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Ellipsoid

An ellipsoid is a smooth, closed surface in three-dimensional that generalizes the sphere by allowing unequal extents along its principal axes. It is mathematically defined by the equation \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, where a, b, and c represent the positive lengths of the semi-axes along the x, y, and z directions, respectively. If a = b = c, the ellipsoid reduces to a sphere; when two semi-axes are equal (e.g., a = b > c), it forms a , such as the oblate spheroid approximating Earth's shape, where the equatorial radius exceeds the polar radius. Ellipsoids exhibit symmetry about their principal axes and possess elliptical cross-sections in planes parallel to the coordinate planes. The surface is bounded, convex, and compact, with a volume given by \frac{4}{3}\pi abc and a surface area that, for the general case, involves elliptic integrals without a simple closed form. In higher dimensions, the concept extends to hyperellipsoids, defined similarly via forms. Beyond , ellipsoids model diverse phenomena across disciplines. In , the ellipsoid describes planetary figures of under and . In physics and , they represent particle shapes in flows, effective mass surfaces in , and confidence regions in statistical . Applications also span for rendering smooth objects, optimization algorithms like the for convex programming, and astronomy for modeling celestial bodies.

Definition and Fundamentals

Definition

An ellipsoid is a surface in three-dimensional that generalizes the sphere, representing a closed, bounded surface symmetric about its center. It arises as the image of a under an , which involves linear scaling along mutually perpendicular directions corresponding to its principal axes. This transformation intuitively stretches or compresses the sphere nonuniformly, producing a , egg-like form without edges or singularities. Ellipsoids are classified based on the lengths of their semi-axes, denoted typically as a, b, and c along the principal directions. A triaxial ellipsoid features three unequal semi-axes, yielding a fully asymmetric shape. In contrast, a spheroid, or ellipsoid of revolution, has two equal semi-axes and results from rotating an ellipse about one of its axes; it is oblate if the axis of rotation is the minor axis (flattened at the poles, as approximated for Earth), and prolate if the axis of rotation is the major axis (elongated along the rotation axis, resembling an American football). When all three semi-axes are equal, the ellipsoid degenerates to a sphere.

Relation to Other Quadrics

The ellipsoid belongs to the family of surfaces, which are defined by second-degree equations in three variables and encompass six non-degenerate types: ellipsoids, elliptic paraboloids, hyperbolic paraboloids, cones (double cones), hyperboloids of one sheet, and hyperboloids of two sheets. Among these, the ellipsoid is distinguished as the only bounded, closed, and quadric surface, arising from a in its defining equation. This ensures that all eigenvalues of the associated are positive, resulting in a compact surface that encloses a finite without extending to infinity. In contrast to other quadrics, the ellipsoid's compactness and convexity set it apart from unbounded surfaces like the of one or two sheets, which extend infinitely in at least one direction due to their indefinite s with both positive and negative eigenvalues. The hyperbolic paraboloid, often called a surface, features a with one positive and two negative eigenvalues (or vice versa), leading to a non-compact, non- shape that is orientable but lacks . Ellipsoids, however, are fully orientable and , sharing the smoothness and algebraic simplicity of all quadrics but uniquely forming a closed manifold homeomorphic to a . Degenerate cases occur when the ellipsoid's parameters cause axis collapse, transforming the surface into lower-dimensional or empty objects. If one semi-axis length approaches zero, the ellipsoid degenerates to an in the spanned by the remaining axes. Further collapse of a second semi-axis reduces it to a single point, while adjustments yielding no real solutions—such as all semi-axes approaching zero or inconsistent signs in the —result in the . These degeneracies highlight the ellipsoid's position at the boundary of classifications, where the positive definite condition begins to fail. The geometric interpretation of the ellipsoid's ties directly to its matrix eigenvalues: the lengths of the semi-axes are the reciprocals of the square roots of these eigenvalues, while the eigenvectors determine the axis directions. For a \mathbf{x}^T A \mathbf{x} = 1 with A positive definite and diagonalized as A = Q \Lambda Q^T, the semi-axis lengths are s_i = \lambda_i^{-1/2} along the directions of the columns of Q. This eigenvalue underscores the ellipsoid's role as a representative of positive definite quadrics, facilitating in linear and optimization.

Mathematical Descriptions

Cartesian Equation

The standard Cartesian equation for an ellipsoid centered at the origin and aligned with the coordinate axes is given by \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, where a \geq b \geq c > 0 are the lengths of the semi-axes along the x-, y-, and z-directions, respectively. This equation describes a bounded surface that extends from -a to a along the x-axis, -b to b along the y-axis, and -c to c along the z-axis. The parameters a, b, and c represent the intercepts of the surface with the respective coordinate axes, providing a direct measure of the ellipsoid's scaling and orientation in the aligned case. These intercepts define the extent of the ellipsoid in each principal direction, with the surface lying entirely within the rectangular box bounded by \pm a, \pm b, and \pm c. In general, the equation of an ellipsoid can be expressed in the quadratic form \mathbf{x}^T A \mathbf{x} = 1, where \mathbf{x} = (x, y, z)^T and A is a symmetric positive . For the axis-aligned case, A is diagonal, specifically A = \operatorname{diag}(1/a^2, 1/b^2, 1/c^2), which directly yields the standard equation upon expansion. The matrix A must be positive definite, meaning all its eigenvalues are positive, to ensure the quadratic form defines a real, bounded ellipsoidal surface rather than an unbounded or imaginary one. This condition guarantees that the level set \mathbf{x}^T A \mathbf{x} = 1 encloses a compact region in \mathbb{R}^3.

Parametric Representation

The surface of an ellipsoid centered at the origin with semi-axes lengths a, b, and c along the x-, y-, and z-axes, respectively, admits a parametric representation using spherical coordinates adapted to the asymmetry: \begin{align*} x &= a \sin \theta \cos \phi, \\ y &= b \sin \theta \sin \phi, \\ z &= c \cos \theta, \end{align*} where \theta \in [0, \pi] is the polar angle and \phi \in [0, 2\pi) is the azimuthal angle. This parameterization arises by applying a linear transformation to the unit sphere's standard parameterization (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta), stretching the coordinates by factors a, b, and c along each axis; this affine mapping preserves the quadratic nature of the surface while aligning it with the principal axes. An alternative rational parameterization, known as the stereographic form, projects from the unit via and then applies the same affine stretch, yielding rational functions in parameters u and v: \begin{align*} x &= a \frac{1 - u^2 - v^2}{1 + u^2 + v^2}, \\ y &= b \frac{2u}{1 + u^2 + v^2}, \\ z &= c \frac{2v}{1 + u^2 + v^2}, \end{align*} with u, v \in \mathbb{R} excluding the projection pole; this form excludes one point on the surface but is advantageous in for generating exact rational points without trigonometric evaluations. For surface integrals over the ellipsoid using the standard parameterization \mathbf{r}(\theta, \phi), the infinitesimal area element is dS = \|\mathbf{r}_\theta \times \mathbf{r}_\phi\| \, d\theta \, d\phi, where \mathbf{r}_\theta and \mathbf{r}_\phi are the partial ; the cross product magnitude provides the factor \sin \theta \sqrt{a^2 b^2 \cos^2 \theta + c^2 (a^2 \cos^2 \phi + b^2 \sin^2 \phi) \sin^2 \theta}, enabling evaluation of integrals such as or area without resolving the implicit .

Geometric Properties

Volume

The volume V enclosed by an ellipsoid with semi-principal axes lengths a, b, and c is given by the formula V = \frac{4}{3} \pi a b c. This expression arises from integrating over the interior region defined by the inequality \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq 1. To derive this, consider the triple integral V = \iiint_R dx\, dy\, dz, where R is the ellipsoidal region. Apply the change of variables u = x/a, v = y/b, w = z/c. The Jacobian determinant of this transformation is abc, and the region R maps to the unit ball u^2 + v^2 + w^2 \leq 1. Thus, the integral simplifies to V = abc \cdot \iiint_{u^2 + v^2 + w^2 \leq 1} du\, dv\, dw = abc times the volume of the unit ball, which is \frac{4}{3} \pi. A special case occurs when a = b = c = r, reducing the ellipsoid to a of r with V = \frac{4}{3} \pi r^3. For a prolate , where the semi-major axis a > b = c, the is V = \frac{4}{3} \pi a b^2; similarly, for an with a = b > c, it is V = \frac{4}{3} \pi a^2 c. This formula highlights the ellipsoidal volume as an affine scaling of the ball's volume by the factor abc, preserving the \pi and cubic structure but adjusting for directional stretches. Under uniform scaling of all axes by a factor k, the volume scales by k^3, consistent with dimensional analysis for three-dimensional regions.

Surface Area

The surface area of a general triaxial ellipsoid with semi-axes a, b, and c (assuming a > b > c > 0) lacks an elementary closed-form expression and is typically computed using elliptic integrals of the first and second kinds. This integral form arises from parameterizing the surface and integrating the magnitude of the cross product of partial derivatives over the parameter domain, which requires numerical evaluation for arbitrary semi-axes. For spheroids, where two semi-axes are equal, closed-form expressions exist in terms of elementary functions. A prolate (with a > b = c) has surface area S = 2\pi b^2 + 2\pi \frac{a b \arcsin e}{e}, where the e = \sqrt{1 - (b/a)^2}. An oblate (with a = b > c) has surface area S = 2\pi a^2 + \pi c^2 \frac{\ln \left( \frac{1+e}{1-e} \right)}{e}, with the same e = \sqrt{1 - (c/a)^2}. These formulas derive from the surface of revolution generated by rotating an about its or . Approximate formulas provide practical alternatives for the triaxial case, balancing accuracy and simplicity. Knud Thomsen's approximation, yielding relative errors under 1.3% across tested cases, is S \approx 4\pi \left( \frac{a^p b^p + a^p c^p + b^p c^p}{3} \right)^{1/p}, with p \approx 1.6075. Historical approximations, such as that proposed by in the late , offer simpler but less precise estimates by expanding the elliptic integrals in series or using geometric bounds. Numerical methods, including techniques for the double integral or series expansions, are commonly employed for high-precision computations in applications requiring exact values.

Plane Sections

The intersection of a with an ellipsoid generally yields an , provided the plane is not (resulting in a point) or misses the ellipsoid entirely (). This fundamental property arises because the ellipsoid is a bounded surface, and plane sections of quadrics are conic sections, with the bounded nature ensuring ellipses rather than unbounded hyperbolas or parabolas. To determine the specific from the intersection, substitute the plane equation into the ellipsoid's , yielding a in the plane's coordinates that represents the . The major and minor axes lengths can then be found by diagonalizing the associated of the projected ; the reciprocals of the square roots of its eigenvalues give the semi-axes lengths. The center of the is the point on the plane closest to the ellipsoid's center, and its orientation aligns with the eigenvectors of the . Formulas for these semi-axes and the center in terms of the ellipsoid's semi-axes and the plane's normal vector provide explicit computations. Parallel projections of an ellipsoid, such as under parallel rays, outline an elliptical boundary formed by projecting the contour curve where tangent planes are parallel to the projection direction. This contour is itself an on the ellipsoid, ensuring the shadow remains elliptical regardless of the direction, with the projected 's eccentricity and size depending on the relative to the principal axes. Special cases include the maximal area plane section, which occurs through the ellipsoid's in the spanned by its two longest semi-axes (the "equatorial" ), yielding an with area \pi a b where a and b are those semi-axes. For spheroids (ellipsoids of with two equal semi-axes), there exist at least two distinct orientations producing circular sections, corresponding to planes whose normals make specific angles with the of , such as \cos \theta = \pm \sqrt{(a^2 - c^2)/a^2} for semi-axes a = b > c.

Constructions and Focal Properties

Pins-and-String Construction

The pins-and-string construction offers a approach to generating an ellipsoid surface, building on the for plane ellipses and extending it to three dimensions through the use of focal conics. In the planar case, the construction begins by placing two pins at the foci of in a principal plane. A of fixed length $2m, where m > c (with $2c the distance between foci), is looped around the pins. A inserted into the loop is moved while keeping the string taut, tracing the ellipse as the set of points where the sum of distances to the foci remains constant at $2m. This technique extends to the 3D ellipsoid using the pair of confocal focal conics—an and a —in the equatorial plane perpendicular to one principal axis. Developed by Otto Staude in , the method replaces the point foci with these curves to generate the full surface. The , now a closed loop of length $2m (chosen greater than the girth of the combined conics), is draped around both the focal ellipse and focal hyperbola. The pencil point is placed within the loop and moved in space, maintaining tautness so the string remains to each conic at one point. The resulting path of the pencil forms the ellipsoid, exploiting the defining property that the sum of tangent lengths from any surface point to the two focal conics equals the constant $2m, tying directly to the confocal parameter of the quadrics family. To practically trace the surface, the configuration is rotated about the principal axis or the is guided to the focal at varying heights, ensuring the string slides freely along the conics. For special cases like spheroids (where two semi-axes are equal), the focal conics simplify—one degenerating toward a —and the construction reduces to tracing the meridional with pins and string before rotating the around the symmetry axis to sweep out the surface. This rotational variant has historical roots in early geometric constructions for spheroids, with generalizations to arbitrary ellipsoids appearing in late 19th-century work. Limitations include the need for smooth conic wires or tracks to guide the string and challenges in maintaining uniform tension over large scales, making it more suitable for models than precise .

Focal Conics and Confocal Systems

In a triaxial ellipsoid, the focal conics consist of an and a lying in mutually perpendicular planes, typically aligned with the principal axes of the ellipsoid. These conics serve as the "foci" in three dimensions, analogous to the two focal points of a , and define the ellipsoid as the locus of points where the sum of the distances to the focal ellipse and focal hyperbola remains constant. Ellipsoids sharing the same pair of focal conics belong to a confocal family of quadrics, which also includes hyperboloids of one and two sheets. Within this confocal pencil, any two distinct quadrics intersect along their common curves, forming a system of triply orthogonal surfaces that coordinate space in elliptic coordinates. This orthogonality property underpins the integrability of various geometric and dynamic systems associated with these surfaces. In the limit case of an ellipsoid of , where two of the semi-axes are equal, the focal conics degenerate: the focal collapses to a along the of revolution, and the focal reduces such that the effective foci coincide at two points on that axis. A key property of ellipsoids defined by these focal conics is their reflection behavior: trajectories inside the ellipsoid remain tangent to fixed confocal quadrics, known as caustics, after each reflection off the boundary, ensuring the motion is integrable. Conversely, any surface generated as the locus of points maintaining a constant sum of distances to a given focal and in perpendicular planes must be a triaxial ellipsoid.

Generalizations and Extensions

Higher Dimensions

In higher dimensions, an ellipsoid generalizes to \mathbb{R}^n as the set of points x satisfying x^T Q^{-1} x \leq 1, where Q is a positive definite symmetric n \times n ; this represents the image of the unit ball under the linear transformation defined by Q^{1/2}. For the principal-axis aligned case, where Q is diagonal with entries a_1^2, \dots, a_n^2 and a_i > 0 are the semi-axis lengths, the bounding surface equation simplifies to \sum_{i=1}^n \frac{x_i^2}{a_i^2} = 1. The volume of the solid n-dimensional ellipsoid is V_n = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2} + 1\right)} \prod_{i=1}^n a_i, which generalizes the formula \frac{4}{3} \pi a b c and follows from the scaling property of volumes under linear transformations, where the factor is \prod a_i times the unit ball . An equivalent form is V_n = \frac{2 \pi^{n/2}}{n \Gamma\left(\frac{n}{2}\right)} \prod_{i=1}^n a_i. The hypersurface area of the n-ellipsoid, measuring the (n-1)-dimensional boundary, lacks a closed-form expression except in special cases (e.g., spheres) and is generally computed via integration over the unit sphere with scaling by the semi-axes; it equals the (n-1)-sphere surface area \frac{2 \pi^{n/2}}{\Gamma(n/2)} times the p-mean of the products of n-1 semi-axes, where p = \frac{n-2}{n-1}. This area is a concave function of the semi-axis lengths and can be bounded using the ellipsoid's volume and mean width. In n dimensions, the Gaussian curvature K of the is the product of its n-1 principal curvatures, which are the eigenvalues of the shape operator; for an ellipsoid, K > 0 everywhere, reflecting its strict convexity, with values determined by the local geometry and semi-axes. The principal curvatures at a point on the surface derive from the second fundamental form of the quadratic . The semi-axes lengths a_i admit a spectral interpretation: they are the square roots of the eigenvalues of Q, or reciprocals of the square roots of the eigenvalues of the quadratic form matrix Q^{-1}, linking the ellipsoid's shape to the eigensystem of its defining matrix.

General Quadrics in Arbitrary Position

In three-dimensional space, quadrics in arbitrary position are described by the general second-degree equation \mathbf{x}^T A \mathbf{x} + \mathbf{b}^T \mathbf{x} + c = 0, where \mathbf{x} = (x, y, z)^T \in \mathbb{R}^3, A is a $3 \times 3 symmetric matrix, \mathbf{b} \in \mathbb{R}^3 is a vector, and c \in \mathbb{R} is a scalar. This form encompasses all possible orientations and positions, unlike the axis-aligned case. For the quadric to represent an ellipsoid, the symmetric matrix A must be positive definite (all eigenvalues positive), ensuring the quadratic form defines a bounded surface after appropriate translation and scaling. To obtain the canonical form, first translate the coordinates to the center of the quadric by solving for the shift \mathbf{x}_0 = -\frac{1}{2} A^{-1} \mathbf{b}, which eliminates the linear terms and yields (\mathbf{x}' )^T A \mathbf{x}' = k in the shifted coordinates \mathbf{x}' = \mathbf{x} - \mathbf{x}_0, where k = \frac{1}{4} \mathbf{b}^T A^{-1} \mathbf{b} - c > 0 for a non-degenerate ellipsoid. Next, diagonalize A via an orthogonal transformation (rotation) to principal axes: since A is symmetric and positive definite, there exists an orthogonal matrix R such that A = R D R^T, with D = \operatorname{diag}(\lambda_1, \lambda_2, \lambda_3) and \lambda_i > 0. Substituting \mathbf{y} = R^T \mathbf{x}' transforms the equation to \sum_{i=1}^3 \lambda_i y_i^2 = k, or equivalently, \frac{y_1^2}{a_1^2} + \frac{y_2^2}{a_2^2} + \frac{y_3^2}{a_3^2} = 1 after scaling with a_i = \sqrt{k / \lambda_i}. This principal axis form reveals the semi-axes lengths and orientations defined by the columns of R. Ellipsoids exhibit affine invariance, meaning the class is closed under invertible affine transformations; specifically, the image of an ellipsoid under an affine map \mathbf{x} \mapsto M \mathbf{x} + \mathbf{d} (with M invertible) is another ellipsoid. This property underscores that all ellipsoids are affine equivalents of the , facilitating analysis in rotated or sheared coordinate systems. In classification, a defined by the general is bounded (an ellipsoid) the A is positive definite and k > 0, distinguishing it from unbounded types like hyperboloids (indefinite A) or paraboloids (degenerate cases).

Applications

Physics and Dynamics

In rigid body dynamics, the ellipsoid serves as a fundamental model for analyzing rotational motion, particularly due to its principal axes aligning with the symmetry of the tensor. For a uniform-density ellipsoid with M and semi-axes a, b, and c along the x, y, and z directions respectively, the tensor is diagonal in the principal frame, with components I_{xx} = \frac{1}{5} M (b^2 + c^2), \quad I_{yy} = \frac{1}{5} M (a^2 + c^2), \quad I_{zz} = \frac{1}{5} M (a^2 + b^2). These expressions arise from integrating the mass distribution over the ellipsoidal , transforming coordinates to a for computation. For torque-free motion of a triaxial ellipsoid (where a \neq b \neq c), Euler's equations govern the evolution of \vec{\omega} = (\omega_1, \omega_2, \omega_3) along the principal axes: I_1 \dot{\omega}_1 + (I_3 - I_2) \omega_2 \omega_3 = 0, \quad I_2 \dot{\omega}_2 + (I_1 - I_3) \omega_3 \omega_1 = 0, \quad I_3 \dot{\omega}_3 + (I_2 - I_1) \omega_1 \omega_2 = 0, where I_1 = I_{xx}, I_2 = I_{yy}, and I_3 = I_{zz}. These equations reveal conserved quantities: the T = \frac{1}{2} (I_1 \omega_1^2 + I_2 \omega_2^2 + I_3 \omega_3^2) and the squared magnitude L^2 = I_1^2 \omega_1^2 + I_2^2 \omega_2^2 + I_3^2 \omega_3^2. Assuming I_1 < I_2 < I_3, rotations about the axes with maximum (I_3) and minimum (I_1) moments are stable, while rotation about the intermediate axis (I_2) is unstable, leading to periodic flips—a phenomenon known as the , first demonstrated experimentally with objects like tennis rackets but theoretically exemplified by the triaxial ellipsoid. In and , ellipsoids model the shapes of rotating bodies and their atmospheres. Due to centrifugal forces from , self-gravitating planets adopt oblate spheroidal forms, flattened at the poles and bulging at the ; , for instance, is an oblate spheroid with an of approximately 21 km relative to the polar radius. This oblate shape influences , with models using to approximate density variations and wind patterns in planetary atmospheres. Prolate spheroids, elongated along one axis, appear in contexts like settling speeds of elongated aerosol particles in Earth's atmosphere, where aspect ratios affect drag and terminal velocities under assumptions. The , representing mean as an surface, is closely approximated by a reference ellipsoid such as WGS84, which captures the oblate form with a of about 1/298.257 while smoothing local gravitational anomalies. The outside a homogeneous ellipsoid also admits an exact analytical solution, crucial for modeling . For a uniform ellipsoid with density \rho, the exterior potential \Phi(\mathbf{r}) can be expressed in using integrals involving the semi-axes, yielding a form that is outside the body and matches Newtonian gravity. This solution, derived through confocal coordinate systems, underpins analyses of Roche ellipsoids in systems and planetary perturbations.

Statistics and Probability

In multivariate statistics, the covariance ellipsoid provides a geometric representation of the dispersion of a random vector, particularly for the multivariate normal distribution. For a p-dimensional random vector \mathbf{X} \sim N_p(\boldsymbol{\mu}, \boldsymbol{\Sigma}), the probability density function is given by f(\mathbf{x}) = (2\pi)^{-p/2} |\boldsymbol{\Sigma}|^{-1/2} \exp\left( -\frac{1}{2} (\mathbf{x} - \boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu}) \right), where the exponent defines elliptical contours. These contours form ellipsoids centered at the mean \boldsymbol{\mu}, with orientation and shape determined by the inverse covariance matrix \boldsymbol{\Sigma}^{-1}; the principal axes align with the eigenvectors of \boldsymbol{\Sigma}, and the semi-axes lengths are proportional to the square roots of its eigenvalues. Confidence ellipsoids arise in inference for the multivariate normal, delineating regions containing the true mean with a specified probability. The squared Mahalanobis distance D^2 = (\mathbf{x} - \boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu}) follows a chi-squared distribution with p degrees of freedom under the centered normal. Thus, the (1 - α) confidence ellipsoid is the set where D^2 \leq \chi^2_{p, 1-\alpha}, scaling the covariance ellipsoid to enclose 100(1 - α)% of the probability mass; for the sample mean \bar{\mathbf{x}}, an analogous region uses the sample covariance \mathbf{S} and scales by the F-distribution to account for estimation uncertainty. This elliptical form accounts for correlations, providing a more accurate depiction of uncertainty than spherical regions. In (), the data ellipsoid from the sample is transformed to reveal underlying structure. The eigenvectors of \boldsymbol{\Sigma} (or \mathbf{S}) define the principal axes, representing uncorrelated directions of maximum to minimum variance, while the eigenvalues \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_p quantify the variance along these axes, with semi-axes lengths scaled by \sqrt{\lambda_i}. This orients the ellipsoid along the data's natural spread, facilitating by retaining components with the largest eigenvalues. Hotelling's T-squared statistic extends univariate t-tests to multivariate hypothesis testing for means, yielding ellipsoidal rejection regions. For testing H_0: \boldsymbol{\mu} = \boldsymbol{\mu}_0, the statistic is T^2 = n (\bar{\mathbf{x}} - \boldsymbol{\mu}_0)^T \mathbf{S}^{-1} (\bar{\mathbf{x}} - \boldsymbol{\mu}_0), which under H_0 follows a scaled ; the rejection region is an ellipsoid centered at \boldsymbol{\mu}_0. Similarly, the (1 - α) for \boldsymbol{\mu} is (\boldsymbol{\mu} - \bar{\mathbf{x}})^T \mathbf{S}^{-1} (\boldsymbol{\mu} - \bar{\mathbf{x}}) \leq \frac{p(n-1)}{n-p} F_{\alpha, p, n-p}, with axes along the eigenvectors of \mathbf{S} and lengths proportional to \sqrt{\lambda_i / n}, where \lambda_i are its eigenvalues. This approach is pivotal for simultaneous inference in multiple dimensions.

Computer Science and Graphics

In mathematical optimization, the is an iterative algorithm for minimizing convex functions over convex sets, notable as the first polynomial-time algorithm for . Developed by in 1979, it maintains a sequence of shrinking ellipsoids containing the , using a separation oracle to cut away infeasible portions until convergence to the optimum. Though theoretically significant for proving the complexity of convex programs, it is less practical than interior-point methods due to high constant factors. In and , ellipsoids serve as effective bounding volumes for approximating the shape of complex objects, enabling efficient . Due to their simplicity and tight fit around polyhedra or robotic components, ellipsoids outperform spheres or axis-aligned boxes in reducing false positives during broad-phase testing. For instance, ellipsoid trees partition solid models into hierarchical bounding ellipsoids, minimizing overlap to accelerate narrow-phase checks in real-time simulations. Oriented bounding ellipsoids (OBEs), analogous to oriented bounding boxes, align axes with the object's principal directions, further optimizing intersection tests in game engines and . Ray-ellipsoid algorithms are fundamental for rendering ellipsoids in ray tracing pipelines, where a is tested against the surface defined by the ellipsoid equation. Substituting the parametric equation \mathbf{p}(t) = \mathbf{o} + t\mathbf{d} into the implicit ellipsoid form \mathbf{x}^T A \mathbf{x} = 1 yields a at^2 + bt + c = 0, solved via the to find points. Positive roots indicate valid hits, with the smaller t selecting the nearest surface for ; this method extends naturally to quadrics like cylinders in scenes. Continuous variants handle by computing time-of-impact intervals, ensuring artifact-free intersections in animated environments. Parametric rendering of ellipsoids involves tessellating the surface using spherical coordinates scaled by semi-axes, generating vertex buffers for GPU pipelines in or . The parametric equations x = a \cos\theta \sin\phi, y = b \sin\theta \sin\phi, z = c \cos\phi (with \theta \in [0, 2\pi], \phi \in [0, \pi]) produce latitude-longitude grids, subdivided into triangles for smooth approximation at varying detail levels. shaders in modern APIs dynamically refine these meshes based on view distance, balancing performance and visual fidelity without precomputing high-resolution models. This approach supports and normal perturbation for realistic materials, as seen in procedural generation. Ellipsoids model planetary bodies in graphics simulations, approximating spheroids for rendering over large scales. Ellipsoidal clipmaps partition the surface into nested rings of adaptive , enabling seamless of Earth-like with geodetic accuracy and minimal seams. In , ellipsoids approximate organ shapes like aneurysms or prostates for and segmentation, providing quick visualizations in /MRI workflows where exact contours are computationally intensive. These approximations facilitate interactive slicing and , aiding diagnostics with errors under 10% for elongated structures.