Mathematics

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.

233 questions
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4 answers

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables $\{x_1,x_2,x_3\}$, calculate the distance from the origin…
Georgy
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3 answers

Definition of an ellipsoid based on its focal points

I have a question concerning the formulation of an (3D) ellipsoid. The most common definition for an ellipsoid seems to be: $$ E = \left\{ x= \left( x_1, \dots x_n \right)^T \in \Bbb R^n: \sum_{i=1}^n \left( \frac{x_i}{r_i} \right)^2 = 1 \right\} $$…
Matthias
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votes
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Best fit ellipsoid

Given a collection of points $P \subset \mathbb R^3$, a crude characterization of the "shape" of $P$ is sometimes given by the principal components. We construct a covariance matrix, e.g., if $P$ is discrete, $C = \displaystyle\sum_{p\in P} (p -…
yasmar
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2 answers

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two $n$ -dimensional ellipsoids? Given an $n$ -vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$ , define the ellipsoid $$E(c,A)=\{x|(A(x-c),x-c)<1\}$$ where $(\cdot,\cdot)$ if the…
sds
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5 answers

Ellipsoid but not quite

I have an ellipsoid centered at the origin. Assume $a,b,c$ are expressed in millimeters. Say I want to cover it with a uniform coat/layer that is $d$ millimeters thick (uniformly). I just realized that in the general case, the new body/solid is not…
peter.petrov
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4 answers

Projection of ellipsoid

Find the projections of the ellipsoid $$ x^2 + y^2 + z^2 -xy -1 = 0$$ on the cordinates plan I have no idea how to do this. I couldn't find much on google to help me with it too. Thanks in advance!
Giiovanna
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2 answers

Why is a positive definite matrix needed in the ellipsoid matrix representation?

An ellipsoid centered at the origin is defined by the solutions $\mathbf{x}$ to the equation $\mathbf{x}^TM\mathbf{x} = 1$ , where $M$ is a positive definite matrix. How can I see why $M$ needs to be positive definite, based on the equation of an…
scip
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votes
2 answers

Ellipsoid contained in another ellipsoid: geometric proof of an inequality?

Let $E_1$ and $E_2$ be two solid ellipsoids in $\mathbb R^n$ , where both $E_1$ and $E_2$ are centered at the origin, $E_1\subset E_2$ $E_1\ne E_2$ Let $0<a_1\le\cdots\le a_n$ be the semi-axes of $E_1$ , and $0<b_1\le \cdots\le b_n$ be the semi-axes…
Lee
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votes
2 answers

Relationship between ellipsoid radii and eigenvalues

I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup. I went through a theorem in the book stating the relationship between ellipsoid's radii and the…
chepukha
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votes
1 answer

How to prove the parallel projection of an ellipsoid is an ellipse?

Take the following ellipsoid in implicit form as an example: $$x^2 + 2 y^2 + 3 z^2 + x y + y z - 2 xz = 5$$ which shows: The parallel projection of the ellipsoid onto $xoy$ coordinate plane can be obtained as: $$ 8 x^2 + 16 x y+23 y^2=60$$ Is it…
user6043040
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2 answers

Test for the intersection of two $N$-dimensional ellipsoids

Let's say I have two $N$-dimensional ellipsoids: $$ \sum_{i=1}^{N} \frac{(x_i - b_i)^2}{c_i^2} = 1 $$ $$ \sum_{i=1}^{N} \frac{(x_i - b'_i)^2}{c_i'^2} = 1 $$ How can I tell if the two intersect? Is there a computationally easy way to do this test?
tskuzzy
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1 answer

Initial ellipsoid for the ellipsoid method for unconstrained minimization

Suppose I would like to solve an unconstrained optimization problem $$ \min_{x \in \mathbb{R}^n} \space f(x), $$ using the ellipsoid method where $n$ is small, i.e. $n = 2$ , and $f$ is a strictly convex function having a minimizer. Is there a good…
Alex Shtoff
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2 answers

Smallest axis-aligned bounding box of hyper-ellipsoid

Let $E$ be the $n$ -dimensional ellipsoid defined by $$E:=\{x \in \mathbb{R}^n: (x-c)^T A (x-c) \le 1\},$$ where $c \in \mathbb{R}^n$ is the center of the ellipsoid, and $A \in \mathbb{R}^{n \times n}$ is a symmetric positive definite matrix.…
Nick Alger
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7
votes
1 answer

Simplifying the Willmore energy of an ellipsoid

Willmore energy measures how "non-spherical" a smooth surface $S$ is. It is defined by $$W(S)=\int_SH^2\,dA$$ where $H$ is the mean curvature. For a torus of revolution with major and minor radii $a$ and $b$ respectively where $a>b$ , if we let…
Parcly Taxel
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7
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1 answer

Maximizing a linear function over an ellipsoid

Let $A \in \mathbb{R}^{n\times n}$ be a positive definite matrix, $x \in \mathbb{R}^n$ and $c \in \mathbb{R} \setminus \{0\}$ . Determine the maximum $$ \max_{y \in \mathbb{R}^n} \left\{ c^T y : y \in \mathcal{E} (A,x) \right\} $$ where $\mathcal{E}…
Thesinus
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