gland using the approximation

8478: 8095: 232: 10169:

generally rotate along their minor axes (as does Earth, which is merely

5222: 3014:, this can be also be expressed in terms of the volume of the ellipsoid 10490: 10479: 9559: 8105: 7353: 220: 5241:

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

5138:

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

1339:{\displaystyle 0\leq \theta \leq \pi ,\qquad 0\leq \varphi <2\pi .} 235:, which means that it may be enclosed in a sufficiently large sphere. 11132: 9544: 7191: 5186: 3769: 2328: 297:. If the three axes have the same length, the ellipsoid is a sphere. 5214: 10514: 10497: 9674: 8168: 8137: 6969: 5248:

is more complicated. First ideas are due to the Scottish physicist

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are orthogonal, parallel to the intersection plane and have length

3997: 2167: 282: 277: 216: 125: 8489: 7139: 610:

are half the length of the principal axes. They correspond to the

268:

If two of the axes have the same length, then the ellipsoid is an

10629: 10242: 10205: 9724: 9608: 9515: 8183:, then the eigenvectors of the matrix are orthogonal (due to the 3765: 2324: 2087: 619: 228: 5218:

Pins-and-string construction of an ellipsoid, blue: focal conics

4232: 10166: 10155: 10143: 9189:. If for simplicity the center of the ellipsoid is the origin, 2279: 2107: 193: 98: 10185:

orbit with their major axis aligned radially to their planet.

10138: 10969:

Computerunterstützte Darstellende und Konstruktive Geometrie.

10571: 9526: 8150: 3992: 26: 10591: 10511:, the gravitational Earth modeled by a best-fitted ellipsoid 8463:{\displaystyle A^{-1/2}\cdot S(\mathbf {0} ,1)+\mathbf {v} } 6282:

are the foci of the ellipse section of the ellipsoid in the

3112:{\displaystyle S=3VR_{G}{\left(a^{-2},b^{-2},c^{-2}\right)}} 9720: 7168: 6821:{\displaystyle \lambda =r_{x}^{2}-{\overline {r}}_{x}^{2}} 9670: 9511:

The ellipsoidal shape finds many practical applications:

8220:, there exists a unique positive definite matrix denoted 7232:

The focal hyperbola intersects the ellipsoid at its four

10868: 10500:, a shell bounded by two concentric, confocal ellipsoids 9202:, the following equation describes the ellipsoid above: 6950:

are valid, which means the two ellipsoids are confocal.

10523:, the smallest ellipsoid containing a given convex set. 9525:, a mathematical figure approximating the shape of the 8921:

form an orthogonal system, the six points with vectors

7222:

is the circle's center; in the central case main point

10517:, a shell bounded by two concentric similar ellipsoids 10396:(which is also the mean vector if the latter exists), 9989: 9914: 9839: 9756: 8851: 8830: 7085: 7040: 6953: 5969: 5916: 5866: 4869: 4742: 4690: 4579:{\displaystyle \mathbf {e} _{0}=\delta \mathbf {m} \;} 4479: 3906:

yields a relative error of at most 1.061%; a value of

2230: 2127: 2019: 1998: 1704: 1665: 1579: 1558: 11048:, Harvard University Press, Cambridge, Massachusetts 10262: 9811: 9788:{\displaystyle m=V\rho ={\tfrac {4}{3}}\pi abc\rho .} 9739: 9211: 8995: 8696: 8537: 8407: 8327: 8302: 8234: 8204: 8000: 7910: 7869: 7626: 7445: 7403: 7373: 7278: 7025: 6840: 6771: 6543: 6371: 6160:

to the given ellipse and the length of the string is

6045: 5742: 5478: 5293: 5002: 4805: 4666: 4602: 4551: 4464: 4341: 4143: 3786: 3592: 3480: 3378: 3197: 3029: 2892: 2856: 2636: 2455: 2344: 2222: 2119: 1822: 1659: 1553: 1389: 1295: 1131: 1084: 1048: 1016: 981: 958: 938: 745: 635: 564: 526: 488: 465: 445: 425: 318: 10768:

Ueber neue Focaleigenschaften der Flächen 2. Grades.

8179:). If the linear transformation is represented by a 7354:

Ellipsoids in higher dimensions and general position

7134: 10883:

Geometric algorithms and combinatorial optimization

9535:, a mathematical figure approximating the shape of 8502:

An ellipsoid is an affine image of the unit sphere.

7343:{\displaystyle r_{x}=a,\quad r_{y}=b,\quad l=3a-c.} 281:. In this case, the ellipsoid is invariant under a 11082: 10361: 10208:shapes can be expected, but these are not stable. 10110: 9787: 9495: 9156: 8886: 8642: 8511:can be represented by a translation with a vector 8462: 8390: 8313: 8288: 8212: 8057: 7973: 7887: 7813: 7568: 7409: 7385: 7342: 7122: 6939: 6820: 6718: 6520: 6124:{\displaystyle Y=(0,r_{y},0),\quad Z=(0,0,r_{z}).} 6123: 6013: 5635: 5464: 5148:The diagrams show an ellipsoid with the semi-axes 5078: 4962: 4775: 4633: 4578: 4531: 4410: 4214: 3888: 3736: 3572: 3460: 3358: 3111: 2992: 2869: 2800: 2616: 2420: 2255: 2155: 2061: 1802: 1617: 1533: 1338: 1275: 1102: 1066: 1034: 999: 964: 944: 921: 728: 588: 550: 512: 471: 451: 431: 408: 10742:Die FadenKonstruktion der Flächen zweiter Ordnung 9588:, a triaxial ellipsoid formed by a rotating fluid 8356: 8104:are the principal axes of the ellipsoid, and the 8029: 7939: 7421:defined by a polynomial of degree two that has a 7159:If one views an ellipsoid from an external point 4634:{\displaystyle \;\rho ={\sqrt {1-\delta ^{2}}}\;} 4332:and the given plane onto the plane with equation 3885: 1799: 1530: 1272: 16:Quadric surface that looks like a deformed sphere 11140: 10228: 7240: 9582:, used to describe a robot's freedom of motion. 8493:ellipsoid as an affine image of the unit sphere 5226:Determination of the semi axis of the ellipsoid 5181: 1380:Measuring from the equator rather than a pole, 4317:transforms the ellipsoid onto the unit sphere 4236:Plane section of the unit sphere (see example) 3003:Simplifying above formula using properties of 10979:Uni Darmstadt (PDF; 3,4 MB), S. 88. 7194:of the focal hyperbola as its direction. The 5687:-coordinates, such that the string runs from 10683:: CS1 maint: multiple names: authors list ( 8484: 5263: 4411:{\displaystyle \ n_{x}au+n_{y}bv+n_{z}cw=d.} 2845:of the first and second kind respectively. 9554:, a geometrical method for visualizing the 7146:3-axial Ellipsoid with its focal hyperbola. 5736:is a point of the ellipsoid with equation 5190:Pins-and-string construction of an ellipse: 4001:Plane section of an ellipsoid (see example) 3163:do not depend on the choice of an order on 5472:with the vertices and foci of the ellipse 5072: 5003: 4630: 4603: 4575: 4528: 4465: 3993:Determining the ellipse of a plane section 249:, called the center of the ellipsoid. The 10755:Ueber Fadenconstructionen des Ellipsoides 10249:. When they exist, the density functions 8362: 8035: 7945: 7872: 7206:on the tangent of the hyperbola at point 6134:The lower part of the diagram shows that 4914: 4863: 3884: 2583: 2539: 2323:the volume of the circumscribed box. The 2156:{\displaystyle V={\tfrac {4}{3}}\pi abc.} 1798: 1529: 1271: 192:is a surface that can be obtained from a 11077: 10988: 10744:, Mathemat. Nachrichten 13, 1955, S. 151 10549: 10137: 9572:for the graphical representation of the 8488: 8079:. For every ellipsoid, there are unique 7183:to disappear into infinity, one gets an 7138: 5221: 5213: 5185: 4231: 3996: 3975: 2256:{\displaystyle V={\tfrac {1}{6}}\pi ABC} 196:by deforming it by means of directional 25: 10563:NIST Handbook of Mathematical Functions 10338: 10304: 9802:of an ellipsoid of uniform density are 9714: 8562: 8358: 8304: 8265: 8245: 8236: 8206: 8031: 7941: 1349:These parameters may be interpreted as 736:, the general ellipsoid is defined as: 11141: 10952:: CS1 maint: archived copy as title ( 10716: 10348: 8937:are the vertices of the ellipsoid and 8350: 8023: 7933: 7358: 7214:is the foot of the perpendicular from 6343: 5675:. The string is kept tight at a point 3775: 2446:of a general (triaxial) ellipsoid is 265:), and the axes are uniquely defined. 10864: 10862: 9642:algorithm of theoretical significance 8520:and a regular 3 × 3 matrix 7249:of confocal ellipsoids determined by 6340:for a pins-and-string construction. 5237:A pins-and-string construction of an 30:Examples of ellipsoids with equation 10817:, Teubner, Leipzig 1861, p. 287 9692:(where 0.52 is an approximation for 4657:(i.e. the plane is horizontal), let 300: 10245:, can be defined in terms of their 9629: 8198:For every positive definite matrix 7988:-dimensional ellipsoid centered at 6954:Limit case, ellipsoid of revolution 6362:with the squares of its semi-axes 6264:From the upper diagram we see that 6152:are the foci of the ellipse in the 4101:, which have an ellipse in common. 1117: 479:are the length of the semi-axes. 13: 10859: 10665:from the original on 3 August 2017 10092: 10089: 10074: 10071: 10052: 10049: 9975: 9972: 9900: 9897: 9825: 9822: 9167:For any ellipsoid there exists an 8314:{\displaystyle {\boldsymbol {A}}.} 7651: 1361:is the azimuth angle of the point 14: 11170: 11107: 10211: 8971:A surface normal vector at point 8676:are the column vectors of matrix 8213:{\displaystyle {\boldsymbol {A}}} 8091:that satisfy the above equation. 7888:{\displaystyle \mathbb {R} ^{n},} 7135:Properties of the focal hyperbola 3965: 11085:Advanced Engineering Mathematics 10841:Analytische Geometrie des Raumes 10826:D. Hilbert & S Cohn-Vossen: 10815:Analytische Geometrie des Raumes 10792:D. Hilbert & S Cohn-Vossen: 10783:Math. Ann. 50, 398 - 428 (1898). 10717:Albert, Abraham Adrian (2016) , 10330: 10313: 10296: 10239:multivariate normal distribution 9653:Ellipsoidal reflector floodlight 9466: 9451: 9436: 9404: 9390: 9375: 9337: 9328: 9314: 9276: 9261: 9252: 9132: 9117: 9084: 9069: 9036: 9021: 8997: 8803: 8770: 8737: 8722: 8698: 8630: 8612: 8594: 8576: 8567: 8548: 8539: 8456: 8439: 8375: 8367: 8340: 8332: 8181:symmetric 3 × 3 matrix 8048: 8040: 8013: 8005: 7958: 7950: 7923: 7915: 7591:by the product of the semi-axes 7427:positive definite quadratic form 7171:, whose axis of rotation is the 5244:The construction of points of a 5053: 5029: 5014: 5005: 4944: 4935: 4921: 4808: 4724: 4672: 4571: 4554: 4467: 4193: 4169: 4154: 4145: 238:An ellipsoid has three pairwise 11059: 11038: 11025: 10982: 10961: 10935:from the original on 2013-06-26 10915: 10846: 10833: 10820: 10807: 10786: 10773: 10760: 10757:. Math. Ann. 20, 147–184 (1882) 10747: 9658:Ellipsoidal reflector spotlight 9506: 8865: 8825: 7835: 7318: 7298: 7057: 6890: 6643: 6593: 6471: 6421: 6083: 5954: 5901: 5595: 5554: 5516: 5421: 4918: 4721: 4593:of the intersection circle and 3760:may again be identified as the 3662: 3284: 2779: 2668: 2431: 2389: 2033: 1593: 1314: 11119:Wolfram Demonstrations Project 10770:Math. Ann. 27, 253–271 (1886). 10734: 10710: 10691: 10652: 10646: 10622: 10606:"Surface Area of an Ellipsoid" 10598: 10584: 10555: 10543: 10343: 10326: 10308: 10292: 10272: 10266: 9233: 9215: 9013: 9001: 8714: 8702: 8543: 8449: 8435: 8379: 8363: 8345: 8328: 8052: 8036: 8018: 8001: 7962: 7946: 7928: 7911: 7269:. For the limit case one gets 6115: 6090: 6077: 6052: 5895: 5877: 5630: 5609: 5589: 5568: 5548: 5530: 5510: 5492: 5415: 5379: 5369: 5363: 5353: 5317: 5307: 5301: 4542:its unit normal vector. Hence 3728: 3716: 3350: 3338: 2649: 2643: 2603: 2597: 2580: 2568: 2559: 2553: 2536: 2524: 2510: 2504: 1787: 1781: 1768: 1762: 1753: 1747: 1734: 1728: 1719: 1713: 1519: 1513: 1484: 1478: 1469: 1463: 1434: 1428: 1419: 1413: 1261: 1255: 1226: 1220: 1211: 1205: 1176: 1170: 1161: 1155: 723: 660: 654: 636: 583: 565: 545: 527: 507: 489: 1: 11071: 10406:which is proportional to the 10229:In probability and statistics 8968:are the semi-principal axes. 7587:can be obtained by replacing 7241:Property of the focal ellipse 5654:Pin one end of the string to 3980:Plane section of an ellipsoid 2110:bounded by the ellipsoid is 1007:, the ellipsoid is a sphere. 10828:Geometry and the Imagination 10794:Geometry and the imagination 10196:(scalene ellipsoid) when in 8189:singular value decomposition 7367:, or ellipsoid of dimension 7167:are the lines of a circular 6897: 6847: 6802: 6478: 6428: 6378: 6027: 5254:Geometry and the imagination 5210:, length of the string (red) 5182:Pins-and-string construction 4071:and the plane with equation 3944:, the area is approximately 3152:, the equations in terms of 2275:when two of them are equal. 7: 11044:Dusenbery, David B. (2009). 10457: 10173:); in addition, because of 9723:of an ellipsoid of uniform 9669:Measurements obtained from 7179:. If one allows the center 6531:then from the equations of 6323: 5647:(in diagram red) of length 3122:Unlike the expression with 627:spherical coordinate system 291:; if it is longer, it is a 215:that may be defined as the 200:, or more generally, of an 19:For spherical ellipse, see 10: 11175: 11089:(3rd ed.), New York: 11035:, (2nd edition) Chapter 5. 11003:10.1016/j.acra.2018.03.014 10796:, Chelsea New York, 1952, 10567:Cambridge University Press 10410:if the latter exists, and 7901:that satisfy the equation 6156:-plane, too. Hence, it is 5163:which is cut by the plane 4644:its radius (see diagram). 3969: 2438:Area of a geodesic polygon 2435: 2166:In terms of the principal 18: 10892:10.1007/978-3-642-78240-4 10504:Geodesics on an ellipsoid 8485:Parametric representation 8321:The ellipsoid defined by 7425:of degree two which is a 5264:Steps of the construction 4973:In any case, the vectors 2101: 1647:is azimuth or longitude. 1035:{\displaystyle a=b\neq c} 10592:"DLMF: 19.2 Definitions" 10537: 10404:positive definite matrix 10235:elliptical distributions 9580:Manipulability ellipsoid 8398:can also be presented as 7855:positive-definite matrix 7008:-axis. The ellipsoid is 5112:are mapped onto vectors 1103:{\displaystyle a=b<c} 1067:{\displaystyle a=b>c} 972:is the azimuthal angle. 965:{\displaystyle \varphi } 11031:Goldstein, H G (1980). 10719:Solid Analytic Geometry 10474:Elliptical distribution 10469:Ellipsoidal coordinates 10237:, which generalize the 10198:hydrostatic equilibrium 10142:Artist's conception of 9566:Lamé's stress ellipsoid 9169:implicit representation 8171:, a consequence of the 7895:then the set of points 7615:volume of a hypersphere 7613:in the formula for the 7190:with the corresponding 5666:and the other to focus 5239:ellipsoid of revolution 3932:In the "flat" limit of 2883:of elliptic integrals: 2881:Carlson symmetric forms 1357:is the polar angle and 952:is the polar angle and 945:{\displaystyle \theta } 589:{\displaystyle (0,0,c)} 551:{\displaystyle (0,b,0)} 513:{\displaystyle (a,0,0)} 10721:, Dover, p. 117, 10363: 10151: 10112: 9789: 9497: 9158: 8888: 8644: 8494: 8471: 8464: 8392: 8315: 8290: 8214: 8059: 7975: 7889: 7846:is a real, symmetric, 7815: 7570: 7434:affine transformation 7411: 7387: 7344: 7152: 7124: 7010:rotationally symmetric 6941: 6822: 6720: 6522: 6125: 6015: 5637: 5466: 5283:, which are a pair of 5260:& S. Vossen, too. 5227: 5219: 5211: 5080: 4964: 4777: 4635: 4580: 4533: 4412: 4237: 4216: 4002: 3981: 3890: 3738: 3574: 3462: 3360: 3113: 2994: 2871: 2802: 2618: 2422: 2257: 2157: 2063: 1804: 1619: 1535: 1340: 1277: 1104: 1068: 1036: 1001: 966: 946: 923: 730: 590: 552: 514: 473: 453: 433: 410: 185: 11046:Living at Micro Scale 10854:Analytische Geometrie 10659:mathworld.wolfram.com 10653:W., Weisstein, Eric. 10572:"Triaxial Ellipsoids" 10364: 10192:(oblate spheroid) or 10146:, a Jacobi-ellipsoid 10141: 10113: 9790: 9498: 9159: 8889: 8645: 8509:affine transformation 8492: 8465: 8400: 8393: 8316: 8291: 8215: 8165:linear transformation 8060: 7976: 7890: 7816: 7571: 7412: 7388: 7345: 7142: 7125: 6942: 6823: 6721: 6523: 6126: 6016: 5638: 5467: 5225: 5217: 5189: 5081: 4965: 4778: 4636: 4581: 4534: 4455:of the new plane and 4413: 4235: 4217: 4000: 3979: 3891: 3739: 3575: 3463: 3361: 3114: 2995: 2872: 2870:{\displaystyle R_{G}} 2803: 2619: 2423: 2258: 2158: 2064: 1805: 1620: 1536: 1351:spherical coordinates 1341: 1278: 1105: 1069: 1037: 1002: 1000:{\displaystyle a=b=c} 967: 947: 924: 731: 591: 553: 515: 474: 454: 434: 411: 307:Cartesian coordinates 245:which intersect at a 202:affine transformation 29: 10878:Schrijver, Alexander 10260: 10253:have the structure: 10150:, with its two moons 9809: 9737: 9715:Dynamical properties 9568:, an alternative to 9209: 8993: 8694: 8535: 8405: 8325: 8300: 8232: 8202: 7998: 7908: 7867: 7624: 7443: 7419:quadric hypersurface 7401: 7371: 7276: 7175:of the hyperbola at 7023: 6838: 6769: 6541: 6369: 6043: 5740: 5476: 5291: 5000: 4803: 4664: 4600: 4549: 4462: 4339: 4141: 3784: 3590: 3478: 3376: 3195: 3186:elementary functions 3027: 2890: 2854: 2634: 2453: 2342: 2220: 2117: 2096:ellipsoidal latitude 1820: 1657: 1551: 1387: 1293: 1129: 1082: 1046: 1014: 979: 956: 936: 743: 633: 562: 524: 486: 463: 443: 423: 316: 11033:Classical Mechanics 10610:analyticphysics.com 10552:, pp. 455–456) 10452:iso-density surface 10443:for quadratic form 10390:with median vector 10376:is a scale factor, 9640:convex optimization 9552:Poinsot's ellipsoid 9533:Reference ellipsoid 8481:around the origin. 8193:polar decomposition 8173:polar decomposition 8065:is also called the 7557: 7542: 7514: 7499: 7477: 7462: 7386:{\displaystyle n-1} 7196:true curve of shape 7188:parallel projection 7102: 6930: 6912: 6880: 6862: 6831:then the equations 6817: 6792: 6676: 6658: 6626: 6608: 6576: 6558: 6511: 6493: 6461: 6443: 6411: 6393: 6344:Confocal ellipsoids 5986: 5933: 5838: 5806: 5774: 4859: 4841: 3776:Approximate formula 2282:of an ellipsoid is 2077:geocentric latitude 1637:parametric latitude 11117:" by Jeff Bryant, 10991:Academic Radiology 10975:2013-11-10 at the 10703:2011-09-30 at the 10655:"Prolate Spheroid" 10359: 10190:Maclaurin spheroid 10152: 10108: 10106: 9998: 9923: 9848: 9800:moments of inertia 9785: 9765: 9623:crystal structures 9605:refractive indices 9556:torque-free motion 9493: 9154: 8884: 8860: 8839: 8640: 8495: 8460: 8388: 8311: 8286: 8210: 8132:). In particular: 8055: 7971: 7885: 7811: 7566: 7543: 7528: 7500: 7485: 7463: 7448: 7407: 7383: 7340: 7153: 7120: 7118: 7088: 7049: 6937: 6916: 6891: 6866: 6841: 6818: 6796: 6778: 6751:of the ellipsoid. 6716: 6662: 6644: 6612: 6594: 6562: 6544: 6518: 6497: 6472: 6447: 6422: 6397: 6372: 6121: 6011: 6009: 6002: 5972: 5949: 5919: 5875: 5824: 5792: 5760: 5633: 5462: 5460: 5246:triaxial ellipsoid 5228: 5220: 5212: 5076: 4960: 4908: 4845: 4827: 4773: 4764: 4712: 4631: 4576: 4529: 4522: 4408: 4238: 4212: 4003: 3982: 3936:much smaller than 3886: 3734: 3570: 3458: 3356: 3109: 2990: 2867: 2843:elliptic integrals 2798: 2614: 2418: 2335:are respectively: 2331:and circumscribed 2253: 2239: 2153: 2136: 2079:on the Earth, and 2059: 2057: 2028: 2007: 1800: 1792: 1687: 1615: 1588: 1567: 1531: 1527: 1377:of the ellipsoid. 1336: 1273: 1269: 1100: 1064: 1032: 997: 962: 942: 919: 726: 586: 548: 510: 469: 449: 429: 406: 259:triaxial ellipsoid 247:center of symmetry 207:An ellipsoid is a 186: 11129:Quadratic Surface 11054:978-0-674-03116-6 10997:(12): 1582–1587. 10901:978-3-642-78242-8 10870:Grötschel, Martin 10728:978-0-486-81026-3 10569:), Section 19.33 10408:covariance matrix 10388:random row vector 10247:density functions 10179:synchronous orbit 10160:moment of inertia 9997: 9922: 9847: 9764: 9619:thermal vibration 9615:Thermal ellipsoid 9576:state at a point. 8859: 8838: 7994:. The expression 7758: 7732: 7731: 7681: 7667: 7647: 7579:The volume of an 7558: 7515: 7478: 7410:{\displaystyle n} 7359:Standard equation 7116: 7048: 6900: 6850: 6805: 6481: 6431: 6381: 6000: 5947: 5874: 5839: 5807: 5775: 5232:pins and a string 4956: 4861: 4860: 4669: 4628: 4453:Hesse normal form 4344: 3879: 3873: 3714: 3709: 3666: 3646: 3600: 3568: 3536: 3512: 3488: 3442: 3386: 3336: 3331: 3288: 3265: 3205: 2976: 2949: 2774: 2663: 2514: 2397: 2375: 2372: 2352: 2303:elliptic cylinder 2238: 2213:), the volume is 2135: 2092:geodetic latitude 2027: 2006: 1982: 1981: 1641:eccentric anomaly 1587: 1566: 908: 863: 802: 472:{\displaystyle c} 452:{\displaystyle b} 432:{\displaystyle a} 395: 368: 341: 301:Standard equation 263:scalene ellipsoid 11166: 11149:Geometric shapes 11103: 11088: 11066: 11063: 11057: 11042: 11036: 11029: 11023: 11022: 10986: 10980: 10965: 10959: 10957: 10951: 10943: 10941: 10940: 10934: 10927: 10919: 10913: 10912: 10866: 10857: 10850: 10844: 10837: 10831: 10824: 10818: 10811: 10805: 10790: 10784: 10777: 10771: 10764: 10758: 10751: 10745: 10738: 10732: 10731: 10714: 10708: 10695: 10689: 10688: 10682: 10674: 10672: 10670: 10650: 10644: 10643: 10641: 10640: 10626: 10620: 10619: 10617: 10616: 10602: 10596: 10595: 10588: 10582: 10581: 10579: 10578: 10559: 10553: 10547: 10527:List of surfaces 10464:Ellipsoidal dome 10446: 10442: 10440: 10438: 10437: 10434: 10431: 10413: 10401: 10395: 10385: 10381: 10375: 10368: 10366: 10365: 10360: 10358: 10354: 10353: 10352: 10351: 10341: 10333: 10325: 10324: 10316: 10307: 10299: 10252: 10241:and are used in 10194:Jacobi ellipsoid 10134: 10117: 10115: 10114: 10109: 10107: 10097: 10096: 10095: 10079: 10078: 10077: 10057: 10056: 10055: 10035: 10031: 10030: 10029: 10017: 10016: 9999: 9990: 9980: 9979: 9978: 9960: 9956: 9955: 9954: 9942: 9941: 9924: 9915: 9905: 9904: 9903: 9885: 9881: 9880: 9879: 9867: 9866: 9849: 9840: 9830: 9829: 9828: 9794: 9792: 9791: 9786: 9766: 9757: 9729: 9709: 9707: 9706: 9703: 9700: 9699: 9691: 9636:Ellipsoid method 9630:Computer science 9586:Jacobi ellipsoid 9537:planetary bodies 9502: 9500: 9499: 9494: 9486: 9485: 9480: 9476: 9475: 9474: 9469: 9460: 9459: 9454: 9445: 9444: 9439: 9418: 9417: 9412: 9408: 9407: 9399: 9398: 9393: 9384: 9383: 9378: 9357: 9356: 9351: 9347: 9346: 9345: 9340: 9331: 9323: 9322: 9317: 9296: 9295: 9290: 9286: 9285: 9284: 9279: 9270: 9269: 9264: 9255: 9201: 9188: 9163: 9161: 9160: 9155: 9141: 9140: 9135: 9126: 9125: 9120: 9093: 9092: 9087: 9078: 9077: 9072: 9045: 9044: 9039: 9030: 9029: 9024: 9000: 8985: 8967: 8965: 8956: 8947: 8936: 8920: 8893: 8891: 8890: 8885: 8861: 8852: 8840: 8831: 8812: 8811: 8806: 8779: 8778: 8773: 8746: 8745: 8740: 8731: 8730: 8725: 8701: 8683: 8675: 8649: 8647: 8646: 8641: 8639: 8638: 8633: 8621: 8620: 8615: 8603: 8602: 8597: 8585: 8584: 8579: 8570: 8565: 8557: 8556: 8551: 8542: 8527: 8519: 8469: 8467: 8466: 8461: 8459: 8442: 8428: 8427: 8423: 8397: 8395: 8394: 8389: 8378: 8370: 8361: 8355: 8354: 8353: 8343: 8335: 8320: 8318: 8317: 8312: 8307: 8295: 8293: 8292: 8287: 8282: 8281: 8277: 8268: 8262: 8261: 8257: 8248: 8239: 8227: 8219: 8217: 8216: 8211: 8209: 8185:spectral theorem 8177:spectral theorem 8158: 8145: 8131: 8125: 8119: 8113: 8103: 8090: 8084: 8078: 8067:ellipsoidal norm 8064: 8062: 8061: 8056: 8051: 8043: 8034: 8028: 8027: 8026: 8016: 8008: 7993: 7980: 7978: 7977: 7972: 7961: 7953: 7944: 7938: 7937: 7936: 7926: 7918: 7900: 7894: 7892: 7891: 7886: 7881: 7880: 7875: 7862: 7853: 7849: 7845: 7827: 7820: 7818: 7817: 7812: 7810: 7809: 7797: 7796: 7787: 7786: 7777: 7776: 7772: 7763: 7759: 7754: 7743: 7733: 7724: 7720: 7715: 7714: 7702: 7701: 7692: 7691: 7682: 7680: 7679: 7675: 7668: 7660: 7649: 7648: 7640: 7634: 7612: 7590: 7582: 7575: 7573: 7572: 7567: 7559: 7556: 7551: 7541: 7536: 7527: 7516: 7513: 7508: 7498: 7493: 7484: 7479: 7476: 7471: 7461: 7456: 7447: 7423:homogeneous part 7416: 7414: 7413: 7408: 7392: 7390: 7389: 7384: 7349: 7347: 7346: 7341: 7308: 7307: 7288: 7287: 7268: 7258: 7234:umbilical points 7229:Umbilical points 7225: 7221: 7217: 7213: 7209: 7205: 7201: 7182: 7178: 7166: 7162: 7129: 7127: 7126: 7121: 7119: 7117: 7115: 7114: 7101: 7096: 7087: 7080: 7079: 7067: 7066: 7050: 7041: 7035: 7034: 7015: 7007: 7003: 6987: 6967: 6946: 6944: 6943: 6938: 6929: 6924: 6911: 6906: 6901: 6893: 6879: 6874: 6861: 6856: 6851: 6843: 6827: 6825: 6824: 6819: 6816: 6811: 6806: 6798: 6791: 6786: 6761: 6760: 6746: 6742: 6725: 6723: 6722: 6717: 6715: 6714: 6702: 6701: 6689: 6688: 6675: 6670: 6657: 6652: 6639: 6638: 6625: 6620: 6607: 6602: 6589: 6588: 6575: 6570: 6557: 6552: 6534: 6527: 6525: 6524: 6519: 6510: 6505: 6492: 6487: 6482: 6474: 6460: 6455: 6442: 6437: 6432: 6424: 6410: 6405: 6392: 6387: 6382: 6374: 6361: 6354:is an ellipsoid 6353: 6352: 6339: 6335: 6331: 6319: 6314: 6313: 6300: 6299: 6286:-plane and that 6285: 6281: 6272: 6260: 6255: 6254: 6241: 6240: 6226: 6212: 6210: 6209: 6206: 6203: 6188: 6181: 6155: 6151: 6142: 6130: 6128: 6127: 6122: 6114: 6113: 6070: 6069: 6035: 6020: 6018: 6017: 6012: 6010: 6003: 6001: 5999: 5998: 5985: 5980: 5971: 5964: 5963: 5950: 5948: 5946: 5945: 5932: 5927: 5918: 5911: 5910: 5876: 5867: 5861: 5860: 5850: 5840: 5837: 5832: 5823: 5822: 5813: 5808: 5805: 5800: 5791: 5790: 5781: 5776: 5773: 5768: 5759: 5758: 5749: 5746: 5735: 5728: 5726: 5712: 5703: 5699: 5695: 5686: 5682: 5678: 5674: 5665: 5650: 5642: 5640: 5639: 5634: 5605: 5604: 5564: 5563: 5526: 5525: 5488: 5487: 5471: 5469: 5468: 5463: 5461: 5457: 5456: 5444: 5443: 5431: 5430: 5282: 5275: 5209: 5207: 5177: 5162: 5134: 5111: 5085: 5083: 5082: 5077: 5062: 5061: 5056: 5038: 5037: 5032: 5023: 5022: 5017: 5008: 4992: 4988: 4969: 4967: 4966: 4961: 4954: 4953: 4952: 4947: 4938: 4930: 4929: 4924: 4913: 4912: 4898: 4897: 4881: 4880: 4862: 4858: 4853: 4840: 4835: 4826: 4822: 4817: 4816: 4811: 4795: 4782: 4780: 4779: 4774: 4769: 4768: 4733: 4732: 4727: 4717: 4716: 4681: 4680: 4675: 4667: 4656: 4640: 4638: 4637: 4632: 4629: 4627: 4626: 4611: 4585: 4583: 4582: 4577: 4574: 4563: 4562: 4557: 4538: 4536: 4535: 4530: 4527: 4526: 4519: 4518: 4505: 4504: 4491: 4490: 4470: 4450: 4417: 4415: 4414: 4409: 4392: 4391: 4373: 4372: 4354: 4353: 4342: 4331: 4316: 4315: 4313: 4312: 4307: 4304: 4291: 4289: 4288: 4283: 4280: 4267: 4265: 4264: 4259: 4256: 4221: 4219: 4218: 4213: 4202: 4201: 4196: 4178: 4177: 4172: 4163: 4162: 4157: 4148: 4133: 4124: 4115: 4100: 4070: 4068: 4066: 4065: 4060: 4057: 4048: 4046: 4045: 4040: 4037: 4028: 4026: 4025: 4020: 4017: 3987:Circular section 3961: 3960:3 ≈ 1.5849625007 3951:, equivalent to 3950: 3943: 3939: 3935: 3928: 3926: 3924: 3923: 3920: 3917: 3905: 3895: 3893: 3892: 3887: 3880: 3878: 3869: 3868: 3867: 3858: 3857: 3845: 3844: 3835: 3834: 3822: 3821: 3812: 3811: 3801: 3800: 3759: 3755: 3743: 3741: 3740: 3735: 3715: 3712: 3710: 3708: 3707: 3698: 3697: 3688: 3677: 3676: 3667: 3664: 3661: 3657: 3647: 3645: 3634: 3621: 3620: 3602: 3601: 3598: 3579: 3577: 3576: 3571: 3569: 3567: 3556: 3545: 3537: 3532: 3531: 3530: 3517: 3510: 3509: 3508: 3490: 3489: 3486: 3467: 3465: 3464: 3459: 3457: 3453: 3443: 3438: 3437: 3436: 3420: 3407: 3406: 3388: 3387: 3384: 3365: 3363: 3362: 3357: 3337: 3334: 3332: 3330: 3329: 3320: 3319: 3310: 3299: 3298: 3289: 3286: 3280: 3276: 3266: 3264: 3263: 3262: 3249: 3248: 3239: 3226: 3225: 3207: 3206: 3203: 3180: 3174: 3168: 3162: 3151: 3136: 3118: 3116: 3115: 3110: 3108: 3107: 3103: 3102: 3101: 3086: 3085: 3070: 3069: 3051: 3050: 3019: 3013: 2999: 2997: 2996: 2991: 2989: 2988: 2984: 2977: 2975: 2974: 2965: 2964: 2955: 2950: 2948: 2947: 2938: 2937: 2928: 2920: 2919: 2878: 2876: 2874: 2873: 2868: 2866: 2865: 2840: 2825: 2807: 2805: 2804: 2799: 2775: 2773: 2772: 2768: 2767: 2766: 2754: 2753: 2739: 2738: 2728: 2727: 2723: 2722: 2721: 2709: 2708: 2694: 2693: 2683: 2678: 2677: 2664: 2656: 2623: 2621: 2620: 2615: 2610: 2606: 2593: 2592: 2549: 2548: 2515: 2513: 2496: 2482: 2477: 2476: 2427: 2425: 2424: 2419: 2399: 2398: 2395: 2376: 2374: 2373: 2368: 2359: 2354: 2353: 2350: 2322: 2320: 2319: 2316: 2313: 2312: 2298:the volume of a 2297: 2295: 2294: 2291: 2288: 2273:prolate spheroid 2262: 2260: 2259: 2254: 2240: 2231: 2212: 2202: 2192: 2182: 2162: 2160: 2159: 2154: 2137: 2128: 2082: 2074: 2068: 2066: 2065: 2060: 2058: 2029: 2020: 2008: 1999: 1990: 1983: 1974: 1973: 1964: 1963: 1954: 1953: 1935: 1934: 1925: 1921: 1914: 1913: 1904: 1903: 1885: 1884: 1875: 1874: 1860: 1859: 1850: 1849: 1838: 1834: 1809: 1807: 1806: 1801: 1797: 1796: 1692: 1691: 1646: 1633:reduced latitude 1630: 1624: 1622: 1621: 1616: 1589: 1580: 1568: 1559: 1540: 1538: 1537: 1532: 1528: 1376: 1360: 1356: 1345: 1343: 1342: 1337: 1282: 1280: 1279: 1274: 1270: 1118:Parameterization 1112:prolate spheroid 1109: 1107: 1106: 1101: 1073: 1071: 1070: 1065: 1041: 1039: 1038: 1033: 1006: 1004: 1003: 998: 971: 969: 968: 963: 951: 949: 948: 943: 928: 926: 925: 920: 909: 907: 906: 897: 890: 889: 880: 879: 869: 864: 862: 861: 852: 845: 844: 829: 828: 819: 818: 808: 803: 801: 800: 791: 784: 783: 768: 767: 758: 757: 747: 735: 733: 732: 727: 609: 595: 593: 592: 587: 557: 555: 554: 549: 519: 517: 516: 511: 478: 476: 475: 470: 458: 456: 455: 450: 438: 436: 435: 430: 415: 413: 412: 407: 396: 394: 393: 384: 383: 374: 369: 367: 366: 357: 356: 347: 342: 340: 339: 330: 329: 320: 294:prolate spheroid 275:, also called a 243:axes of symmetry 177: 170: 163: 146: 139: 116: 92: 90: 88: 87: 82: 79: 70: 68: 67: 62: 59: 50: 48: 47: 42: 39: 11174: 11173: 11169: 11168: 11167: 11165: 11164: 11163: 11139: 11138: 11110: 11101: 11079:Kreyszig, Erwin 11074: 11069: 11064: 11060: 11043: 11039: 11030: 11026: 10987: 10983: 10977:Wayback Machine 10966: 10962: 10945: 10944: 10938: 10936: 10932: 10925: 10923:"Archived copy" 10921: 10920: 10916: 10902: 10867: 10860: 10851: 10847: 10838: 10834: 10825: 10821: 10812: 10808: 10791: 10787: 10778: 10774: 10765: 10761: 10752: 10748: 10739: 10735: 10729: 10715: 10711: 10705:Wayback Machine 10696: 10692: 10676: 10675: 10668: 10666: 10651: 10647: 10638: 10636: 10628: 10627: 10623: 10614: 10612: 10604: 10603: 10599: 10590: 10589: 10585: 10576: 10574: 10570: 10560: 10556: 10548: 10544: 10540: 10476:, in statistics 10460: 10444: 10435: 10432: 10427: 10426: 10424: 10415: 10411: 10397: 10391: 10383: 10377: 10373: 10347: 10346: 10342: 10337: 10329: 10317: 10312: 10311: 10303: 10295: 10291: 10287: 10261: 10258: 10257: 10250: 10231: 10214: 10154:Ellipsoids and 10122: 10105: 10104: 10088: 10087: 10083: 10070: 10069: 10065: 10058: 10048: 10047: 10043: 10040: 10039: 10025: 10021: 10012: 10008: 10007: 10003: 9988: 9981: 9971: 9970: 9966: 9964: 9950: 9946: 9937: 9933: 9932: 9928: 9913: 9906: 9896: 9895: 9891: 9889: 9875: 9871: 9862: 9858: 9857: 9853: 9838: 9831: 9821: 9820: 9816: 9812: 9810: 9807: 9806: 9755: 9738: 9735: 9734: 9727: 9717: 9704: 9701: 9697: 9696: 9695: 9693: 9678: 9673:imaging of the 9632: 9601:Index ellipsoid 9594:Crystallography 9523:Earth ellipsoid 9509: 9481: 9470: 9465: 9464: 9455: 9450: 9449: 9440: 9435: 9434: 9433: 9429: 9428: 9413: 9403: 9394: 9389: 9388: 9379: 9374: 9373: 9372: 9368: 9367: 9352: 9341: 9336: 9335: 9327: 9318: 9313: 9312: 9311: 9307: 9306: 9291: 9280: 9275: 9274: 9265: 9260: 9259: 9251: 9250: 9246: 9245: 9210: 9207: 9206: 9196: 9190: 9171: 9136: 9131: 9130: 9121: 9116: 9115: 9088: 9083: 9082: 9073: 9068: 9067: 9040: 9035: 9034: 9025: 9020: 9019: 8996: 8994: 8991: 8990: 8972: 8964: 8958: 8955: 8949: 8946: 8940: 8938: 8935: 8928: 8922: 8919: 8912: 8905: 8899: 8898:If the vectors 8850: 8829: 8807: 8802: 8801: 8774: 8769: 8768: 8741: 8736: 8735: 8726: 8721: 8720: 8697: 8695: 8692: 8691: 8677: 8674: 8667: 8660: 8654: 8634: 8629: 8628: 8616: 8611: 8610: 8598: 8593: 8592: 8580: 8575: 8574: 8566: 8561: 8552: 8547: 8546: 8538: 8536: 8533: 8532: 8521: 8518: 8512: 8487: 8455: 8438: 8419: 8412: 8408: 8406: 8403: 8402: 8374: 8366: 8357: 8349: 8348: 8344: 8339: 8331: 8326: 8323: 8322: 8303: 8301: 8298: 8297: 8273: 8269: 8264: 8263: 8253: 8249: 8244: 8243: 8235: 8233: 8230: 8229: 8221: 8205: 8203: 8200: 8199: 8154: 8141: 8127: 8121: 8115: 8109: 8099: 8086: 8080: 8070: 8047: 8039: 8030: 8022: 8021: 8017: 8012: 8004: 7999: 7996: 7995: 7989: 7957: 7949: 7940: 7932: 7931: 7927: 7922: 7914: 7909: 7906: 7905: 7896: 7876: 7871: 7870: 7868: 7865: 7864: 7863:is a vector in 7858: 7851: 7847: 7841: 7838: 7825: 7805: 7801: 7792: 7788: 7782: 7778: 7768: 7764: 7744: 7742: 7738: 7737: 7719: 7710: 7706: 7697: 7693: 7687: 7683: 7659: 7658: 7654: 7650: 7639: 7635: 7633: 7625: 7622: 7621: 7610: 7604: 7598: 7592: 7588: 7580: 7552: 7547: 7537: 7532: 7526: 7509: 7504: 7494: 7489: 7483: 7472: 7467: 7457: 7452: 7446: 7444: 7441: 7440: 7402: 7399: 7398: 7395:Euclidean space 7372: 7369: 7368: 7361: 7356: 7303: 7299: 7283: 7279: 7277: 7274: 7273: 7265: 7260: 7250: 7243: 7223: 7219: 7215: 7211: 7207: 7203: 7202:and main point 7199: 7180: 7176: 7164: 7160: 7147: 7137: 7110: 7106: 7097: 7092: 7086: 7084: 7075: 7071: 7062: 7058: 7039: 7030: 7026: 7024: 7021: 7020: 7013: 7005: 7002: 6995: 6989: 6986: 6979: 6973: 6959: 6956: 6925: 6920: 6907: 6902: 6892: 6875: 6870: 6857: 6852: 6842: 6839: 6836: 6835: 6812: 6807: 6797: 6787: 6782: 6770: 6767: 6766: 6756: 6755: 6744: 6730: 6710: 6706: 6697: 6693: 6684: 6680: 6671: 6666: 6653: 6648: 6634: 6630: 6621: 6616: 6603: 6598: 6584: 6580: 6571: 6566: 6553: 6548: 6542: 6539: 6538: 6532: 6506: 6501: 6488: 6483: 6473: 6456: 6451: 6438: 6433: 6423: 6406: 6401: 6388: 6383: 6373: 6370: 6367: 6366: 6359: 6350: 6349: 6346: 6337: 6333: 6329: 6326: 6312: 6307: 6306: 6305: 6298: 6293: 6292: 6291: 6287: 6283: 6280: 6274: 6271: 6265: 6253: 6248: 6247: 6246: 6239: 6234: 6233: 6232: 6228: 6207: 6204: 6201: 6200: 6198: 6195: 6190: 6187: 6183: 6170: 6161: 6153: 6150: 6144: 6141: 6135: 6109: 6105: 6065: 6061: 6044: 6041: 6040: 6033: 6030: 6008: 6007: 5994: 5990: 5981: 5976: 5970: 5968: 5959: 5955: 5941: 5937: 5928: 5923: 5917: 5915: 5906: 5902: 5865: 5856: 5852: 5848: 5847: 5833: 5828: 5818: 5814: 5812: 5801: 5796: 5786: 5782: 5780: 5769: 5764: 5754: 5750: 5748: 5743: 5741: 5738: 5737: 5733: 5722: 5716: 5714: 5711: 5705: 5701: 5697: 5694: 5688: 5684: 5680: 5676: 5673: 5667: 5664: 5658: 5648: 5600: 5596: 5559: 5555: 5521: 5517: 5483: 5479: 5477: 5474: 5473: 5459: 5458: 5452: 5448: 5439: 5435: 5426: 5422: 5372: 5357: 5356: 5310: 5294: 5292: 5289: 5288: 5280: 5273: 5266: 5234:(see diagram). 5206: 5200: 5194: 5192: 5191: 5184: 5164: 5149: 5133: 5126: 5119: 5113: 5110: 5103: 5096: 5090: 5057: 5052: 5051: 5033: 5028: 5027: 5018: 5013: 5012: 5004: 5001: 4998: 4997: 4990: 4987: 4980: 4974: 4948: 4943: 4942: 4934: 4925: 4920: 4919: 4907: 4906: 4900: 4899: 4893: 4889: 4883: 4882: 4876: 4872: 4865: 4864: 4854: 4849: 4836: 4831: 4821: 4812: 4807: 4806: 4804: 4801: 4800: 4792: 4787: 4763: 4762: 4756: 4755: 4749: 4748: 4738: 4737: 4728: 4723: 4722: 4711: 4710: 4704: 4703: 4697: 4696: 4686: 4685: 4676: 4671: 4670: 4665: 4662: 4661: 4653: 4648: 4622: 4618: 4610: 4601: 4598: 4597: 4570: 4558: 4553: 4552: 4550: 4547: 4546: 4521: 4520: 4514: 4510: 4507: 4506: 4500: 4496: 4493: 4492: 4486: 4482: 4475: 4474: 4466: 4463: 4460: 4459: 4443: 4435: 4427: 4422: 4387: 4383: 4368: 4364: 4349: 4345: 4340: 4337: 4336: 4318: 4308: 4305: 4300: 4299: 4297: 4284: 4281: 4276: 4275: 4273: 4260: 4257: 4252: 4251: 4249: 4244: 4197: 4192: 4191: 4173: 4168: 4167: 4158: 4153: 4152: 4144: 4142: 4139: 4138: 4132: 4126: 4123: 4117: 4114: 4108: 4093: 4085: 4077: 4072: 4061: 4058: 4053: 4052: 4050: 4041: 4038: 4033: 4032: 4030: 4021: 4018: 4013: 4012: 4010: 4009: 3995: 3974: 3968: 3959: 3952: 3945: 3941: 3937: 3933: 3921: 3918: 3915: 3914: 3912: 3907: 3900: 3874: 3863: 3859: 3853: 3849: 3840: 3836: 3830: 3826: 3817: 3813: 3807: 3803: 3802: 3799: 3785: 3782: 3781: 3778: 3757: 3754: 3748: 3713: and 3711: 3703: 3699: 3693: 3689: 3687: 3672: 3668: 3663: 3638: 3633: 3626: 3622: 3616: 3612: 3597: 3593: 3591: 3588: 3587: 3557: 3546: 3544: 3526: 3522: 3518: 3516: 3504: 3500: 3485: 3481: 3479: 3476: 3475: 3432: 3428: 3421: 3419: 3412: 3408: 3402: 3398: 3383: 3379: 3377: 3374: 3373: 3335: and 3333: 3325: 3321: 3315: 3311: 3309: 3294: 3290: 3285: 3258: 3254: 3250: 3244: 3240: 3238: 3231: 3227: 3221: 3217: 3202: 3198: 3196: 3193: 3192: 3176: 3170: 3164: 3161: 3153: 3138: 3123: 3094: 3090: 3078: 3074: 3062: 3058: 3057: 3053: 3052: 3046: 3042: 3028: 3025: 3024: 3015: 3012: 3004: 2970: 2966: 2960: 2956: 2954: 2943: 2939: 2933: 2929: 2927: 2926: 2922: 2921: 2915: 2911: 2891: 2888: 2887: 2861: 2857: 2855: 2852: 2851: 2849: 2841:are incomplete 2827: 2812: 2762: 2758: 2749: 2745: 2744: 2740: 2734: 2730: 2729: 2717: 2713: 2704: 2700: 2699: 2695: 2689: 2685: 2684: 2682: 2673: 2669: 2655: 2635: 2632: 2631: 2588: 2584: 2544: 2540: 2520: 2516: 2497: 2483: 2481: 2472: 2468: 2454: 2451: 2450: 2440: 2434: 2394: 2390: 2367: 2363: 2358: 2349: 2345: 2343: 2340: 2339: 2317: 2314: 2310: 2309: 2308: 2306: 2292: 2289: 2286: 2285: 2283: 2229: 2221: 2218: 2217: 2204: 2194: 2184: 2170: 2126: 2118: 2115: 2114: 2104: 2080: 2072: 2056: 2055: 2018: 1997: 1988: 1987: 1969: 1965: 1959: 1955: 1949: 1945: 1930: 1926: 1909: 1905: 1899: 1895: 1880: 1876: 1870: 1866: 1865: 1861: 1855: 1851: 1839: 1837: 1835: 1833: 1823: 1821: 1818: 1817: 1791: 1790: 1772: 1771: 1738: 1737: 1700: 1699: 1686: 1685: 1679: 1678: 1672: 1671: 1661: 1660: 1658: 1655: 1654: 1644: 1628: 1578: 1557: 1552: 1549: 1548: 1526: 1525: 1497: 1491: 1490: 1447: 1441: 1440: 1397: 1390: 1388: 1385: 1384: 1362: 1358: 1354: 1294: 1291: 1290: 1268: 1267: 1239: 1233: 1232: 1189: 1183: 1182: 1139: 1132: 1130: 1127: 1126: 1120: 1083: 1080: 1079: 1076:oblate spheroid 1047: 1044: 1043: 1015: 1012: 1011: 980: 977: 976: 957: 954: 953: 937: 934: 933: 902: 898: 885: 881: 875: 871: 870: 868: 857: 853: 840: 836: 824: 820: 814: 810: 809: 807: 796: 792: 779: 775: 763: 759: 753: 749: 748: 746: 744: 741: 740: 634: 631: 630: 616:semi-minor axis 612:semi-major axis 597: 563: 560: 559: 525: 522: 521: 487: 484: 483: 464: 461: 460: 444: 441: 440: 424: 421: 420: 389: 385: 379: 375: 373: 362: 358: 352: 348: 346: 335: 331: 325: 321: 319: 317: 314: 313: 303: 288:oblate spheroid 209:quadric surface 184: 172: 165: 158: 141: 130: 103: 83: 80: 75: 74: 72: 63: 60: 55: 54: 52: 43: 40: 35: 34: 32: 31: 24: 21:spherical conic 17: 12: 11: 5: 11172: 11162: 11161: 11156: 11151: 11137: 11136: 11122: 11109: 11108:External links 11106: 11105: 11104: 11099: 11073: 11070: 11068: 11067: 11058: 11037: 11024: 10981: 10960: 10914: 10900: 10874:Lovász, László 10858: 10845: 10832: 10819: 10806: 10785: 10772: 10759: 10746: 10733: 10727: 10709: 10690: 10645: 10621: 10597: 10583: 10554: 10550:Kreyszig (1972 10541: 10539: 10536: 10535: 10534: 10532:Superellipsoid 10529: 10524: 10521:John ellipsoid 10518: 10512: 10509:Geodetic datum 10506: 10501: 10495: 10482:, also called 10477: 10471: 10466: 10459: 10456: 10370: 10369: 10357: 10350: 10345: 10340: 10336: 10332: 10328: 10323: 10320: 10315: 10310: 10306: 10302: 10298: 10294: 10290: 10286: 10283: 10280: 10277: 10274: 10271: 10268: 10265: 10230: 10227: 10223:microorganisms 10213: 10212:Fluid dynamics 10210: 10119: 10118: 10103: 10100: 10094: 10091: 10086: 10082: 10076: 10073: 10068: 10064: 10061: 10059: 10054: 10051: 10046: 10042: 10041: 10038: 10034: 10028: 10024: 10020: 10015: 10011: 10006: 10002: 9996: 9993: 9987: 9984: 9982: 9977: 9974: 9969: 9965: 9963: 9959: 9953: 9949: 9945: 9940: 9936: 9931: 9927: 9921: 9918: 9912: 9909: 9907: 9902: 9899: 9894: 9890: 9888: 9884: 9878: 9874: 9870: 9865: 9861: 9856: 9852: 9846: 9843: 9837: 9834: 9832: 9827: 9824: 9819: 9815: 9814: 9796: 9795: 9784: 9781: 9778: 9775: 9772: 9769: 9763: 9760: 9754: 9751: 9748: 9745: 9742: 9716: 9713: 9712: 9711: 9666: 9665: 9661: 9660: 9655: 9649: 9648: 9644: 9643: 9631: 9628: 9627: 9626: 9612: 9597: 9596: 9590: 9589: 9583: 9577: 9563: 9558:of a rotating 9548: 9547: 9541: 9540: 9530: 9519: 9518: 9508: 9505: 9504: 9503: 9492: 9489: 9484: 9479: 9473: 9468: 9463: 9458: 9453: 9448: 9443: 9438: 9432: 9427: 9424: 9421: 9416: 9411: 9406: 9402: 9397: 9392: 9387: 9382: 9377: 9371: 9366: 9363: 9360: 9355: 9350: 9344: 9339: 9334: 9330: 9326: 9321: 9316: 9310: 9305: 9302: 9299: 9294: 9289: 9283: 9278: 9273: 9268: 9263: 9258: 9254: 9249: 9244: 9241: 9238: 9235: 9232: 9229: 9226: 9223: 9220: 9217: 9214: 9194: 9165: 9164: 9153: 9150: 9147: 9144: 9139: 9134: 9129: 9124: 9119: 9114: 9111: 9108: 9105: 9102: 9099: 9096: 9091: 9086: 9081: 9076: 9071: 9066: 9063: 9060: 9057: 9054: 9051: 9048: 9043: 9038: 9033: 9028: 9023: 9018: 9015: 9012: 9009: 9006: 9003: 8999: 8962: 8957:|, | 8953: 8948:|, | 8944: 8933: 8926: 8917: 8910: 8903: 8896: 8895: 8883: 8880: 8877: 8874: 8871: 8868: 8864: 8858: 8855: 8849: 8846: 8843: 8837: 8834: 8828: 8824: 8821: 8818: 8815: 8810: 8805: 8800: 8797: 8794: 8791: 8788: 8785: 8782: 8777: 8772: 8767: 8764: 8761: 8758: 8755: 8752: 8749: 8744: 8739: 8734: 8729: 8724: 8719: 8716: 8713: 8710: 8707: 8704: 8700: 8672: 8665: 8658: 8651: 8650: 8637: 8632: 8627: 8624: 8619: 8614: 8609: 8606: 8601: 8596: 8591: 8588: 8583: 8578: 8573: 8569: 8564: 8560: 8555: 8550: 8545: 8541: 8516: 8505: 8504: 8486: 8483: 8458: 8454: 8451: 8448: 8445: 8441: 8437: 8434: 8431: 8426: 8422: 8418: 8415: 8411: 8387: 8384: 8381: 8377: 8373: 8369: 8365: 8360: 8352: 8347: 8342: 8338: 8334: 8330: 8310: 8306: 8285: 8280: 8276: 8272: 8267: 8260: 8256: 8252: 8247: 8242: 8238: 8208: 8163:An invertible 8161: 8160: 8147: 8054: 8050: 8046: 8042: 8038: 8033: 8025: 8020: 8015: 8011: 8007: 8003: 7982: 7981: 7970: 7967: 7964: 7960: 7956: 7952: 7948: 7943: 7935: 7930: 7925: 7921: 7917: 7913: 7884: 7879: 7874: 7837: 7834: 7830:gamma function 7822: 7821: 7808: 7804: 7800: 7795: 7791: 7785: 7781: 7775: 7771: 7767: 7762: 7757: 7753: 7750: 7747: 7741: 7736: 7730: 7727: 7723: 7718: 7713: 7709: 7705: 7700: 7696: 7690: 7686: 7678: 7674: 7671: 7666: 7663: 7657: 7653: 7646: 7643: 7638: 7632: 7629: 7608: 7602: 7596: 7585:hyperellipsoid 7577: 7576: 7565: 7562: 7555: 7550: 7546: 7540: 7535: 7531: 7525: 7522: 7519: 7512: 7507: 7503: 7497: 7492: 7488: 7482: 7475: 7470: 7466: 7460: 7455: 7451: 7406: 7382: 7379: 7376: 7365:hyperellipsoid 7360: 7357: 7355: 7352: 7351: 7350: 7339: 7336: 7333: 7330: 7327: 7324: 7321: 7317: 7314: 7311: 7306: 7302: 7297: 7294: 7291: 7286: 7282: 7263: 7242: 7239: 7238: 7237: 7230: 7227: 7226:is the center. 7157: 7136: 7133: 7132: 7131: 7113: 7109: 7105: 7100: 7095: 7091: 7083: 7078: 7074: 7070: 7065: 7061: 7056: 7053: 7047: 7044: 7038: 7033: 7029: 7000: 6993: 6984: 6977: 6955: 6952: 6948: 6947: 6936: 6933: 6928: 6923: 6919: 6915: 6910: 6905: 6899: 6896: 6889: 6886: 6883: 6878: 6873: 6869: 6865: 6860: 6855: 6849: 6846: 6829: 6828: 6815: 6810: 6804: 6801: 6795: 6790: 6785: 6781: 6777: 6774: 6727: 6726: 6713: 6709: 6705: 6700: 6696: 6692: 6687: 6683: 6679: 6674: 6669: 6665: 6661: 6656: 6651: 6647: 6642: 6637: 6633: 6629: 6624: 6619: 6615: 6611: 6606: 6601: 6597: 6592: 6587: 6583: 6579: 6574: 6569: 6565: 6561: 6556: 6551: 6547: 6529: 6528: 6517: 6514: 6509: 6504: 6500: 6496: 6491: 6486: 6480: 6477: 6470: 6467: 6464: 6459: 6454: 6450: 6446: 6441: 6436: 6430: 6427: 6420: 6417: 6414: 6409: 6404: 6400: 6396: 6391: 6386: 6380: 6377: 6345: 6342: 6325: 6322: 6308: 6294: 6278: 6269: 6249: 6235: 6227:; furthermore 6193: 6185: 6182:. Solving for 6168: 6148: 6139: 6132: 6131: 6120: 6117: 6112: 6108: 6104: 6101: 6098: 6095: 6092: 6089: 6086: 6082: 6079: 6076: 6073: 6068: 6064: 6060: 6057: 6054: 6051: 6048: 6029: 6026: 6025: 6024: 6021: 6006: 5997: 5993: 5989: 5984: 5979: 5975: 5967: 5962: 5958: 5953: 5944: 5940: 5936: 5931: 5926: 5922: 5914: 5909: 5905: 5900: 5897: 5894: 5891: 5888: 5885: 5882: 5879: 5873: 5870: 5864: 5859: 5855: 5851: 5849: 5846: 5843: 5836: 5831: 5827: 5821: 5817: 5811: 5804: 5799: 5795: 5789: 5785: 5779: 5772: 5767: 5763: 5757: 5753: 5747: 5745: 5730: 5720: 5709: 5692: 5679:with positive 5671: 5662: 5652: 5632: 5629: 5626: 5623: 5620: 5617: 5614: 5611: 5608: 5603: 5599: 5594: 5591: 5588: 5585: 5582: 5579: 5576: 5573: 5570: 5567: 5562: 5558: 5553: 5550: 5547: 5544: 5541: 5538: 5535: 5532: 5529: 5524: 5520: 5515: 5512: 5509: 5506: 5503: 5500: 5497: 5494: 5491: 5486: 5482: 5455: 5451: 5447: 5442: 5438: 5434: 5429: 5425: 5420: 5417: 5414: 5411: 5408: 5405: 5402: 5399: 5396: 5393: 5390: 5387: 5384: 5381: 5378: 5375: 5373: 5371: 5368: 5365: 5362: 5359: 5358: 5355: 5352: 5349: 5346: 5343: 5340: 5337: 5334: 5331: 5328: 5325: 5322: 5319: 5316: 5313: 5311: 5309: 5306: 5303: 5300: 5297: 5296: 5265: 5262: 5204: 5198: 5183: 5180: 5131: 5124: 5117: 5108: 5101: 5094: 5087: 5086: 5075: 5071: 5068: 5065: 5060: 5055: 5050: 5047: 5044: 5041: 5036: 5031: 5026: 5021: 5016: 5011: 5007: 4985: 4978: 4971: 4970: 4959: 4951: 4946: 4941: 4937: 4933: 4928: 4923: 4917: 4911: 4905: 4902: 4901: 4896: 4892: 4888: 4885: 4884: 4879: 4875: 4871: 4870: 4868: 4857: 4852: 4848: 4844: 4839: 4834: 4830: 4825: 4820: 4815: 4810: 4790: 4784: 4783: 4772: 4767: 4761: 4758: 4757: 4754: 4751: 4750: 4747: 4744: 4743: 4741: 4736: 4731: 4726: 4720: 4715: 4709: 4706: 4705: 4702: 4699: 4698: 4695: 4692: 4691: 4689: 4684: 4679: 4674: 4651: 4642: 4641: 4625: 4621: 4617: 4614: 4609: 4606: 4587: 4586: 4573: 4569: 4566: 4561: 4556: 4540: 4539: 4525: 4517: 4513: 4509: 4508: 4503: 4499: 4495: 4494: 4489: 4485: 4481: 4480: 4478: 4473: 4469: 4441: 4433: 4425: 4419: 4418: 4407: 4404: 4401: 4398: 4395: 4390: 4386: 4382: 4379: 4376: 4371: 4367: 4363: 4360: 4357: 4352: 4348: 4223: 4222: 4211: 4208: 4205: 4200: 4195: 4190: 4187: 4184: 4181: 4176: 4171: 4166: 4161: 4156: 4151: 4147: 4130: 4121: 4112: 4107:Three vectors 4091: 4083: 4075: 3994: 3991: 3967: 3966:Plane sections 3964: 3957: 3897: 3896: 3883: 3877: 3872: 3866: 3862: 3856: 3852: 3848: 3843: 3839: 3833: 3829: 3825: 3820: 3816: 3810: 3806: 3798: 3795: 3792: 3789: 3777: 3774: 3752: 3745: 3744: 3733: 3730: 3727: 3724: 3721: 3718: 3706: 3702: 3696: 3692: 3686: 3683: 3680: 3675: 3671: 3660: 3656: 3653: 3650: 3644: 3641: 3637: 3632: 3629: 3625: 3619: 3615: 3611: 3608: 3605: 3596: 3581: 3580: 3566: 3563: 3560: 3555: 3552: 3549: 3543: 3540: 3535: 3529: 3525: 3521: 3515: 3507: 3503: 3499: 3496: 3493: 3484: 3469: 3468: 3456: 3452: 3449: 3446: 3441: 3435: 3431: 3427: 3424: 3418: 3415: 3411: 3405: 3401: 3397: 3394: 3391: 3382: 3367: 3366: 3355: 3352: 3349: 3346: 3343: 3340: 3328: 3324: 3318: 3314: 3308: 3305: 3302: 3297: 3293: 3283: 3279: 3275: 3272: 3269: 3261: 3257: 3253: 3247: 3243: 3237: 3234: 3230: 3224: 3220: 3216: 3213: 3210: 3201: 3157: 3120: 3119: 3106: 3100: 3097: 3093: 3089: 3084: 3081: 3077: 3073: 3068: 3065: 3061: 3056: 3049: 3045: 3041: 3038: 3035: 3032: 3008: 3001: 3000: 2987: 2983: 2980: 2973: 2969: 2963: 2959: 2953: 2946: 2942: 2936: 2932: 2925: 2918: 2914: 2910: 2907: 2904: 2901: 2898: 2895: 2864: 2860: 2809: 2808: 2797: 2794: 2791: 2788: 2785: 2782: 2778: 2771: 2765: 2761: 2757: 2752: 2748: 2743: 2737: 2733: 2726: 2720: 2716: 2712: 2707: 2703: 2698: 2692: 2688: 2681: 2676: 2672: 2667: 2662: 2659: 2654: 2651: 2648: 2645: 2642: 2639: 2625: 2624: 2613: 2609: 2605: 2602: 2599: 2596: 2591: 2587: 2582: 2579: 2576: 2573: 2570: 2567: 2564: 2561: 2558: 2555: 2552: 2547: 2543: 2538: 2535: 2532: 2529: 2526: 2523: 2519: 2512: 2509: 2506: 2503: 2500: 2495: 2492: 2489: 2486: 2480: 2475: 2471: 2467: 2464: 2461: 2458: 2433: 2430: 2429: 2428: 2417: 2414: 2411: 2408: 2405: 2402: 2393: 2388: 2385: 2382: 2379: 2371: 2366: 2362: 2357: 2348: 2265: 2264: 2252: 2249: 2246: 2243: 2237: 2234: 2228: 2225: 2164: 2163: 2152: 2149: 2146: 2143: 2140: 2134: 2131: 2125: 2122: 2103: 2100: 2070: 2069: 2054: 2051: 2048: 2045: 2042: 2039: 2036: 2032: 2026: 2023: 2017: 2014: 2011: 2005: 2002: 1996: 1993: 1991: 1989: 1986: 1980: 1977: 1972: 1968: 1962: 1958: 1952: 1948: 1944: 1941: 1938: 1933: 1929: 1924: 1920: 1917: 1912: 1908: 1902: 1898: 1894: 1891: 1888: 1883: 1879: 1873: 1869: 1864: 1858: 1854: 1848: 1845: 1842: 1836: 1832: 1829: 1826: 1825: 1811: 1810: 1795: 1789: 1786: 1783: 1780: 1777: 1774: 1773: 1770: 1767: 1764: 1761: 1758: 1755: 1752: 1749: 1746: 1743: 1740: 1739: 1736: 1733: 1730: 1727: 1724: 1721: 1718: 1715: 1712: 1709: 1706: 1705: 1703: 1698: 1695: 1690: 1684: 1681: 1680: 1677: 1674: 1673: 1670: 1667: 1666: 1664: 1626: 1625: 1614: 1611: 1608: 1605: 1602: 1599: 1596: 1592: 1586: 1583: 1577: 1574: 1571: 1565: 1562: 1556: 1542: 1541: 1524: 1521: 1518: 1515: 1512: 1509: 1506: 1503: 1500: 1498: 1496: 1493: 1492: 1489: 1486: 1483: 1480: 1477: 1474: 1471: 1468: 1465: 1462: 1459: 1456: 1453: 1450: 1448: 1446: 1443: 1442: 1439: 1436: 1433: 1430: 1427: 1424: 1421: 1418: 1415: 1412: 1409: 1406: 1403: 1400: 1398: 1396: 1393: 1392: 1347: 1346: 1335: 1332: 1329: 1326: 1323: 1320: 1317: 1313: 1310: 1307: 1304: 1301: 1298: 1284: 1283: 1266: 1263: 1260: 1257: 1254: 1251: 1248: 1245: 1242: 1240: 1238: 1235: 1234: 1231: 1228: 1225: 1222: 1219: 1216: 1213: 1210: 1207: 1204: 1201: 1198: 1195: 1192: 1190: 1188: 1185: 1184: 1181: 1178: 1175: 1172: 1169: 1166: 1163: 1160: 1157: 1154: 1151: 1148: 1145: 1142: 1140: 1138: 1135: 1134: 1119: 1116: 1099: 1096: 1093: 1090: 1087: 1063: 1060: 1057: 1054: 1051: 1031: 1028: 1025: 1022: 1019: 996: 993: 990: 987: 984: 961: 941: 930: 929: 918: 915: 912: 905: 901: 896: 893: 888: 884: 878: 874: 867: 860: 856: 851: 848: 843: 839: 835: 832: 827: 823: 817: 813: 806: 799: 795: 790: 787: 782: 778: 774: 771: 766: 762: 756: 752: 725: 722: 719: 716: 713: 710: 707: 704: 701: 698: 695: 692: 689: 686: 683: 680: 677: 674: 671: 668: 665: 662: 659: 656: 653: 650: 647: 644: 641: 638: 585: 582: 579: 576: 573: 570: 567: 547: 544: 541: 538: 535: 532: 529: 509: 506: 503: 500: 497: 494: 491: 468: 448: 428: 417: 416: 405: 402: 399: 392: 388: 382: 378: 372: 365: 361: 355: 351: 345: 338: 334: 328: 324: 302: 299: 255:principal axes 211:; that is, a 183: 182: 152: 122: 94: 15: 9: 6: 4: 3: 2: 11171: 11160: 11157: 11155: 11152: 11150: 11147: 11146: 11144: 11134: 11130: 11126: 11123: 11120: 11116: 11112: 11111: 11102: 11100:0-471-50728-8 11096: 11092: 11087: 11086: 11080: 11076: 11075: 11062: 11055: 11051: 11047: 11041: 11034: 11028: 11020: 11016: 11012: 11008: 11004: 11000: 10996: 10992: 10985: 10978: 10974: 10971: 10970: 10964: 10955: 10949: 10931: 10924: 10918: 10911: 10907: 10903: 10897: 10893: 10889: 10885: 10884: 10879: 10875: 10871: 10865: 10863: 10855: 10852:W. Blaschke: 10849: 10843:, p. 301 10842: 10836: 10829: 10823: 10816: 10810: 10803: 10802:0-8284-1087-9 10799: 10795: 10789: 10782: 10776: 10769: 10763: 10756: 10750: 10743: 10737: 10730: 10724: 10720: 10713: 10706: 10702: 10699: 10698:Final answers 10694: 10686: 10680: 10664: 10660: 10656: 10649: 10635: 10634:dlmf.nist.gov 10631: 10625: 10611: 10607: 10601: 10593: 10587: 10573: 10568: 10564: 10558: 10551: 10546: 10542: 10533: 10530: 10528: 10525: 10522: 10519: 10516: 10513: 10510: 10507: 10505: 10502: 10499: 10496: 10493: 10492: 10487: 10486: 10481: 10478: 10475: 10472: 10470: 10467: 10465: 10462: 10461: 10455: 10453: 10448: 10430: 10422: 10418: 10409: 10405: 10400: 10394: 10389: 10386:-dimensional 10380: 10355: 10334: 10321: 10318: 10300: 10288: 10284: 10281: 10278: 10275: 10269: 10263: 10256: 10255: 10254: 10248: 10244: 10240: 10236: 10226: 10224: 10219: 10218:creeping flow 10209: 10207: 10203: 10199: 10195: 10191: 10186: 10184: 10180: 10176: 10175:tidal locking 10172: 10168: 10163: 10161: 10157: 10149: 10145: 10140: 10136: 10133: 10129: 10125: 10101: 10098: 10084: 10080: 10066: 10062: 10060: 10044: 10036: 10032: 10026: 10022: 10018: 10013: 10009: 10004: 10000: 9994: 9991: 9985: 9983: 9967: 9961: 9957: 9951: 9947: 9943: 9938: 9934: 9929: 9925: 9919: 9916: 9910: 9908: 9892: 9886: 9882: 9876: 9872: 9868: 9863: 9859: 9854: 9850: 9844: 9841: 9835: 9833: 9817: 9805: 9804: 9803: 9801: 9782: 9779: 9776: 9773: 9770: 9767: 9761: 9758: 9752: 9749: 9746: 9743: 9740: 9733: 9732: 9731: 9726: 9722: 9689: 9685: 9681: 9676: 9672: 9668: 9667: 9663: 9662: 9659: 9656: 9654: 9651: 9650: 9646: 9645: 9641: 9637: 9634: 9633: 9624: 9620: 9616: 9613: 9610: 9606: 9602: 9599: 9598: 9595: 9592: 9591: 9587: 9584: 9581: 9578: 9575: 9571: 9570:Mohr's circle 9567: 9564: 9561: 9557: 9553: 9550: 9549: 9546: 9543: 9542: 9538: 9534: 9531: 9528: 9524: 9521: 9520: 9517: 9514: 9513: 9512: 9490: 9487: 9482: 9477: 9471: 9461: 9456: 9446: 9441: 9430: 9425: 9422: 9419: 9414: 9409: 9400: 9395: 9385: 9380: 9369: 9364: 9361: 9358: 9353: 9348: 9342: 9332: 9324: 9319: 9308: 9303: 9300: 9297: 9292: 9287: 9281: 9271: 9266: 9256: 9247: 9242: 9239: 9236: 9230: 9227: 9224: 9221: 9218: 9212: 9205: 9204: 9203: 9200: 9193: 9186: 9182: 9178: 9174: 9170: 9151: 9148: 9145: 9142: 9137: 9127: 9122: 9112: 9109: 9106: 9103: 9100: 9097: 9094: 9089: 9079: 9074: 9064: 9061: 9058: 9055: 9052: 9049: 9046: 9041: 9031: 9026: 9016: 9010: 9007: 9004: 8989: 8988: 8987: 8983: 8979: 8975: 8969: 8961: 8952: 8943: 8932: 8925: 8916: 8909: 8902: 8881: 8878: 8875: 8872: 8869: 8866: 8862: 8856: 8853: 8847: 8844: 8841: 8835: 8832: 8826: 8822: 8819: 8816: 8813: 8808: 8798: 8795: 8792: 8789: 8786: 8783: 8780: 8775: 8765: 8762: 8759: 8756: 8753: 8750: 8747: 8742: 8732: 8727: 8717: 8711: 8708: 8705: 8690: 8689: 8688: 8685: 8682: 8681: 8671: 8664: 8657: 8635: 8625: 8622: 8617: 8607: 8604: 8599: 8589: 8586: 8581: 8571: 8558: 8553: 8531: 8530: 8529: 8526: 8525: 8515: 8510: 8503: 8500: 8499: 8498: 8491: 8482: 8480: 8476: 8470: 8452: 8446: 8443: 8432: 8429: 8424: 8420: 8416: 8413: 8409: 8399: 8385: 8382: 8371: 8336: 8308: 8283: 8278: 8274: 8270: 8258: 8254: 8250: 8240: 8228:, such that 8226: 8225: 8196: 8194: 8190: 8186: 8182: 8178: 8174: 8170: 8166: 8157: 8152: 8148: 8144: 8139: 8135: 8134: 8133: 8130: 8124: 8118: 8112: 8107: 8102: 8097: 8092: 8089: 8083: 8077: 8073: 8068: 8044: 8009: 7992: 7987: 7968: 7965: 7954: 7919: 7904: 7903: 7902: 7899: 7882: 7877: 7861: 7856: 7844: 7833: 7831: 7806: 7802: 7798: 7793: 7789: 7783: 7779: 7773: 7769: 7765: 7760: 7755: 7751: 7748: 7745: 7739: 7734: 7728: 7725: 7721: 7716: 7711: 7707: 7703: 7698: 7694: 7688: 7684: 7676: 7672: 7669: 7664: 7661: 7655: 7644: 7641: 7636: 7630: 7627: 7620: 7619: 7618: 7616: 7611: 7601: 7595: 7586: 7583:-dimensional 7563: 7560: 7553: 7548: 7544: 7538: 7533: 7529: 7523: 7520: 7517: 7510: 7505: 7501: 7495: 7490: 7486: 7480: 7473: 7468: 7464: 7458: 7453: 7449: 7439: 7438: 7437: 7435: 7430: 7428: 7424: 7420: 7404: 7397:of dimension 7396: 7380: 7377: 7374: 7366: 7337: 7334: 7331: 7328: 7325: 7322: 7319: 7315: 7312: 7309: 7304: 7300: 7295: 7292: 7289: 7284: 7280: 7272: 7271: 7270: 7266: 7257: 7253: 7248: 7235: 7231: 7228: 7197: 7193: 7189: 7186: 7174: 7170: 7158: 7155: 7154: 7150: 7145: 7141: 7111: 7107: 7103: 7098: 7093: 7089: 7081: 7076: 7072: 7068: 7063: 7059: 7054: 7051: 7045: 7042: 7036: 7031: 7027: 7019: 7018: 7017: 7011: 6999: 6992: 6983: 6976: 6971: 6966: 6962: 6951: 6934: 6931: 6926: 6921: 6917: 6913: 6908: 6903: 6894: 6887: 6884: 6881: 6876: 6871: 6867: 6863: 6858: 6853: 6844: 6834: 6833: 6832: 6813: 6808: 6799: 6793: 6788: 6783: 6779: 6775: 6772: 6765: 6764: 6763: 6759: 6752: 6750: 6743:as ellipsoid 6741: 6737: 6733: 6711: 6707: 6703: 6698: 6694: 6690: 6685: 6681: 6677: 6672: 6667: 6663: 6659: 6654: 6649: 6645: 6640: 6635: 6631: 6627: 6622: 6617: 6613: 6609: 6604: 6599: 6595: 6590: 6585: 6581: 6577: 6572: 6567: 6563: 6559: 6554: 6549: 6545: 6537: 6536: 6535: 6515: 6512: 6507: 6502: 6498: 6494: 6489: 6484: 6475: 6468: 6465: 6462: 6457: 6452: 6448: 6444: 6439: 6434: 6425: 6418: 6415: 6412: 6407: 6402: 6398: 6394: 6389: 6384: 6375: 6365: 6364: 6363: 6357: 6341: 6321: 6318: 6311: 6304: 6297: 6290: 6277: 6268: 6262: 6259: 6252: 6245: 6238: 6231: 6224: 6220: 6216: 6196: 6179: 6175: 6171: 6164: 6159: 6147: 6138: 6118: 6110: 6106: 6102: 6099: 6096: 6093: 6087: 6084: 6080: 6074: 6071: 6066: 6062: 6058: 6055: 6049: 6046: 6039: 6038: 6037: 6022: 6004: 5995: 5991: 5987: 5982: 5977: 5973: 5965: 5960: 5956: 5951: 5942: 5938: 5934: 5929: 5924: 5920: 5912: 5907: 5903: 5898: 5892: 5889: 5886: 5883: 5880: 5871: 5868: 5862: 5857: 5853: 5844: 5841: 5834: 5829: 5825: 5819: 5815: 5809: 5802: 5797: 5793: 5787: 5783: 5777: 5770: 5765: 5761: 5755: 5751: 5731: 5725: 5719: 5708: 5691: 5670: 5661: 5657: 5653: 5646: 5627: 5624: 5621: 5618: 5615: 5612: 5606: 5601: 5597: 5592: 5586: 5583: 5580: 5577: 5574: 5571: 5565: 5560: 5556: 5551: 5545: 5542: 5539: 5536: 5533: 5527: 5522: 5518: 5513: 5507: 5504: 5501: 5498: 5495: 5489: 5484: 5480: 5453: 5449: 5445: 5440: 5436: 5432: 5427: 5423: 5418: 5412: 5409: 5406: 5403: 5400: 5397: 5394: 5391: 5388: 5385: 5382: 5376: 5374: 5366: 5360: 5350: 5347: 5344: 5341: 5338: 5335: 5332: 5329: 5326: 5323: 5320: 5314: 5312: 5304: 5298: 5286: 5279: 5272: 5268: 5267: 5261: 5259: 5255: 5251: 5250:J. C. Maxwell 5247: 5242: 5240: 5235: 5233: 5224: 5216: 5203: 5197: 5188: 5179: 5175: 5171: 5167: 5160: 5156: 5152: 5147: 5143: 5141: 5136: 5130: 5123: 5116: 5107: 5100: 5093: 5073: 5069: 5066: 5063: 5058: 5048: 5045: 5042: 5039: 5034: 5024: 5019: 5009: 4996: 4995: 4994: 4984: 4977: 4957: 4949: 4939: 4931: 4926: 4915: 4909: 4903: 4894: 4890: 4886: 4877: 4873: 4866: 4855: 4850: 4846: 4842: 4837: 4832: 4828: 4823: 4818: 4813: 4799: 4798: 4797: 4793: 4770: 4765: 4759: 4752: 4745: 4739: 4734: 4729: 4718: 4713: 4707: 4700: 4693: 4687: 4682: 4677: 4660: 4659: 4658: 4654: 4645: 4623: 4619: 4615: 4612: 4607: 4604: 4596: 4595: 4594: 4592: 4567: 4564: 4559: 4545: 4544: 4543: 4523: 4515: 4511: 4501: 4497: 4487: 4483: 4476: 4471: 4458: 4457: 4456: 4454: 4449: 4445: 4437: 4429: 4405: 4402: 4399: 4396: 4393: 4388: 4384: 4380: 4377: 4374: 4369: 4365: 4361: 4358: 4355: 4350: 4346: 4335: 4334: 4333: 4329: 4325: 4321: 4311: 4303: 4295: 4287: 4279: 4271: 4263: 4255: 4247: 4242: 4234: 4230: 4228: 4209: 4206: 4203: 4198: 4188: 4185: 4182: 4179: 4174: 4164: 4159: 4149: 4137: 4136: 4135: 4129: 4120: 4116:(center) and 4111: 4106: 4102: 4099: 4095: 4087: 4079: 4064: 4056: 4044: 4036: 4024: 4016: 4007: 3999: 3990: 3988: 3978: 3973: 3972:Earth section 3963: 3955: 3949: 3930: 3910: 3903: 3881: 3875: 3870: 3864: 3860: 3854: 3850: 3846: 3841: 3837: 3831: 3827: 3823: 3818: 3814: 3808: 3804: 3796: 3793: 3790: 3787: 3780: 3779: 3773: 3771: 3767: 3763: 3751: 3731: 3725: 3722: 3719: 3704: 3700: 3694: 3690: 3684: 3681: 3678: 3673: 3669: 3658: 3654: 3651: 3648: 3642: 3639: 3635: 3630: 3627: 3623: 3617: 3613: 3609: 3606: 3603: 3594: 3586: 3585: 3584: 3564: 3561: 3558: 3553: 3550: 3547: 3541: 3538: 3533: 3527: 3523: 3519: 3513: 3505: 3501: 3497: 3494: 3491: 3482: 3474: 3473: 3472: 3454: 3450: 3447: 3444: 3439: 3433: 3429: 3425: 3422: 3416: 3413: 3409: 3403: 3399: 3395: 3392: 3389: 3380: 3372: 3371: 3370: 3353: 3347: 3344: 3341: 3326: 3322: 3316: 3312: 3306: 3303: 3300: 3295: 3291: 3281: 3277: 3273: 3270: 3267: 3259: 3255: 3251: 3245: 3241: 3235: 3232: 3228: 3222: 3218: 3214: 3211: 3208: 3199: 3191: 3190: 3189: 3187: 3182: 3179: 3173: 3167: 3160: 3156: 3149: 3145: 3141: 3134: 3130: 3126: 3104: 3098: 3095: 3091: 3087: 3082: 3079: 3075: 3071: 3066: 3063: 3059: 3054: 3047: 3043: 3039: 3036: 3033: 3030: 3023: 3022: 3021: 3018: 3011: 3007: 2985: 2981: 2978: 2971: 2967: 2961: 2957: 2951: 2944: 2940: 2934: 2930: 2923: 2916: 2912: 2908: 2905: 2902: 2899: 2896: 2893: 2886: 2885: 2884: 2882: 2879:, one of the 2862: 2858: 2846: 2844: 2838: 2834: 2830: 2823: 2819: 2815: 2795: 2792: 2789: 2786: 2783: 2780: 2776: 2769: 2763: 2759: 2755: 2750: 2746: 2741: 2735: 2731: 2724: 2718: 2714: 2710: 2705: 2701: 2696: 2690: 2686: 2679: 2674: 2670: 2665: 2660: 2657: 2652: 2646: 2640: 2637: 2630: 2629: 2628: 2611: 2607: 2600: 2594: 2589: 2585: 2577: 2574: 2571: 2565: 2562: 2556: 2550: 2545: 2541: 2533: 2530: 2527: 2521: 2517: 2507: 2501: 2498: 2493: 2490: 2487: 2484: 2478: 2473: 2469: 2465: 2462: 2459: 2456: 2449: 2448: 2447: 2445: 2439: 2415: 2412: 2409: 2406: 2403: 2400: 2396:circumscribed 2391: 2386: 2383: 2380: 2377: 2369: 2364: 2360: 2355: 2346: 2338: 2337: 2336: 2334: 2330: 2326: 2304: 2301: 2300:circumscribed 2281: 2276: 2274: 2270: 2250: 2247: 2244: 2241: 2235: 2232: 2226: 2223: 2216: 2215: 2214: 2211: 2207: 2201: 2197: 2191: 2187: 2181: 2177: 2173: 2169: 2150: 2147: 2144: 2141: 2138: 2132: 2129: 2123: 2120: 2113: 2112: 2111: 2109: 2099: 2097: 2093: 2089: 2084: 2078: 2052: 2049: 2046: 2043: 2040: 2037: 2034: 2030: 2024: 2021: 2015: 2012: 2009: 2003: 2000: 1994: 1992: 1984: 1978: 1975: 1970: 1966: 1960: 1956: 1950: 1946: 1942: 1939: 1936: 1931: 1927: 1922: 1918: 1915: 1910: 1906: 1900: 1896: 1892: 1889: 1886: 1881: 1877: 1871: 1867: 1862: 1856: 1852: 1846: 1843: 1840: 1830: 1827: 1816: 1815: 1814: 1793: 1784: 1778: 1775: 1765: 1759: 1756: 1750: 1744: 1741: 1731: 1725: 1722: 1716: 1710: 1707: 1701: 1696: 1693: 1688: 1682: 1675: 1668: 1662: 1653: 1652: 1651: 1648: 1642: 1638: 1634: 1612: 1609: 1606: 1603: 1600: 1597: 1594: 1590: 1584: 1581: 1575: 1572: 1569: 1563: 1560: 1554: 1547: 1546: 1545: 1522: 1516: 1510: 1507: 1504: 1501: 1499: 1494: 1487: 1481: 1475: 1472: 1466: 1460: 1457: 1454: 1451: 1449: 1444: 1437: 1431: 1425: 1422: 1416: 1410: 1407: 1404: 1401: 1399: 1394: 1383: 1382: 1381: 1378: 1374: 1370: 1366: 1352: 1333: 1330: 1327: 1324: 1321: 1318: 1315: 1311: 1308: 1305: 1302: 1299: 1296: 1289: 1288: 1287: 1264: 1258: 1252: 1249: 1246: 1243: 1241: 1236: 1229: 1223: 1217: 1214: 1208: 1202: 1199: 1196: 1193: 1191: 1186: 1179: 1173: 1167: 1164: 1158: 1152: 1149: 1146: 1143: 1141: 1136: 1125: 1124: 1123: 1115: 1113: 1097: 1094: 1091: 1088: 1085: 1077: 1061: 1058: 1055: 1052: 1049: 1029: 1026: 1023: 1020: 1017: 1008: 994: 991: 988: 985: 982: 973: 959: 939: 916: 913: 910: 903: 899: 894: 891: 886: 882: 876: 872: 865: 858: 854: 849: 846: 841: 837: 833: 830: 825: 821: 815: 811: 804: 797: 793: 788: 785: 780: 776: 772: 769: 764: 760: 754: 750: 739: 738: 737: 720: 717: 714: 711: 708: 705: 702: 699: 696: 693: 690: 687: 684: 681: 678: 675: 672: 669: 666: 663: 657: 651: 648: 645: 642: 639: 628: 623: 621: 617: 613: 608: 604: 600: 580: 577: 574: 571: 568: 542: 539: 536: 533: 530: 504: 501: 498: 495: 492: 480: 466: 446: 426: 403: 400: 397: 390: 386: 380: 376: 370: 363: 359: 353: 349: 343: 336: 332: 326: 322: 312: 311: 310: 308: 298: 296: 295: 290: 289: 284: 280: 279: 274: 273: 270:ellipsoid of 266: 264: 260: 256: 252: 251:line segments 248: 244: 241: 240:perpendicular 236: 234: 230: 227:is either an 226: 225:cross section 222: 218: 214: 210: 205: 203: 199: 195: 191: 181: 175: 168: 161: 156: 153: 150: 144: 137: 133: 128: 127: 123: 120: 114: 110: 106: 101: 100: 96: 95: 86: 78: 66: 58: 46: 38: 28: 22: 11084: 11061: 11045: 11040: 11032: 11027: 10994: 10990: 10984: 10968: 10963: 10937:. Retrieved 10917: 10882: 10853: 10848: 10840: 10835: 10827: 10822: 10814: 10809: 10793: 10788: 10780: 10779:Staude, O.: 10775: 10767: 10766:Staude, O.: 10762: 10754: 10753:Staude, O.: 10749: 10741: 10736: 10718: 10712: 10693: 10667:. Retrieved 10658: 10648: 10637:. Retrieved 10633: 10624: 10613:. Retrieved 10609: 10600: 10586: 10575:. Retrieved 10562: 10557: 10545: 10489: 10483: 10449: 10428: 10420: 10416: 10398: 10392: 10378: 10371: 10232: 10215: 10187: 10164: 10153: 10148:dwarf planet 10131: 10127: 10123: 10120: 9797: 9718: 9687: 9683: 9679: 9621:of atoms in 9510: 9507:Applications 9198: 9191: 9184: 9180: 9176: 9172: 9166: 8981: 8977: 8973: 8970: 8959: 8950: 8941: 8930: 8923: 8914: 8907: 8900: 8897: 8686: 8679: 8678: 8669: 8662: 8655: 8652: 8523: 8522: 8513: 8506: 8501: 8496: 8474: 8472: 8401: 8223: 8222: 8197: 8162: 8155: 8142: 8128: 8122: 8116: 8110: 8100: 8096:eigenvectors 8093: 8087: 8081: 8075: 8071: 8066: 7990: 7985: 7983: 7897: 7859: 7842: 7839: 7836:As a quadric 7823: 7606: 7599: 7593: 7584: 7578: 7431: 7364: 7362: 7261: 7255: 7251: 7244: 7195: 7173:tangent line 7148: 7143: 6997: 6990: 6981: 6974: 6964: 6960: 6957: 6949: 6830: 6762:and defines 6757: 6753: 6749:focal curves 6748: 6739: 6735: 6731: 6728: 6530: 6347: 6327: 6316: 6309: 6302: 6295: 6288: 6275: 6266: 6263: 6257: 6250: 6243: 6236: 6229: 6222: 6218: 6214: 6191: 6177: 6173: 6166: 6162: 6145: 6136: 6133: 6031: 5723: 5717: 5706: 5689: 5668: 5659: 5644: 5285:focal conics 5277: 5270: 5253: 5245: 5243: 5236: 5229: 5201: 5195: 5173: 5169: 5165: 5158: 5154: 5150: 5145: 5144: 5137: 5128: 5121: 5114: 5105: 5098: 5091: 5088: 4982: 4975: 4972: 4788: 4785: 4649: 4646: 4643: 4590: 4588: 4541: 4447: 4439: 4431: 4423: 4420: 4327: 4323: 4319: 4309: 4301: 4293: 4285: 4277: 4269: 4261: 4253: 4245: 4243:The scaling 4240: 4239: 4224: 4127: 4118: 4109: 4104: 4103: 4097: 4089: 4081: 4073: 4062: 4054: 4042: 4034: 4022: 4014: 4005: 4004: 3983: 3953: 3947: 3931: 3908: 3901: 3898: 3762:eccentricity 3749: 3746: 3582: 3470: 3368: 3183: 3177: 3171: 3165: 3158: 3154: 3147: 3143: 3139: 3132: 3128: 3124: 3121: 3016: 3009: 3005: 3002: 2847: 2836: 2832: 2828: 2821: 2817: 2813: 2810: 2626: 2444:surface area 2441: 2432:Surface area 2277: 2266: 2209: 2205: 2199: 2195: 2189: 2185: 2179: 2175: 2171: 2165: 2105: 2085: 2071: 1812: 1649: 1627: 1543: 1379: 1372: 1368: 1364: 1348: 1285: 1121: 1009: 974: 931: 624: 606: 602: 598: 481: 418: 304: 292: 286: 276: 269: 267: 262: 258: 254: 237: 206: 189: 187: 180:bottom right 179: 173: 166: 159: 154: 148: 142: 135: 131: 124: 118: 112: 108: 104: 97: 84: 76: 64: 56: 44: 36: 10485:ellipticity 10177:, moons in 9539:in general. 8479:unit sphere 8477:,1) is the 8175:(also, see 8106:eigenvalues 7012:around the 6972:) one gets 6958:In case of 5256:written by 3665:where 3287:where 1074:, it is an 482:The points 157:ellipsoid, 149:bottom left 11143:Categories 11072:References 10958:pp. 17–18. 10939:2013-10-12 10839:O. Hesse: 10813:O. Hesse: 10639:2024-07-23 10615:2024-07-23 10577:2012-01-08 10491:oblateness 10480:Flattening 9560:rigid body 7185:orthogonal 7156:True curve 7016:-axis and 5269:Choose an 5258:D. Hilbert 4008:Ellipsoid 3970:See also: 2811:and where 2436:See also: 1110:, it is a 629:for which 272:revolution 221:polynomial 11133:MathWorld 11125:Ellipsoid 11115:Ellipsoid 10804:, p. 20 . 10740:W. Böhm: 10423:) = exp(− 10339:μ 10335:− 10319:− 10314:Σ 10305:μ 10301:− 10282:⋅ 9780:ρ 9768:π 9750:ρ 9545:Mechanics 9426:⁡ 9420:− 9365:⁡ 9304:⁡ 9243:⁡ 9149:θ 9146:⁡ 9128:× 9110:φ 9107:⁡ 9101:θ 9098:⁡ 9080:× 9062:φ 9059:⁡ 9053:θ 9050:⁡ 9032:× 9011:φ 9005:θ 8882:π 8873:φ 8870:≤ 8854:π 8845:θ 8833:π 8827:− 8820:θ 8817:⁡ 8796:φ 8793:⁡ 8787:θ 8784:⁡ 8763:φ 8760:⁡ 8754:θ 8751:⁡ 8712:φ 8706:θ 8544:↦ 8430:⋅ 8414:− 8372:− 8337:− 8045:− 8010:− 7955:− 7920:− 7799:⋯ 7752:π 7735:⋅ 7726:π 7717:≈ 7704:⋯ 7652:Γ 7637:π 7521:⋯ 7378:− 7332:− 7192:asymptote 7104:− 6935:λ 6932:− 6898:¯ 6885:λ 6882:− 6848:¯ 6803:¯ 6794:− 6773:λ 6691:− 6660:− 6610:− 6560:− 6516:λ 6513:− 6479:¯ 6466:λ 6463:− 6429:¯ 6416:λ 6413:− 6379:¯ 6028:Semi-axes 5988:− 5935:− 5884:− 5613:− 5572:− 5446:− 5413:ψ 5410:⁡ 5392:ψ 5389:⁡ 5367:ψ 5345:φ 5342:⁡ 5330:φ 5327:⁡ 5305:φ 5278:hyperbola 5067:⁡ 5043:⁡ 4940:× 4887:− 4824:ρ 4753:ρ 4694:ρ 4620:δ 4616:− 4605:ρ 4568:δ 4241:Solution: 4207:⁡ 4183:⁡ 3797:π 3791:≈ 3770:Mathworld 3685:− 3652:⁡ 3610:π 3562:− 3542:⁡ 3520:π 3498:π 3448:⁡ 3426:− 3396:π 3307:− 3271:⁡ 3215:π 3096:− 3080:− 3064:− 2903:π 2790:≥ 2784:≥ 2756:− 2711:− 2647:φ 2641:⁡ 2601:φ 2595:⁡ 2572:φ 2557:φ 2551:⁡ 2528:φ 2508:φ 2502:⁡ 2488:π 2466:π 2351:inscribed 2329:inscribed 2242:π 2168:diameters 2139:π 2075:would be 2050:π 2041:λ 2038:≤ 2022:π 2016:≤ 2013:γ 2010:≤ 2001:π 1995:− 1979:γ 1976:⁡ 1940:γ 1937:⁡ 1919:λ 1916:⁡ 1890:λ 1887:⁡ 1785:γ 1779:⁡ 1766:λ 1760:⁡ 1751:γ 1745:⁡ 1732:λ 1726:⁡ 1717:γ 1711:⁡ 1610:π 1601:λ 1598:≤ 1582:π 1576:≤ 1573:θ 1570:≤ 1561:π 1555:− 1517:θ 1511:⁡ 1482:λ 1476:⁡ 1467:θ 1461:⁡ 1432:λ 1426:⁡ 1417:θ 1411:⁡ 1331:π 1322:φ 1319:≤ 1309:π 1306:≤ 1303:θ 1300:≤ 1259:θ 1253:⁡ 1224:φ 1218:⁡ 1209:θ 1203:⁡ 1174:φ 1168:⁡ 1159:θ 1153:⁡ 1027:≠ 960:φ 940:θ 895:θ 892:⁡ 850:φ 847:⁡ 834:θ 831:⁡ 789:φ 786:⁡ 773:θ 770:⁡ 721:θ 718:⁡ 706:φ 703:⁡ 697:θ 694:⁡ 682:φ 679:⁡ 673:θ 670:⁡ 190:ellipsoid 155:Tri-axial 11159:Quadrics 11154:Surfaces 11081:(1972), 11011:29609953 10973:Archived 10948:cite web 10930:Archived 10880:(1993), 10856:, p. 125 10701:Archived 10679:cite web 10669:25 March 10663:Archived 10515:Homoeoid 10498:Focaloid 10458:See also 10202:piriform 10181:such as 9675:prostate 9664:Medicine 9647:Lighting 8473:where S( 8169:rotation 8138:diameter 6970:spheroid 6356:confocal 6324:Converse 6158:confocal 5146:Example: 4589:is the 3904:≈ 1.6075 1353:, where 283:rotation 278:spheroid 261:(rarely 217:zero set 198:scalings 126:Spheroid 11121:, 2007. 11019:4621745 10910:1261419 10830:, p. 24 10439:⁠ 10425:⁠ 10243:finance 10206:oviform 10156:cuboids 9725:density 9708:⁠ 9694:⁠ 9609:crystal 9516:Geodesy 7828:is the 7824:(where 7417:, is a 7149:Bottom: 6211:⁠ 6199:⁠ 6189:yields 5271:ellipse 5140:ellipse 4451:be the 4314:⁠ 4298:⁠ 4290:⁠ 4274:⁠ 4266:⁠ 4250:⁠ 4227:ellipse 4105:Wanted: 4067:⁠ 4051:⁠ 4047:⁠ 4031:⁠ 4027:⁠ 4011:⁠ 3925:⁠ 3913:⁠ 3766:ellipse 3599:prolate 2877:⁠ 2850:⁠ 2327:of the 2325:volumes 2321:⁠ 2307:⁠ 2296:⁠ 2284:⁠ 2183:(where 2088:geodesy 1631:is the 620:ellipse 233:bounded 229:ellipse 213:surface 89:⁠ 73:⁠ 69:⁠ 53:⁠ 49:⁠ 33:⁠ 11097: 11052: 11017: 11009: 10908: 10898: 10800: 10725: 10399:Σ 10393:μ 10382:is an 10372:where 10171:oblate 10167:Haumea 10144:Haumea 9690:× 0.52 9574:stress 8982:φ 8978:θ 8966:| 8939:| 8653:where 7984:is an 7857:, and 7826:Γ 7247:pencil 5732:Then: 5727:| 5715:| 5683:- and 5656:vertex 5645:string 5643:and a 5276:and a 5208:| 5193:| 4991:ρ 4955: 4796:, let 4786:Where 4668: 4647:Where 4591:center 4448:δ 4343: 4006:Given: 3753:oblate 3649:arcsin 3511: 3487:oblate 3445:artanh 3385:oblate 3268:artanh 3204:oblate 3175:, and 2627:where 2311:π 2305:, and 2280:volume 2269:oblate 2108:volume 2102:Volume 2090:, the 2081:λ 2073:γ 1813:where 1645:λ 1629:θ 1544:where 1359:φ 1355:θ 1286:where 932:where 618:of an 419:where 194:sphere 99:Sphere 11091:Wiley 11015:S2CID 10933:(PDF) 10926:(PDF) 10538:Notes 10402:is a 10183:Mimas 9607:in a 9527:Earth 9187:) = 0 8934:1,2,3 8151:width 7393:in a 5157:= 5, 5153:= 4, 4225:(see 3956:= log 3927:= 1.6 3899:Here 3583:and 2333:boxes 1639:, or 1078:; if 1010:When 975:When 219:of a 162:= 4.5 11127:and 11095:ISBN 11050:ISBN 11007:PMID 10954:link 10896:ISBN 10798:ISBN 10723:ISBN 10685:link 10671:2018 10488:and 10233:The 10121:For 9798:The 9721:mass 9719:The 9638:, a 8876:< 8848:< 8842:< 8191:and 8149:The 8136:The 8126:and 8094:The 8085:and 7850:-by- 7259:for 7169:cone 7144:Top: 6988:and 6273:and 6143:and 5407:sinh 5386:cosh 4794:≠ ±1 4655:= ±1 4421:Let 3940:and 3723:> 3345:< 3137:and 2826:and 2442:The 2278:The 2106:The 2044:< 1643:and 1604:< 1325:< 1095:< 1059:> 614:and 558:and 459:and 309:as: 10999:doi 10888:doi 10204:or 9730:is 9671:MRI 9423:det 9362:det 9301:det 9240:det 9143:sin 9104:sin 9095:cos 9056:cos 9047:cos 8986:is 8814:sin 8790:sin 8781:cos 8757:cos 8748:cos 8507:An 8108:of 8098:of 8069:of 7840:If 7832:). 7605:... 7267:→ 0 7210:. ( 6968:(a 6358:to 6348:If 6172:+ ( 6165:= 2 5704:to 5696:to 5339:sin 5324:cos 5176:= 5 5161:= 3 5064:sin 5040:cos 4330:= 1 4229:). 4204:sin 4180:cos 4069:= 1 3989:). 3471:or 3369:or 3181:. 2638:cos 2586:cos 2542:sin 2499:sin 2271:or 2208:= 2 2198:= 2 2188:= 2 2086:In 1967:sin 1928:cos 1907:sin 1878:cos 1776:sin 1757:sin 1742:cos 1723:cos 1708:cos 1508:sin 1473:sin 1458:cos 1423:cos 1408:cos 1250:cos 1215:sin 1200:sin 1165:cos 1150:sin 883:cos 838:sin 822:sin 777:cos 761:sin 715:cos 700:sin 691:sin 676:cos 667:sin 625:In 188:An 176:= 3 169:= 6 145:= 3 138:= 5 119:top 115:= 4 91:= 1 11145:: 11131:, 11093:, 11013:. 11005:. 10995:25 10993:. 10950:}} 10946:{{ 10928:. 10906:MR 10904:, 10894:, 10876:; 10872:; 10861:^ 10681:}} 10677:{{ 10661:. 10657:. 10632:. 10608:. 10447:. 10225:. 10130:= 10126:= 10102:0. 9686:× 9682:× 9197:= 9183:, 9179:, 8980:, 8929:± 8913:, 8906:, 8684:. 8668:, 8661:, 8528:: 8120:, 8074:- 7617:: 7564:1. 7429:. 7363:A 7254:, 6996:= 6980:= 6963:= 6738:, 6734:, 6336:, 6332:, 6320:. 6315:− 6301:= 6284:xz 6261:. 6256:− 6242:= 6221:+ 6217:− 6197:= 6176:− 6154:xy 6036:: 5287:: 5178:. 5172:+ 5168:+ 5142:. 5127:, 5120:, 5104:, 5097:, 4981:, 4446:= 4438:+ 4430:+ 4326:+ 4322:+ 4296:= 4292:, 4272:= 4268:, 4248:= 4125:, 4096:= 4088:+ 4080:+ 4049:+ 4029:+ 3962:. 3948:ab 3946:2π 3911:= 3772:. 3539:ln 3188:: 3169:, 3146:, 3131:, 3020:: 2835:, 2820:, 2203:, 2193:, 2178:, 2174:, 2098:. 1635:, 1371:, 1367:, 1114:. 622:. 605:, 601:, 520:, 439:, 204:. 178:, 171:; 164:, 147:, 140:, 134:= 129:, 117:, 111:= 107:= 102:, 93:: 71:+ 51:+ 11135:. 11113:" 11056:. 11021:. 11001:: 10956:) 10942:. 10890:: 10687:) 10673:. 10642:. 10618:. 10594:. 10580:. 10565:( 10445:z 10441:) 10436:2 10433:/ 10429:z 10421:z 10419:( 10417:g 10412:g 10384:n 10379:x 10374:k 10356:) 10349:T 10344:) 10331:x 10327:( 10322:1 10309:) 10297:x 10293:( 10289:( 10285:g 10279:k 10276:= 10273:) 10270:x 10267:( 10264:f 10251:f 10132:c 10128:b 10124:a 10099:= 10093:x 10090:z 10085:I 10081:= 10075:z 10072:y 10067:I 10063:= 10053:y 10050:x 10045:I 10037:, 10033:) 10027:2 10023:b 10019:+ 10014:2 10010:a 10005:( 10001:m 9995:5 9992:1 9986:= 9976:z 9973:z 9968:I 9962:, 9958:) 9952:2 9948:a 9944:+ 9939:2 9935:c 9930:( 9926:m 9920:5 9917:1 9911:= 9901:y 9898:y 9893:I 9887:, 9883:) 9877:2 9873:c 9869:+ 9864:2 9860:b 9855:( 9851:m 9845:5 9842:1 9836:= 9826:x 9823:x 9818:I 9783:. 9777:c 9774:b 9771:a 9762:3 9759:4 9753:= 9747:V 9744:= 9741:m 9728:ρ 9710:) 9705:6 9702:/ 9698:π 9688:H 9684:W 9680:L 9625:. 9611:. 9562:. 9529:. 9491:0 9488:= 9483:2 9478:) 9472:3 9467:f 9462:, 9457:2 9452:f 9447:, 9442:1 9437:f 9431:( 9415:2 9410:) 9405:x 9401:, 9396:2 9391:f 9386:, 9381:1 9376:f 9370:( 9359:+ 9354:2 9349:) 9343:3 9338:f 9333:, 9329:x 9325:, 9320:1 9315:f 9309:( 9298:+ 9293:2 9288:) 9282:3 9277:f 9272:, 9267:2 9262:f 9257:, 9253:x 9248:( 9237:= 9234:) 9231:z 9228:, 9225:y 9222:, 9219:x 9216:( 9213:F 9199:0 9195:0 9192:f 9185:z 9181:y 9177:x 9175:( 9173:F 9152:. 9138:2 9133:f 9123:1 9118:f 9113:+ 9090:1 9085:f 9075:3 9070:f 9065:+ 9042:3 9037:f 9027:2 9022:f 9017:= 9014:) 9008:, 9002:( 8998:n 8984:) 8976:( 8974:x 8963:3 8960:f 8954:2 8951:f 8945:1 8942:f 8931:f 8927:0 8924:f 8918:3 8915:f 8911:2 8908:f 8904:1 8901:f 8894:. 8879:2 8867:0 8863:, 8857:2 8836:2 8823:, 8809:3 8804:f 8799:+ 8776:2 8771:f 8766:+ 8743:1 8738:f 8733:+ 8728:0 8723:f 8718:= 8715:) 8709:, 8703:( 8699:x 8680:A 8673:3 8670:f 8666:2 8663:f 8659:1 8656:f 8636:3 8631:f 8626:z 8623:+ 8618:2 8613:f 8608:y 8605:+ 8600:1 8595:f 8590:x 8587:+ 8582:0 8577:f 8572:= 8568:x 8563:A 8559:+ 8554:0 8549:f 8540:x 8524:A 8517:0 8514:f 8475:0 8457:v 8453:+ 8450:) 8447:1 8444:, 8440:0 8436:( 8433:S 8425:2 8421:/ 8417:1 8410:A 8386:1 8383:= 8380:) 8376:v 8368:x 8364:( 8359:A 8351:T 8346:) 8341:v 8333:x 8329:( 8309:. 8305:A 8284:; 8279:2 8275:/ 8271:1 8266:A 8259:2 8255:/ 8251:1 8246:A 8241:= 8237:A 8224:A 8207:A 8159:. 8156:A 8146:. 8143:A 8129:c 8123:b 8117:a 8111:A 8101:A 8088:v 8082:A 8076:v 8072:x 8053:) 8049:v 8041:x 8037:( 8032:A 8024:T 8019:) 8014:v 8006:x 8002:( 7991:v 7986:n 7969:1 7966:= 7963:) 7959:v 7951:x 7947:( 7942:A 7934:T 7929:) 7924:v 7916:x 7912:( 7898:x 7883:, 7878:n 7873:R 7860:v 7852:n 7848:n 7843:A 7807:n 7803:a 7794:2 7790:a 7784:1 7780:a 7774:2 7770:/ 7766:n 7761:) 7756:n 7749:e 7746:2 7740:( 7729:n 7722:1 7712:n 7708:a 7699:2 7695:a 7689:1 7685:a 7677:) 7673:1 7670:+ 7665:2 7662:n 7656:( 7645:2 7642:n 7631:= 7628:V 7609:n 7607:a 7603:2 7600:a 7597:1 7594:a 7589:R 7581:n 7561:= 7554:2 7549:n 7545:a 7539:2 7534:n 7530:x 7524:+ 7518:+ 7511:2 7506:2 7502:a 7496:2 7491:2 7487:x 7481:+ 7474:2 7469:1 7465:a 7459:2 7454:1 7450:x 7405:n 7381:1 7375:n 7338:. 7335:c 7329:a 7326:3 7323:= 7320:l 7316:, 7313:b 7310:= 7305:y 7301:r 7296:, 7293:a 7290:= 7285:x 7281:r 7264:z 7262:r 7256:b 7252:a 7236:. 7224:H 7220:O 7216:V 7212:H 7208:V 7204:H 7200:V 7181:V 7177:V 7165:V 7161:V 7130:. 7112:2 7108:c 7099:2 7094:x 7090:r 7082:= 7077:z 7073:r 7069:= 7064:y 7060:r 7055:, 7052:l 7046:2 7043:1 7037:= 7032:x 7028:r 7014:x 7006:x 7001:2 6998:F 6994:2 6991:S 6985:1 6982:F 6978:1 6975:S 6965:c 6961:a 6927:2 6922:z 6918:r 6914:= 6909:2 6904:z 6895:r 6888:, 6877:2 6872:y 6868:r 6864:= 6859:2 6854:y 6845:r 6814:2 6809:x 6800:r 6789:2 6784:x 6780:r 6776:= 6758:l 6745:E 6740:c 6736:b 6732:a 6712:2 6708:b 6704:= 6699:2 6695:c 6686:2 6682:a 6678:= 6673:2 6668:z 6664:r 6655:2 6650:y 6646:r 6641:, 6636:2 6632:a 6628:= 6623:2 6618:z 6614:r 6605:2 6600:x 6596:r 6591:, 6586:2 6582:c 6578:= 6573:2 6568:y 6564:r 6555:2 6550:x 6546:r 6533:E 6508:2 6503:z 6499:r 6495:= 6490:2 6485:z 6476:r 6469:, 6458:2 6453:y 6449:r 6445:= 6440:2 6435:y 6426:r 6419:, 6408:2 6403:x 6399:r 6395:= 6390:2 6385:x 6376:r 6360:E 6351:E 6338:l 6334:b 6330:a 6317:a 6310:x 6303:r 6296:z 6289:r 6279:2 6276:S 6270:1 6267:S 6258:c 6251:x 6244:r 6237:y 6230:r 6225:) 6223:c 6219:a 6215:l 6213:( 6208:2 6205:/ 6202:1 6194:x 6192:r 6186:x 6184:r 6180:) 6178:c 6174:a 6169:x 6167:r 6163:l 6149:2 6146:F 6140:1 6137:F 6119:. 6116:) 6111:z 6107:r 6103:, 6100:0 6097:, 6094:0 6091:( 6088:= 6085:Z 6081:, 6078:) 6075:0 6072:, 6067:y 6063:r 6059:, 6056:0 6053:( 6050:= 6047:Y 6034:P 6005:. 5996:2 5992:a 5983:2 5978:x 5974:r 5966:= 5961:z 5957:r 5952:, 5943:2 5939:c 5930:2 5925:x 5921:r 5913:= 5908:y 5904:r 5899:, 5896:) 5893:c 5890:+ 5887:a 5881:l 5878:( 5872:2 5869:1 5863:= 5858:x 5854:r 5845:1 5842:= 5835:2 5830:z 5826:r 5820:2 5816:z 5810:+ 5803:2 5798:y 5794:r 5788:2 5784:y 5778:+ 5771:2 5766:x 5762:r 5756:2 5752:x 5734:P 5724:P 5721:1 5718:S 5710:2 5707:F 5702:P 5698:P 5693:1 5690:S 5685:z 5681:y 5677:P 5672:2 5669:F 5663:1 5660:S 5651:. 5649:l 5631:) 5628:0 5625:, 5622:0 5619:, 5616:a 5610:( 5607:= 5602:2 5598:S 5593:, 5590:) 5587:0 5584:, 5581:0 5578:, 5575:c 5569:( 5566:= 5561:2 5557:F 5552:, 5549:) 5546:0 5543:, 5540:0 5537:, 5534:c 5531:( 5528:= 5523:1 5519:F 5514:, 5511:) 5508:0 5505:, 5502:0 5499:, 5496:a 5493:( 5490:= 5485:1 5481:S 5454:2 5450:b 5441:2 5437:a 5433:= 5428:2 5424:c 5419:, 5416:) 5404:b 5401:, 5398:0 5395:, 5383:c 5380:( 5377:= 5370:) 5364:( 5361:H 5354:) 5351:0 5348:, 5336:b 5333:, 5321:a 5318:( 5315:= 5308:) 5302:( 5299:E 5281:H 5274:E 5205:2 5202:S 5199:1 5196:S 5174:z 5170:y 5166:x 5159:c 5155:b 5151:a 5132:2 5129:f 5125:1 5122:f 5118:0 5115:f 5109:2 5106:e 5102:1 5099:e 5095:0 5092:e 5074:. 5070:t 5059:2 5054:e 5049:+ 5046:t 5035:1 5030:e 5025:+ 5020:0 5015:e 5010:= 5006:u 4986:2 4983:e 4979:1 4976:e 4958:. 4950:1 4945:e 4936:m 4932:= 4927:2 4922:e 4916:, 4910:] 4904:0 4895:u 4891:m 4878:v 4874:m 4867:[ 4856:2 4851:v 4847:m 4843:+ 4838:2 4833:u 4829:m 4819:= 4814:1 4809:e 4791:w 4789:m 4771:. 4766:] 4760:0 4746:0 4740:[ 4735:= 4730:2 4725:e 4719:, 4714:] 4708:0 4701:0 4688:[ 4683:= 4678:1 4673:e 4652:w 4650:m 4624:2 4613:1 4608:= 4572:m 4565:= 4560:0 4555:e 4524:] 4516:w 4512:m 4502:v 4498:m 4488:u 4484:m 4477:[ 4472:= 4468:m 4444:w 4442:w 4440:m 4436:v 4434:v 4432:m 4428:u 4426:u 4424:m 4406:. 4403:d 4400:= 4397:w 4394:c 4389:z 4385:n 4381:+ 4378:v 4375:b 4370:y 4366:n 4362:+ 4359:u 4356:a 4351:x 4347:n 4328:w 4324:v 4320:u 4310:c 4306:/ 4302:z 4294:w 4286:b 4282:/ 4278:y 4270:v 4262:a 4258:/ 4254:x 4246:u 4210:t 4199:2 4194:f 4189:+ 4186:t 4175:1 4170:f 4165:+ 4160:0 4155:f 4150:= 4146:x 4131:2 4128:f 4122:1 4119:f 4113:0 4110:f 4098:d 4094:z 4092:z 4090:n 4086:y 4084:y 4082:n 4078:x 4076:x 4074:n 4063:c 4059:/ 4055:z 4043:b 4039:/ 4035:y 4023:a 4019:/ 4015:x 3958:2 3954:p 3942:b 3938:a 3934:c 3922:5 3919:/ 3916:8 3909:p 3902:p 3882:. 3876:p 3871:3 3865:p 3861:c 3855:p 3851:b 3847:+ 3842:p 3838:c 3832:p 3828:a 3824:+ 3819:p 3815:b 3809:p 3805:a 3794:4 3788:S 3758:e 3750:S 3732:, 3729:) 3726:a 3720:c 3717:( 3705:2 3701:c 3695:2 3691:a 3682:1 3679:= 3674:2 3670:e 3659:) 3655:e 3643:e 3640:a 3636:c 3631:+ 3628:1 3624:( 3618:2 3614:a 3607:2 3604:= 3595:S 3565:e 3559:1 3554:e 3551:+ 3548:1 3534:e 3528:2 3524:c 3514:+ 3506:2 3502:a 3495:2 3492:= 3483:S 3455:) 3451:e 3440:e 3434:2 3430:e 3423:1 3417:+ 3414:1 3410:( 3404:2 3400:a 3393:2 3390:= 3381:S 3354:, 3351:) 3348:a 3342:c 3339:( 3327:2 3323:a 3317:2 3313:c 3304:1 3301:= 3296:2 3292:e 3282:, 3278:) 3274:e 3260:2 3256:a 3252:e 3246:2 3242:c 3236:+ 3233:1 3229:( 3223:2 3219:a 3212:2 3209:= 3200:S 3178:c 3172:b 3166:a 3159:G 3155:R 3150:) 3148:k 3144:φ 3142:( 3140:E 3135:) 3133:k 3129:φ 3127:( 3125:F 3105:) 3099:2 3092:c 3088:, 3083:2 3076:b 3072:, 3067:2 3060:a 3055:( 3048:G 3044:R 3040:V 3037:3 3034:= 3031:S 3017:V 3010:G 3006:R 2986:) 2982:1 2979:, 2972:2 2968:a 2962:2 2958:c 2952:, 2945:2 2941:a 2935:2 2931:b 2924:( 2917:G 2913:R 2909:c 2906:b 2900:4 2897:= 2894:S 2863:G 2859:R 2839:) 2837:k 2833:φ 2831:( 2829:E 2824:) 2822:k 2818:φ 2816:( 2814:F 2796:, 2793:c 2787:b 2781:a 2777:, 2770:) 2764:2 2760:c 2751:2 2747:a 2742:( 2736:2 2732:b 2725:) 2719:2 2715:c 2706:2 2702:b 2697:( 2691:2 2687:a 2680:= 2675:2 2671:k 2666:, 2661:a 2658:c 2653:= 2650:) 2644:( 2612:, 2608:) 2604:) 2598:( 2590:2 2581:) 2578:k 2575:, 2569:( 2566:F 2563:+ 2560:) 2554:( 2546:2 2537:) 2534:k 2531:, 2525:( 2522:E 2518:( 2511:) 2505:( 2494:b 2491:a 2485:2 2479:+ 2474:2 2470:c 2463:2 2460:= 2457:S 2416:. 2413:c 2410:b 2407:a 2404:8 2401:= 2392:V 2387:, 2384:c 2381:b 2378:a 2370:3 2365:3 2361:8 2356:= 2347:V 2318:6 2315:/ 2293:3 2290:/ 2287:2 2263:. 2251:C 2248:B 2245:A 2236:6 2233:1 2227:= 2224:V 2210:c 2206:C 2200:b 2196:B 2190:a 2186:A 2180:C 2176:B 2172:A 2151:. 2148:c 2145:b 2142:a 2133:3 2130:4 2124:= 2121:V 2053:. 2047:2 2035:0 2031:, 2025:2 2004:2 1985:, 1971:2 1961:2 1957:b 1951:2 1947:a 1943:+ 1932:2 1923:) 1911:2 1901:2 1897:a 1893:+ 1882:2 1872:2 1868:b 1863:( 1857:2 1853:c 1847:c 1844:b 1841:a 1831:= 1828:R 1794:] 1788:) 1782:( 1769:) 1763:( 1754:) 1748:( 1735:) 1729:( 1720:) 1714:( 1702:[ 1697:R 1694:= 1689:] 1683:z 1676:y 1669:x 1663:[ 1613:, 1607:2 1595:0 1591:, 1585:2 1564:2 1523:, 1520:) 1514:( 1505:c 1502:= 1495:z 1488:, 1485:) 1479:( 1470:) 1464:( 1455:b 1452:= 1445:y 1438:, 1435:) 1429:( 1420:) 1414:( 1405:a 1402:= 1395:x 1375:) 1373:z 1369:y 1365:x 1363:( 1334:. 1328:2 1316:0 1312:, 1297:0 1265:, 1262:) 1256:( 1247:c 1244:= 1237:z 1230:, 1227:) 1221:( 1212:) 1206:( 1197:b 1194:= 1187:y 1180:, 1177:) 1171:( 1162:) 1156:( 1147:a 1144:= 1137:x 1098:c 1092:b 1089:= 1086:a 1062:c 1056:b 1053:= 1050:a 1030:c 1024:b 1021:= 1018:a 995:c 992:= 989:b 986:= 983:a 917:, 914:1 911:= 904:2 900:c 887:2 877:2 873:r 866:+ 859:2 855:b 842:2 826:2 816:2 812:r 805:+ 798:2 794:a 781:2 765:2 755:2 751:r 724:) 712:r 709:, 688:r 685:, 664:r 661:( 658:= 655:) 652:z 649:, 646:y 643:, 640:x 637:( 607:c 603:b 599:a 584:) 581:c 578:, 575:0 572:, 569:0 566:( 546:) 543:0 540:, 537:b 534:, 531:0 528:( 508:) 505:0 502:, 499:0 496:, 493:a 490:( 467:c 447:b 427:a 404:, 401:1 398:= 391:2 387:c 381:2 377:z 371:+ 364:2 360:b 354:2 350:y 344:+ 337:2 333:a 327:2 323:x 174:c 167:b 160:a 151:; 143:c 136:b 132:a 121:; 113:c 109:b 105:a 85:c 81:/ 77:z 65:b 61:/ 57:y 45:a 41:/ 37:x 23:.

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