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For a finite-dimensional normed vector space $E$, represent a finite family by its column operator $A:{\ell }_2^m\to E$. Its operator norm is the family's weak square norm. Since $A=AQ$ for the orthogonal projection onto $(\ker A{)}^{\perp }$, summing $\Vert A{e}_j{\Vert }^2\le \Vert A{\Vert }^2\Vert Q{e}_j{\Vert }_2^2$ proves the upper bound $\sqrt{\dim E}$. In coordinates adapted to a maximal inscribed Euclidean or Hermitian ellipsoid, the John contact decomposition gives ${\sum }_r{c}_r{u}_r{u}_r^{\ast }=I$ and ${\sum }_r{c}_r=\dim E$. The vectors $\sqrt{c_r}{u}_r$ have total squared norm $\dim E$ and weak square norm one, giving equality. This holds over both real and complex scalars.

The two-state calculation is not a general formula for a ferromagnet. Exchange interactions, magnetic domains, magnetic domain-wall pinning, magnetic anisotropy, and hysteresis can determine which response is accessible. Both spin angular momentum and orbital angular momentum can contribute; a single constant gyromagnetic ratio may be only a restricted effective description. Rotation sets an internal bias, while the material and its preparation determine magnetization. The Barkhausen effect illustrates why a smooth bias can produce jumps instead of the toy smooth curve.

For a limited, reversible, isotropic linear model, suppose the same intrinsic volume magnetic susceptibility ${\chi }_v$ responds to the applied bias and to $B_{\Omega }/{\mu }_0$. For a uniformly magnetized ellipsoid along a principal axis, include its demagnetizing factor $N_d$:Solving this linear equation yieldsHere $H_{\mathrm{int}}$ and $H_{\mathrm{app}}$ are actual magnetic field intensities; the added rotational bias is an equivalent drive in the constitutive response. In SI, dimensionless ${\chi }_v$ multiplies an intensity in $A/m$, so $M_z={\chi }_v\Omega /\gamma$ would omit ${\mu }_0$ and have the wrong units. A susceptibility defined instead as $\partial M/\partial B$ has different units. State the convention before comparing formulas.

At zero applied magnetic field intensity, a stipulated ${\chi }_v=3$ gives prefactor $3/2$ in a sphere, with $N_d=1/3$, and a prefactor approaching $3$ in a long axial ellipsoid. This is a geometry check of the assumed linear model. A real cylinder does not have an exactly uniform demagnetizing field, and an irreversible ferromagnet cannot in general be represented by this single linear equation. Inferring an intrinsic gyromagnetic ratio requires an independently calibrated material response and a geometry-dependent conversion of the measured stray magnetic field.

Cardiac image foreshortening occurs when a cardiac view represents a shorter apparent base-to-apex extent than the intended anatomical chamber length, often because the imaging plane misses the true cardiac apex. It can create an apparently thick false apex and affect derived ejection fraction and deformation measurements.

A primary echocardiography study acquired optimal and deliberately foreshortened views in 72 participants using a GE Vivid E9 system. The altered acquisition window was one rib-space above the optimal window; two analysis systems were used to compare derived quantities.^[ref] This tests acquisition geometry while keeping the participant the same, rather than interpreting every shorter-looking chamber as a biological change.

An original ellipsoid model distinguishes a missed apex from a simple projected length. Letwith the true cardiac long axis along $z$. The central long-axis plane $y=0$ includes both axial tips and has full length $2a$. A parallel offset plane $y=d$ instead intersects an ellipse whose apparent axial extent isAt $d=0.6b$, $L(d)=0.8(2a)$. A 10 cm true extent therefore appears as 8 cm in this model even though the slice was translated rather than rotated. Its apparent tips are not the original chamber tips.

This model is a geometrical explanation, not a clinical correction formula. A shorter outline can alter measured volume differently at end-diastolic volume and end-systolic volume; the ejection fraction error need not follow the length error by one fixed percentage. Consistent anatomical landmarks and independent views help distinguish actual chamber shape from the chosen section. Improving the repeatability of an outline inside an incorrect plane does not recover anatomy that was never acquired.

For nonzero real $a_i$, scale the ambient coordinates by $q_i=a_i{r}_i$. On the half where $\epsilon q_i>0$, omitting $q_i$ is a manifold chart onto the open unit ball; its inverse restores $q_i=\epsilon \sqrt{1-{\sum }_{j\ne i}{q}_j^2}$. These $2(n+1)$ charts form a smooth atlas of the ellipsoid, including for negative $a_i$.

A demagnetizing factor relates a component of demagnetizing field to magnetization. The three principal factors of a uniformly magnetized ellipsoid sum to one in the International System of Units. Factors quoted in a different unit convention need conversion; state the convention and measurement geometry before applying them.

The demagnetizing field is the contribution to magnetic field intensity produced by the sample's own magnetization. Inside a uniformly magnetized ellipsoid, it is uniform and has the form ${\mathbf{H}}_d=-\mathsf{N}\mathbf{M}$; the demagnetizing factor along a principal axis gives its longitudinal component. These conditions make the scalar correction in field conventions for magnetic susceptibility useful.

A cylinder generally has spatially varying magnetization and demagnetizing field. A volume-sensitive instrument and a short central pickup coil consequently need different effective corrections: one averages over the sample, while the other samples central flux. A shape factor copied from an ellipsoid is not an exact correction for every cylinder, especially with nonlinear response or hysteresis. ^[ref]

The electric field produced by the bound charge of a polarized body is its depolarization field. For a uniformly polarized ellipsoid in vacuum, the internal depolarization field is uniform and opposes the relevant component of the electric polarization: $E_{\mathrm{dep},i}=-N_i{P}_i/{ϵ}_0$. Its dimensionless geometric factors obey $N_x+N_y+N_z=1$.

Let $A$ be a real positive-definite matrix and let the columns of $U$ be an orthonormal basis of a two-dimensional linear subspace in three-dimensional Euclidean space. Points in that central plane are $x=Uq$, with $|x{|}^2=|q{|}^2$. Restricting the ellipsoid $x^{T}Ax=1$ gives $q^T(U^{T}AU)q=1$. The compressed matrix $U^{T}AU$ is a positive-definite matrix, and its extreme eigenvalues give the distance formulas in the title by the Rayleigh quotient. The two extremal axes are the images under $U$ of the corresponding eigenvectors. This restriction reduces a problem with two Lagrange multipliers to a two-dimensional quadratic form.

An ellipsoidal admissibility test for overlap data checks whether every point of a specified joint calibration ellipsoid permits finite Gaussian radius inversion. For the inversion of a symmetric Gaussian overlap log-ratio, write $x=(D,h{)}^{\mathsf{T}}$ with $D$ the measured logarithmic ratio and $h$ the positive radius-setting parameter. The admissible set is $D-4h<0$ and $-D-4h<0$. Together these strict inequalities already imply $h>0$.

Suppose the joint set isFor nonsingular $\Sigma$ this is equivalently $(x-\mu {)}^{\mathsf{T}}{\Sigma }^{-1}(x-\mu )\le k^2$. The parameterized form also accommodates a singular covariance matrix or a lower-dimensional deterministic set. Its support function in direction $l$ isIndeed, $l^{\mathsf{T}}Ay=(A^{\mathsf{T}}l{)}^{\mathsf{T}}y$, so the Cauchy-Schwarz inequality gives the bound, attained by a parallel $y$ when $A^{\mathsf{T}}l\ne 0$.

Let $\mu =(D_0,h_0{)}^{\mathsf{T}}$, marginal scales $s_D,s_h$, and off-diagonal entry $c_{Dh}$. Substituting $l_{+}=(1,-4{)}^{\mathsf{T}}$ and $l_{-}=(-1,-4{)}^{\mathsf{T}}$ gives two exact tests:Both must hold. Equality means that the closed joint set touches a boundary where no finite positive radius ratio exists; testing only the center would miss that point.

For an original example, take $D_0=2.4$, $h_0=0.7$, $s_D=0.1$, $s_h=0.03$ and $k=2$. Zero covariance gives upper-boundary excursion $2\sqrt{0.0244}\approx 0.3124$, smaller than the central margin $0.4$. The lower-boundary test also passes. With correlation coefficient $-0.8$, however, $c_{Dh}=-0.0024$ and the upper excursion becomes $2\sqrt{0.0436}\approx 0.4176$: the joint set crosses the forbidden region despite identical marginal scales.

The offending point need not be found by grid search. When $l^{\mathsf{T}}\Sigma l>0$, the maximizing point isFor the correlated example and $l=l_{+}$, this gives approximately $(D,h)=(2.5877,0.6425)$, with $D-4h\approx 0.0176>0$. It is a concrete incompatible calibration pair.

The ellipsoid here is a stated joint set. Interpreting it as a confidence region requires a separate sampling or uncertainty model; choosing $k=2$ alone supplies no universal coverage probability. JCGM 101 distinguishes assignment of joint input distributions from propagation and summaries.^[ref] If the set fails, report the incompatible region and reconsider the input model or measurement. Clipping it before inversion changes the set being propagated.

For a real positive-definite matrix $A$ in finite dimension, write $A=Q\Lambda Q^T$ by the spectral theorem for real symmetric matrices. In the coordinates in an orthonormal basis $w=Q^{T}x$, the positive level set $q(x)=k$ becomesThis is an ellipsoid, with semi-major and semi-minor axes in two dimensions forming an ellipse; each semi-axis has length $r_j=\sqrt{k/{\lambda }_j}$ and direction given by the corresponding eigenvector. In the opening example, the shorter semi-axis is $\sqrt{k}/2$ along $e_{+}$, while the longer is $\sqrt{k/2}$ along $e_{-}$. A stiffer direction, meaning a larger eigenvalue, reaches the same energy at a smaller displacement. The ratio of longest to shortest semi-axis is $\sqrt{{\lambda }_{\max }/{\lambda }_{\min }}$, the square root of the spectral condition number of a positive-definite matrix.^[ref]

Strict positivity matters because ${\lambda }_{\min }>0$ givesHence a fixed positive level set cannot extend arbitrarily far. The Cholesky decomposition $A=L{L}^T$ gives another view: $q(x)=\Vert L^{T}x{\Vert }^2$, so the ellipsoid is the inverse image of a sphere under an invertible linear map. A singular positive semidefinite matrix has a nonzero kernel; adding a kernel vector preserves its quadratic form, so any nonempty positive level set is unbounded. These statements concern finite dimension; the positive definite symmetric operator entry explains why strict positivity alone need not give a uniform lower bound on an infinite-dimensional Hilbert space.

In 2006, Yilong Han and colleagues dispersed plastic ellipsoids in water inside a thin glass cell, using a very dilute suspension to limit particle interactions. A camera on an optical microscope recorded 30 frames per second. Image analysis extracted centre positions and long-axis angles, with reported angular resolution about $1^{\circ }$. Their shape supplied the observable axis. Reflected interference colours calibrated cell thickness; nearly constant particle interference colour checked vertical wandering. One tracked particle had dimensions $2.4\times 0.3\times 0.3\mu \mathrm{m}$ in an $846\mathrm{nm}$ gap. Angular displacement variance grew linearly with lag; displacement resolved along the particle's axes and in the laboratory showed how turning washes out translational anisotropy. These are observations of confined particles: walls alter drag, and slight out-of-plane tumbling affected positional resolution. They do not establish the bulk sphere formula. The paper's Figure 1A–B shows simulations; Figure 1F shows a measured trajectory.^[ref]

For interpreting such images, the director distinction is essential: identical ends give an angle modulo $\pi$. A continuously unwrapped trace can follow small successive turns, but it does not make the ends physically distinguishable. The invariant planar axis autocorrelation is $⟨\cos (2\Delta \varphi )⟩$, and its decay determines $D_r$ through $e^{-4D_{r}t}$. Separately measuring position and rigid body orientation is what makes the connection between translational anisotropy and loss of angular memory experimentally accessible.

The feasible-set boundary in overlap inversion separates joint measurements admitting a finite positive Gaussian radius ratio from measurements incompatible with that forward model. For radius-setting parameter $h>0$ and log-ratio $D$, the admissible region isAt its upper boundary the inferred radius ratio tends to zero; at its lower boundary it becomes arbitrarily large. The joint radius-setting and log-ratio bounds must therefore be checked against the sloping joint boundary, not against a fixed limit evaluated only at the central $h$.

An original ellipsoid calculation gives an exact test for a common joint-set model. Let $x=(D,h{)}^{\mathsf{T}}$ lie in $(x-\mu {)}^{\mathsf{T}}{\Sigma }^{-1}(x-\mu )\le k^2$, with $\Sigma$ positive definite. For any vector $l$, the maximum of $l^{\mathsf{T}}x$ isTo see this, write $x=\mu +{\Sigma }^{1/2}y$, $\Vert y\Vert \le k$, and apply the Cauchy-Schwarz inequality; equality is attained along the corresponding transformed direction.

Use $l_{+}=(1,-4{)}^{\mathsf{T}}$ and $l_{-}=(-1,-4{)}^{\mathsf{T}}$. The entire ellipse is strictly admissible if both displayed maxima are negative and its smallest $h$ is positive. This ellipsoidal admissibility test for overlap data checks every point of the stated joint set. It does not require sampling a coarse grid that might miss the closest approach to the boundary.

For stipulated means $(D_0,h_0)=(2.5,0.7)$, standard-scale parameters $s_D=0.1$, $s_h=0.03$ and $k=2$, the central distance $4h_0-D_0$ is 0.3. With zero covariance, the upper-boundary excursion isso the ellipse crosses that boundary. With correlation $+0.8$, the covariance is 0.0024 and the excursion becomesThat ellipse stays below the upper boundary; the lower boundary and positive-$h$ condition are also satisfied. Correlation changes the geometry even though both marginal uncertainty scales are unchanged.

If a set intersects the forbidden region, report the incompatibility and examine calibration, model adequacy and the meaning of the input bounds. Clipping points and applying the inverse to the survivors changes the propagated set or probability distribution. JCGM 101 separates input assignment, propagation and output summaries and gives conditions for valid Monte Carlo propagation.^[ref] An illustrative ellipse is a deterministic set here; interpreting it as a confidence region requires an additional statistical model.

In vacuum, magnetization is zero and magnetic flux density obeys $\mathbf{B}={\mu }_0\mathbf{H}$. The applied quantities ${\mathbf{H}}_{\mathrm{app}}$ and ${\mathbf{B}}_0={\mu }_0{\mathbf{H}}_{\mathrm{app}}$ mean the field of the external sources at the sample position with the sample absent. The internal magnetic field intensity includes the sample's demagnetizing field; its internal magnetic flux density additionally includes ${\mu }_0\mathbf{M}$. A field quoted in the tesla is a $B$ quantity, even when an instrument informally calls it “$H$”.

For a uniformly magnetized ellipsoid, along a principal axis with demagnetizing factor $N_d$ in the International System of Units,For a sphere, $N_d=1/3$. The axial value approaches zero in the long slender ellipsoid limit and the normal value approaches one in the thin flat ellipsoid limit. Thus a stipulated intrinsic volume magnetic susceptibility ${\chi }_v=3$ would yield apparent values $1.5$ in a sphere and approximately $3$ in a long axial sample. These are calculated geometry effects, not measured material values. For fixed $N_d$, implicit differentiation gives the same conversion for local differential magnetic susceptibility, $dM/d{H}_{\mathrm{app}}={\chi }_{\mathrm{diff}}/(1+N_d{\chi }_{\mathrm{diff}})$.

A floating-pinion ellipsoidal uncertainty bound gives the smallest two-contact compression margin when the uncertain parameter vector lies in a specified ellipsoid. It retains shared parameter effects, as does the floating-pinion affine uncertainty bound, but uses a different admissible-set shape.

Let approach and imbalance be affine functionsand describe the set by $u=u_0+Lz$ with $\Vert z{\Vert }_2\le 1$. Put $\bar{\delta }={\delta }_0+a^{\mathsf{T}}{u}_0$ and $\bar{b}=b_0+c^{\mathsf{T}}{u}_0$. The floating-pinion robust contact margin then satisfies the exact formulaTo prove it, write $\delta -|b|=\min (\delta -b,\delta +b)$. For either linear coefficient vector $q$, the Cauchy-Schwarz inequality givesIf $L^{\mathsf{T}}q\ne 0$, equality is attained at $z=-L^{\mathsf{T}}q/\Vert L^{\mathsf{T}}q{\Vert }_2$. Thus the result supplies both a margin and an attaining direction. The image representation also handles rank-deficient sets without inverting a shape matrix.

For an original example in micrometres, take $u_0=0$, $L$ the identity andOver $u_1^2+u_2^2\le 1$, the two branch minima are $4.5-\sqrt{0.1}$ and $5.5-\sqrt{0.1}$, so the margin is about $4.1838\mu \mathrm{m}$. The first branch attains it at $u=(-1,-3)/\sqrt{10}$. The encompassing box $|u_1|,|u_2|\le 1$ instead gives $4.1\mu \mathrm{m}$ because it admits combinations excluded by the disk.

A covariance-derived ellipse does not automatically certify an absolute engineering bound. Its probability content requires distributional assumptions and an appropriate radius; a hard guarantee requires a defensible admissible set. NASA's face-gear mounting and slow-roll load-sharing observations motivate the physical contact question, not the particular coefficients or set used here.^[ref] The two-contact force law must also remain valid; contact opening requires the floating-pinion contact-loss check.

For isotropic turning in three dimensions, the tip of a marked unit vector $\mathbf{u}$ wanders over the unit sphere. Its probability density function $f$ obeys the orientational Fokker-Planck equationHere ${\nabla }_{\mathbf{u}}^2$ is the Laplace-Beltrami operator on the unit sphere. The factor two follows from the sphere surface derivative identities: ${\nabla }_{\mathbf{u}}^2{u}_i=-2u_i$, so each mean vector component decays at rate $2D_r$. This is a different orientation space from the circle of planar rotational diffusion, whose corresponding rate is $D_r$. For the same illustrative $D_r=0.25{\mathrm{s}}^{-1}$ and $t=2\mathrm{s}$, this three-dimensional autocorrelation is $e^{-1}\simeq 0.368$.

One unit vector records a pointing direction, not every aspect of a rigid body's rigid body orientation: rotation about that vector remains unspecified. Also, $C_1$ requires an identifiable arrow. An unmarked ellipsoid's long axis alone does not supply one, and projecting a three-dimensional trajectory onto a camera plane does not automatically produce the planar model.

The John ellipsoid of a convex body is its inscribed ellipsoid of largest volume. For the unit ball $K$ of a $d$-dimensional normed vector space, change coordinates so that this ellipsoid is the Euclidean unit ball. The resulting coordinates are called John position. In these coordinates the John contact decomposition implies $B_2^d\subseteq K\subseteq \sqrt{d}{B}_2^d$, or $\Vert x{\Vert }_E\le |x|\le \sqrt{d}\Vert x{\Vert }_E$.

A researcher whose joint work measured the Brownian motion of an ellipsoid using both a laboratory frame and a frame that rotates with the particle. The two descriptions expose different displacement statistics. This provides an experimental example of why a random trajectory must be analyzed with its changing orientation, rather than position alone.^[ref]

Every finite-dimensional vector subspace $E$ of a Banach space $X$ is the range of a bounded linear projection with the displayed norm bound. In maximal inscribed Euclidean or Hermitian ellipsoid coordinates, write the John contact decomposition as ${\sum }_r{c}_r{u}_r{u}_r^{\ast }=I_E$. Its contact functionals ${\varphi }_r(x)=u_r^{\ast }x$ have norm one on $E$. Extend them to $X$ by the Hahn-Banach theorem and set $Px={\sum }_r{c}_r{\tilde{\varphi }}_r(x)u_r$. This restricts to the identity on $E$. The contact synthesis operator has Euclidean norm one, the weights sum to $\dim E$, and the Euclidean unit ball is contained in the norm unit ball; these facts give $\Vert Px{\Vert }_E\le \sqrt{\dim E}\Vert x{\Vert }_X$.

For a real symmetric matrix $A$ with nonnegative eigenvalues, the finite-dimensional spectral theorem changes the equation to ${\sum }_i{\lambda }_i{q}_i^2=E$ in an orthonormal eigenbasis. If $E>0$ and $A$ has positive rank, the positive-eigenvalue coordinates form an ellipsoid, while the zero-eigenvalue coordinates are arbitrary. The level set is therefore a cylinder over that ellipsoid, with axis directions given by the kernel of a linear map of $A$. For rank two in three dimensions, it is an elliptic cylinder. At $E=0$ the level set is exactly the kernel of a linear map; for $E<0$ it is empty. This distinguishes a positive semidefinite matrix from a positive-definite matrix, whose positive level sets are bounded ellipsoids.

The mass second-moment tensor about a centre of mass is $I_{jk}=\int y_j{y}_{k}dm$. It measures the shape of a mass distribution and appears in tidal torque theory. It differs from the mechanical inertia tensor, which is ${\mathcal{I}}_{jk}=\mathrm{tr}(I){\delta }_{jk}-I_{jk}$. For a uniform ellipsoid of mass $M$ and semiaxes $a_1,a_2,a_3$, its principal values are $M{a}_j^2/5$.