Questions tagged [ellipsoids]
An ellipsoid is a convex set defined by $\mathcal{E} := \left\{ x \in \mathbb R^n \mid (x - x_c)^T P^{-1} (x - x_c) \leq 1 \right\}$ where matrix $P$ is symmetric and positive definite.
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Volume-preserving continuous deformation of an arbitrary ellipsoid centered at the origin into a sphere centered at the origin (graph included)
I want to know if the equation I provide actually does indeed accurately model the transformation that I desire to model
Model the ellipsoid by
$$f\left(u,v\right)=(A\cos\left(u\right)\sin\left(v\...
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Smallest ellipsoid which includes two equal radii balls
Consider $C_0, v \in \mathbb{R}^n$ with $\|v\| = 1$ and $\epsilon > 0$, then $C_1 = C_0 + \epsilon \cdot v$ and $C_2 = C_0 - \epsilon\cdot v$.
What is the smallest volume ellipsoid containing $\...
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Plane through feet of normals to ellipsoid
The problem
If $P,Q,R,P',Q',R'$ be the feet of six normals drawn from a point to the ellipsoid
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} - 1=0$$
and the plane $PQR$ is represented by $lx+...
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Poisson Equation for a perturbed sphere - both exterior and interior solutions
I am trying to solve a Heat transfer problem in a slightly ellipsoidal geometry using the Poission Equation, and Cauchy matching conditions on the boundary between the interior (which is a finite ...
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Equation of the tangent plane to the ellipsoid
I need help finding the equation of the tangent plane to the ellipsoid ${x^2\over{a^2}} + {y^2\over{b^2}} + {z^2\over{c^2}} = 1$, such that the sum of intercepts on the coordinate axes is minimized. I ...
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Use Covariance Matrix to Convert Ellipsoid to Sphere?
If I have an unbiased ellipsoid with a known covariance, is there a way I can use that covariance matrix to transform all known points on the given ellipsoid to trace a sphere instead?
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Orthogonal vector to ellipsoid surface is... $\vec{0}$?
I was looking at this ellipsoid:
$$
\frac{x^2}{25}+\frac{y^2}{25}+\frac{z^2}{9}=1
$$
I tried parametrizing it as such:
$$
\gamma\left(\theta, \varphi\right)=\left(5\cos\left(\theta\right)\sin\left(\...
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Eigenvalues and Eigenfunctions of the Laplace operator an ellipsoid
I am currently trying to find the spectrum of the Laplace operator for ellipsoids in $\mathbb{R}^{3}$ with Dirichlet boundary conditions, i.e., I am looking for solutions to the following PDE,
$$
\...
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How to find the largest ellipsoid under certain constraints?
Given a symmetric positive definite matrix $\bf Q$ and a bounded set $\mathcal X$, what is the following maximum?
$$ \max_{{\bf x} \in \mathcal X} {\bf x}' {\bf Q} \, {\bf x} $$
Using a Matlab program ...
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Determine if a point is located within a rotated ellipsoid
I'm attempting to implement this algorithm https://arxiv.org/pdf/1711.00748.pdf. Part of this algorithm requires determining whether a point $p \in \mathbb{R}^d$ is located within the volume of a ...
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Randomly generate points uniformly on an ellipsoid in general position
I want to uniformly generate points at random on the boundary of an ellipsoid in any dimension.
The method given by @elhuhdron here is very nice and it can be straightforwardly generalized to any ...
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Is it a valid way to uniformly generate points in/on an ellipsoid?
I found this method somewhere (I don't remember where) and I'm wondering whether it is correct. One has an ellipsoid of any dimension and one wants to uniformly generate some points at random on its ...
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Is it ture that any zonoid (not a trivial zonotope), must be a affine image of a unit ball (l2)?
I am trying to find some literature discussing zonoid in 3-dimensional space. And a seemly simple question that I am considered specifically is what are those non-trivial (not being a zonohedron) ...
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Ellipsoid set definitions
In Boyd & Vandenberghe's Convex Optimization, one can find two definitions for the ellipsoid.
$$ \mathcal{E} = \left \{ x \mid (x-x_c)^\mathsf{T}P^{-1}(x-x_c) \leq 1 \right\} $$
and
$$ \mathcal{E} ...
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Smallest N-dimensional ellipsoid containing a given ellipsoid and a point
Warning: I am not a mathematician, so excuse me if the problem at hand seems trivial or incomplete.
So, I am working on the following problem: I have to find the smallest (i.e smallest volume) N-...
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Find the largest cube that can be inscribed in this ellipsoid
I am trying to answer the following question:
Using Lagrange multipliers, find the largest cube (with faces parallel to the coordinate axes) that can be inscribed in the ellipsoid $$x^2 + \frac14 y^2 ...
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a straight line divides the ellipse into two parts, find the distance between the centers of mass of its parts
$3x+3y-9=0$ divides the ellipse $\frac{x^2}{25}+\frac{y^2}{4} = 1$ into two parts. Find the distance between the centers of mass of its parts
I realized that I need to make several double integrals, ...
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Vector field flux through ellipsoid
Calculate the flux of the vector field $F(x, y, z) = (2x, 3y, 4z)$
through the surface of the ellipsoid $9x^2 + 4y^2 + z^2 ≤ 36$.
The divergence of $F$ is $9$. And then I changed coordinates (using ...
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How can I show that these two ellipses are the same?
Through testing, I have convinced myself that these two shapes are identical:
$$(x,y) = \frac{1}{\sqrt{a\cos^2\theta + b\sin^2\theta}}( a\cos\theta, b\sin\theta)$$
$$r(\phi) = \sqrt{\frac{ab}{a\sin^2\...
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References for volume of hyperellipsoid
I am looking for a reference that I can use for calculating the volume of a high-dimensional ellipsoid.
Here is the ellipsoid equation:
$$ \mathcal{E}^n(r) = ({X}-\mu)^\text{T}{\Sigma}^{-1} ({X}-\mu) =...
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formula for segment of an ellipsoid
I'm an artist trying to sculpt shapes that I would describe as ellipsoids. I am wanting to sew together fabric to form these shapes, and although I know the dimensions of the ellipsoid I want to form, ...
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Properties of the ellipse inscribed in a convex pentagon
Recently I have learned that every convex pentagon has a unique ellipse inscribed. I am curious about the properties of this ellipse and its relationships with the pentagon where it is inscribed.
...
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Outer approximation of union of ellipsoids via $\log\det$ objective and LMI constraint
Given $A_i$, $b_i$, $c_i$ for $i=1,2,..p$ ($p$ is known), where $A_i\in \mathbb{R^{3\times3}}$ are symmetric positive definite matrices, $b_i\in \mathbb{R^3}$ and $c_i\in \mathbb{R}$. Let the ...
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Given two ellipsoids each specified by the center and shape matrix, how to find the union (minimum volume ellipsoid) of these two ellipsoids?
An ellipsoid in $\mathbb{R}^n$ is defined as
$${\cal E}_{\mathbb{R}^n}(\hat{x}, P)=\left\{x\in {\mathbb{R}}^{n}\vert (x-\hat{x})^{T}P^{-1}(x-\hat{x})\leq 1\right\}$$
where $\hat{x}\in \mathbb{R}^{n}$, ...
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Volume of half an ellipsoid
I want to calculate the volume of the solid defined as follows.
$$ K := \left\{ (x,y,z) \in \Bbb R^3 : x^2 + y^2 + 3z^2 \le 4, z > 0 \right\} $$
I know that this is a half solid ellipsoid and that ...
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Boyd & Vandenberghe, section 8.4.2 — Maximum volume ellipsoid in an intersection of ellipsoids
I originally asked this on Stack Overflow, it was suggested that I ask here.
I need to find the maximum volume inscribed ellipsoid subject to linear inequality constraints and a constraining ...
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Perspective projection ellipse and ellipsoid still ellipse?
I know that in perspective projection the circle turns into an ellipse. but does it convert to ellipse and ellipsoid to ellipse in perspective projection?
Especially in perspective ellipse is still ...
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What's the difference between the ellipsoid described by $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ vs $<1$?
I have to find the volume of the ellipsoid described by the set $ E = \{(x,y,z) \in \mathbb{R}^3 |\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}<1\}$. I have a few ideas and there is a bit of ...
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Converting Ellipsoid Equation to Canonical Form Parameters
Solving a least squares problem to try and get a best fit ellipsoid (or a general 4D conic section) to a cloud of data points such as the database of nearby stars.
Can solve the linear equation ...
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How can we prove that the Cholesky decomposition of an ellipsoid transforms that ellipsoid onto the unit sphere
Say I have a collection of points $x_i$ that define the surface of a fully general ellipsoid in three dimensions, except let's assume that ellipsoid is centered at the origin. I know that the ...
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what is the size of the smallest box that includes two ellipses/elipsoids of the same center?
How to find the size of the smallest box that includes two ellipses of the same center but arbitrarily tilted ? in $n$ dimensions.
The maximum is $(2b_1\times2b_2 \times ... \times 2b_n)$, but Im ...
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Tangency points of an ellipsoid with the coordinate planes
Suppose you have an ellipsoid with known semi-axes, say $a,b,c$. The ellipsoid is then rotated so that the three axes of the ellipsoid are aligned with the three columns of a known rotation matrix $R$...
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Algebra and convex geometry for proving uniqueness of John ellipsoids
I am looking at the proof about the uniqueness of John ellipsoids. The book I am looking at starts by supposing that there are two different ellipsoids $\epsilon:=S(\mathbb{B}_2^n)+x$ and $\epsilon^{'}...
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Random uniform points on the surface of (hyper) ellipsoid
How to generate random uniform points on (hyper) ellipsoid?
Note: Rejection methods are very inefficient specially in higher dimension. Check the ratio of volumes (box VS its inscripted ellipsoid) as ...
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Slices of an ellipsoid are ellipses: Can you slice an oblate ellipsoid to get an ellipse that is more oblate?
I'm struggling visualizing this. Is it possible to slice an ellipsoid with minor to major axis = $F$ and get an ellipse with axis ratios less than $F$?
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Ellipsoid Tait-Bryan orientation from Eigenbasis
I can parameterize any arbitrary ellipsoids as a set of $P$, $D$ and $R$ (respectively being the offset from origin, the squared inverse of the semi-axis and the eigenbasis) from a set of points lying ...
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Write inner product of two vectors as a function of the angle between them
Suppose $x_i,x_j$ vectors whose starting point is on the original $(0,0)$ and end point are on the unit circle in $\mathbb{R}^2$, then $x_i\cdot x_j$ can be written as
$\cos(\theta_i-\theta_j)$, where ...
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Projecting an ellipse defined on a sphere onto the XY plane
I want to find the equation for and the area of the ellipse defined by $\Theta_H$ and $\Psi_H$ when it gets projected onto the XY plane.
The following diagram shows these ellipses.
Let us suppose we ...
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How to check if a surface normal faces inwards or outwards an ellipsoid?
I have a half ellipsoid parametrised by $a,b,c,\theta,\phi$ and centered at origin. I have an optimization problem where I estimate the three components of a 3D direction vector $\hat{d}=(x,y,z)$ ...
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How to find a matrix that produced a symmetric matrix?
$$M=\begin{bmatrix} m_{00} & m_{01} & m_{02}\\ m_{10} & m_{11} & m_{12} \\ m_{20} & m_{21}& m_{22} \end{bmatrix} $$
$$ M^{T}M = \begin{bmatrix} a & d & e\\ d & b &...
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Must a function link every input value to a single output value?
I'm wondering whether equations like $x^2 + y^2 = 4$ can describe a function or not. The reason is that a function should normally link every input value to a single output value. However, in case of $...
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Determine an ellipsoid from $3$ perspective images of it
There is an ellipsoid of unknown dimensions and unknown orientation hanging somewhere (also unknown) in $3$-dimensional space. You are given three perspective images of it from three distinct ...
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Equation of a cylinder with a profile / ellipsoid
I have a geometry of a cylinder that is curved along both the lengths where there is generally a height of a cylinder. I am aware of the equation of a cylinder. I wanted to know what could the ...
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Immersed volume calculation
Im trying to solve the following problem :
Given the ellipsoid represented by the matrix $\widehat{A}$ and knowing the coordinates of each point A,B,C,D,E,F,G,H, calculate the volume of intersection (...
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An integral Identity about the Elliptic Integrals and Ellipsoids
I was reading this article: https://webspace.science.uu.nl/~maas0131/files/MaasJCAM1994.pdf
I came to the equation (24):
The author says... "We thus find, by the same argument,":
$$\int_0^1 ...
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Geodesic curve on an ellipsoid
I cannot find my mistake in the statement below, neither computational nor conceptual. I cannot prove that the velocity vector $\dot{x}$ is constant along the acclaimed geodesic. Could you review it ...
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Simulation and fitting 3D ellipsoid
I would like to simulate ellipsoid fitting.
In the first step I had ellipsoid with centre in 0,0,0 with specific length of axes a, b, c described by eq. $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{...
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Parametrically enlarge one ellipsoid to fit another one
I'm trying to figure out the smallest enlargement factor which I need to apply to one ellipsoid $E_1$ in order to fit another one $E_2$. Precisely, let $E(c, S) := \{x | (x-c)^T S (x-c) \leq 1\}$ be ...
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Convert Ellipsoid from Cov and Mean to Quadric Representation
Question
I have an ellipsoid represented as a Covariance $\Sigma \in \mathbb{R}^{3\times3}$ and a mean centroid $\mu\in \mathbb{R}^{3}$. I want to represent it as a homogenous quadric, which is a $4\...
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Orthogonal projection of an ellipsoïd from N to 2 dimensional space
Suppose we have a $N\times N$ symmetric-positive-definite matrix $A$, representing an ellipsoïd in $N$ dimensional space. How to find the matrix $A_{xy}$ corresponding to orthogonal projection of ...
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