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.
Questions tagged [ellipsoids]
226 questions
32
votes
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…
Georgy
- 1,475
20
votes
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:
$$ E = \left\{ x= \left( x_1, \dots x_n \right)^T \in \Bbb R^n: \sum_{i=1}^n \left( \frac{x_i}{r_i} \right)^2 = 1 \right\}…
Matthias
- 603
16
votes
3 answers
Best fit ellipsoid
Given a collection of points $P \subset \mathbb R^3$, a crude characterization of the "shape" of $P$ is sometimes given by the principal components. We construct a covariance matrix, e.g., if $P$ is discrete, $C = \displaystyle\sum_{p\in P} (p -…
yasmar
- 1,174
14
votes
2 answers
Volume of the intersection of ellipsoids
How do I compute the volume of the intersection of two $n$-dimensional ellipsoids?
Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid $$E(c,A)=\{x|(A(x-c),x-c)<1\}$$ where $(\cdot,\cdot)$ if the…
sds
- 4,613
14
votes
5 answers
Ellipsoid but not quite
I have an ellipsoid centered at the origin. Assume $a,b,c$ are expressed in millimeters. Say I want to cover it with a uniform coat/layer that is $d$ millimeters thick (uniformly).
I just realized that in the general case, the new body/solid is not…
peter.petrov
- 12,963
12
votes
4 answers
Projection of ellipsoid
Find the projections of the ellipsoid
$$ x^2 + y^2 + z^2 -xy -1 = 0$$
on the cordinates plan
I have no idea how to do this. I couldn't find much on google to help me with it too.
Thanks in advance!
Giiovanna
- 3,257
12
votes
2 answers
Why is a positive definite matrix needed in the ellipsoid matrix representation?
An ellipsoid centered at the origin is defined by the solutions $\mathbf{x}$ to the equation $\mathbf{x}^TM\mathbf{x} = 1$, where $M$ is a positive definite matrix. How can I see why $M$ needs to be positive definite, based on the equation of an…
scip
- 969
9
votes
2 answers
Relationship between ellipsoid radii and eigenvalues
I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup.
I went through a theorem in the book stating the relationship between ellipsoid's radii and the…
chepukha
- 231
9
votes
1 answer
How to prove the parallel projection of an ellipsoid is an ellipse?
Take the following ellipsoid in implicit form as an example:
$$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$$
Is it…
user6043040
- 712
7
votes
1 answer
Initial ellipsoid for the ellipsoid method for unconstrained minimization
Suppose I would like to solve an unconstrained optimization problem
$$
\min_{x \in \mathbb{R}^n} \space f(x),
$$
using the ellipsoid method where $n$ is small, i.e. $n = 2$, and $f$ is a strictly convex function having a minimizer. Is there a good…
Alex Shtoff
- 168
7
votes
2 answers
Smallest axis-aligned bounding box of hyper-ellipsoid
Let $E$ be the $n$-dimensional ellipsoid defined by
$$E:=\{x \in \mathbb{R}^n: (x-c)^T A (x-c) \le 1\},$$
where $c \in \mathbb{R}^n$ is the center of the ellipsoid, and $A \in \mathbb{R}^{n \times n}$ is a symmetric positive definite…
Nick Alger
- 19,977
7
votes
1 answer
Simplifying the Willmore energy of an ellipsoid
Willmore energy measures how "non-spherical" a smooth surface $S$ is. It is defined by
$$W(S)=\int_SH^2\,dA$$
where $H$ is the mean curvature.
For a torus of revolution with major and minor radii $a$ and $b$ respectively where $a>b$, if we let…
Parcly Taxel
- 105,904
7
votes
1 answer
Maximizing a linear function over an ellipsoid
Let $A \in \mathbb{R}^{n\times n}$ be a positive definite matrix, $x \in \mathbb{R}^n$ and $c \in \mathbb{R} \setminus \{0\}$. Determine the maximum $$ \max_{y \in \mathbb{R}^n} \left\{ c^T y : y \in \mathcal{E} (A,x) \right\} $$ where $\mathcal{E}…
Thesinus
- 1,262
- 9
- 17
5
votes
0 answers
Can ellipsoids pack better than spheres?
It is known that same-sized spheres can be packed at a density of $\pi/3\surd2$. If we uniformly stretch or compress the packed configuration as a whole, in any direction, the packing density does not change, because the gaps are rescaled in the…
John Bentin
- 20,004
5
votes
4 answers
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 $\mathcal{B}(C_1, R) \cup \mathcal{B}(C_2,R)$ for some…
C Marius
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