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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Calculate an ellipsoid given two points and a arc length.

I have a problem and I would like some insight into if people think it is possible. If they think it is possible do they also have an idea of how to solve it. Given two points $A$, $B$ in the ...
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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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Create a mesh of points on an ellipsoid

I have given 4 vectors $m, a, b, c \in \mathbb{R}^3$ (the center of an ellipsoid and its 3 main axes). I am looking for an clever way to compute a mesh of points on the ellipse, which is not to ...
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Tangency point between a sphere and an ellipsoid

I am trying to inscribe a sphere, in the first octant, inside the ellipsoid $$ \dfrac{x^2}{a^2 } + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1 \tag{1}$$ such that the sphere is tangent to the three ...
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How to formulate the definition of an ellipsoid using $\| Ax-b\|_2$?

An ellipsoid in $\mathbf{R}^n$ has the following form \begin{equation}\label{ellipsoid} \mathcal E = \left\{ x \mid (x - x_c)^T P^{-1} (x - x_c) \leq 1 \right\} = \left\{ x_c + Ax \mid \left \|x \...
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Simple Way to Calculate the Volume of solid enclosed by Quadric (E.g. Ellipsoid) Surface

I am trying to apply ellipsoid specific fitting to a set of points with measured coordinates with respect to a reference point. Following that, I need to estimate the volume of the fitted ellipsoid. ...
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Prove that half-ellipsoid is convex

Let $$ S := \left\{ (x , y, z) \in \mathbb{R}^3 : x^2 + 2y^2 + 3z^2 \leq 6, x \geq 0 \right\} $$ Prove that $S$ is convex. I have chosen two points: $\mathbf{(x_1, y_1, z_1)}$ and $\mathbf{(x_2, y_2, ...
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Distance from an arbitrary point to an ellipsoid

I have a collection of points in $\mathbb{R}^3$ which I would like to fit an ellipsoid to. My approach is to frame it as an optimization problem, where for $$\frac{(x-x_0)^2}{a} + \frac{(y-y_0)^2}{b} +...
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Reflection across an Ellipsoid

A reflective ellipsoid is given by $$ \dfrac{x^2}{16} + \dfrac{y^2}{9} + \dfrac{(z - 2)^2}{4} = 1 $$ A light source, emitting rays in all directions, is placed at $A=(10,4,3)$. Find the point $C=(x,y,...
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What does $\mbox{cholesky}(P)^{-1} [\cos(\theta), \sin(\theta)]$ correspond to?

I'm trying to understand a function, and its name isn't really helping me (I've found resources similar to this and this but this is either R or not exactly "ELLIPLOT" and I'm not sure I ...
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Is the 2D projection of the maximum volume inscribed ellipsoid still inside the 2D projection of the polyhedron?

I have a set of points for which I computed the convex hull (which is a Polyhedron). I then computed the maximum volume inscribed ellipsoid of it. Because this problem is six-dimensional, plotting for ...
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2D projections of n-D hyper-ellipsoid in matrix representation

I have a hyper-ellipsoid defined as the transformation of the unit ball: $$ \mathcal{E} = \left\{ T x + d \mid \left\Vert x \right\Vert_2 \leq 1 \right\} $$ where $T \in \mathbb{R}^{n \times n}$ is ...
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Find the parametric equation for the tangent line to the intersection curve between an ellipsoid and a paraboloid?

We got various problems in this site asking for similar problems btw an ellipsoid and a plane. What if it's btw an ellipsoid and a paraboloid? I got the equation of both surfaces: $4x^2 + y^2 + z^2 = ...
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Why is radius of gyration for an ellipsoid derived from a tensor different than the formula?

The first method one can use to solve for radius of gyration, $Rg$, of an ellipsoid with semi-axes $a$, $b$, and $c$ involves using the formula: $$ Rg=\sqrt{\frac{a^2+b^2+c^2}{5}} $$ However, it ...
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Find opposite side length of right triangle inside ellipsoid [closed]

Given an ellipsoid and angle theta (illustrated below), what is the equation to find the length of the opposite side, shown in red. The hypotenuse connects at the ellipsoid center and perimeter. For ...
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Intuition on using norm induced by ellipsoid in optimization problems.

Watching Boyd's lectures on convex optimization, I was confused on the use of an ellipsoidal induced norm. Specifically, he talks about justifying, e.g., $2$-norm on pitch-rate in the objective in ...
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Orientation of ellipsoid after a matrix $T$ act on a sphere described by all the unit vector $\vec{x}$?

I had this problem when I read this paper. It states that(around eq.(1)) for all the vectors $\vec{x}$ in the unit sphere, $T\vec{x}$ will lead the unit sphere into an ellipsoid if $T$ has full rank ...
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What is the Condition Number of an Ellipsoid?

I would like to know how to calculate the condition number of an ellipsoid, in the Convex Optimization book (Stephen Boyd) is calculated as something like below: $$ \mathcal{E} = \left\{x \ | \ (x-x_0)...
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How to derive the parametric equations of the intersection curve of cylinder to ellipsoid

If person wants to derive the intersection curve of the rotated cylinder with offset to ellipsoid $x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$. The equations of rotated cylinder around $y$ with angle $\phi$ plus ...
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Name or Adjective for Ellipse with very different (or very similar) scales

I am looking for an adjective or word to describe an ellipse (or ellipsoid, in more dims) where the length of the principal axes are of roughly the same order of magnitude $\mathcal{O}(a) = \mathcal{O}...
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Determine value of a limit envolving integer solutions $(x,y,z)$ to $2x^2+3y^2+5z^2 = R$, where $R$ is a positive integer.

The exercise is the following: Evaluate the limit $\lim \limits_{R \to \infty} \frac{n(1)+...n(R)}{R^{3/2}}$, given that $R>0$ is an integer and $n(R)$ is the number of integer solutions $(x,y,z) \...
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Distance between the ellipsoid and the integer lattice

Let $r_1, r_2, \dots, r_n > 0$ be positive real numbers and let $$ E: \Big(\frac{x_1}{r_1}\Big)^2+\Big(\frac{x_2}{r_2}\Big)^2+\dots+\Big(\frac{x_n}{r_n}\Big)^2 = 1 $$ be the corresponding ellipsoid ...
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What is the mean curvature of the hyper-ellipsoid $\{x \in \mathbb R^n \mid x^\top B x = r\}$ as a function of $r$ and the condition number of $B$?

Let $B$ be an $n \times n$ positive-definite matrix and let $r>0$. Consider the hyper-ellipsoid $E_B \subseteq \mathbb R^n$ defined by $$ E_B(r) := \left\{x \in \mathbb R^n \mid x^\top B x = r \...
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Legendre's formula for the surface area of scalene ellipsoid

I am looking for a derivation of the formula $$S=2\pi c^2 +\frac{2\pi a b}{\sin \varphi}\left(E(\varphi, k)\sin ^2 \varphi +F(\varphi,k)\cos^2 \varphi\right)$$ for the surface area of the ellipsoid $$\...
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Confusion regarding geodesics and (linear) transformations

I'm having an embarrassingly hard time reconciling some basic calculations that I think are correct (but given my confusion, I won't make a warranty) and the discrepancy in pheneomenology of the ...
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3 votes
1 answer
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Equation to calculate the cap area of an oblate spheroid

I am trying to write a code that calculates the cap area of prolate and oblate spheroids, while avoiding integrals. Through this online calculator I got the equation for a prolate spheroid (i.e. c >...
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Show that a very small ellipsoid can be covered by a very small strip.

Let $E = \{x | (x-z)^TD^{-1}(x-z) \leq 1\}$ be an ellipsoid such that volume of $E$ is at most $2^{-cn^2}$ for some large enough constant $c > 0$. Show that there exists $w \in \mathbb{R}^n$ such ...
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Representation of an ellipse as function of point

Has anyone come across this representation of an ellipse: $\mathcal E := \left\{ x \mid f(x) \leq 0 \right\}$, where $$ \begin{aligned} f(x) &= x^{\top} c x+2 d^{\top} x + e\\ &=\left[\begin{...
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Euclidean Chebyshev center of intersection of ellipsoids. In Boyd & Vandenberghe's Convex Optimization, why does Formula 8.17 on Page 418 hold?

A screenshot of P418 of B&V's CVX OPT Is it necessary to solve the $\sup$ in $g_i(x,R)$. If yes, how can we solve it? I noticed that $A_{i} \in \mathbf{S}_{++}^{n}$, but there is a $\sup$ over $\...
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What is the symbol after the second matrix? [duplicate]

I know this is a very short question but the symbol doesn't pop up with the keywords I've tried. Could it be some sort of "greater than" for matrices? Meaning that the matrix is non-zero?
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Find x, y, z maximum and minimum points of rotated ellipsoid

I have a question very similar to a previously asked question: [Link]Find $x$, $y$, $z$ maximum and minimum points of ellipsoid. However, I have an ellipsoid in the form: $ Ax^2 + By^2 + Cz^2 + Dxy + ...
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1 vote
1 answer
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Plane formed by three mutually perpendicular semi diameters of an ellipsoid.touches a fixed sphere

I have the following ellipsoid before me: $(x^2/a^2)+(y^2/b^2)+(z^2/c^2)=1$. There are three points $P$, $Q$ and $R$ on the ellipsoid which when joined with the origin form a set of three mutually ...
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Volume of $\{(x-\mu)^T S^{-1} (x-\mu) ≤ c^2\} = k_p |S|^{1/2} c^p$

This is following in multivariate analysis book. The coordinates $x^T=[x_1,x_2, \ldots, x_p]$ of the points a constant distance $c$ from $\bar x$ satisfy $(x-\bar x)^T S^{-1} (x-\bar x) = c^2 $ When $...
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Integral over surface of quarter-ellipsoid $\int_{x\in \mathcal{E}: x_1,x_2>0} x_1 x_2$

Let $\mathcal{E} = \{x\in\Re^n:x^\top P^{-1} x=1\}$ denote an ellipsoid that is centered at origin. The goal is to integrate over the surface of ellipsoid, but only in positive half-spaces $x_1,x_2>...
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Surface integral over a half-ellipsoid

Suppose the surface $S$ is made up of the half-ellipsoid $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, \quad z\geqslant 0,$$ together with its base in the $xy$-plane, i.e., $x^2/a^2 + y^2/...
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Shared volume of two overlapping elipsoids

Is there a function to compute the exact (or approximate) shared volume between two ellipsoids with arbitrary rotation and translation?
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Intuition for Ellipsoid Semi-axis Equation

Background: I am trying to develop an understanding of the equation for semi-axes of an ellipsoid. Consider the general form of an ellipsoid with a center at $\vec{s}$: $$ \left[ \begin{matrix} ...
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Check if ellipse lies inside rectangle

I have Ellipse center Cx, Cy and radius (major radius Rx and minor radius Ry) with an angle of α (or α = rotation). Rectangle cordinates are (x1,y1), (x2,y2), (x3,y3) and (x4,y4). The Ellipse can be ...
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Meaning of defining interior of solid object as negative

A sometimes offered definition of the "interior" of a solid object is that it is negative. For example, the following definition is given for the interior of two ellipsoids $\mathcal{A}$ and ...
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Ellipsoid expression effect on characteristic equation

In "An algebraic condition for the separation of two ellipsoids" (https://i.cs.hku.hk/~ykchoi/quadrics/Ellipsoid_Separation.pdf) , the authors offer conditions for ellipsoid intersection ...
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how to find two separate centroids in cross section ellipsoid

How can I find the centroid of left and right side part of ellipsoid to cross section plane. Here is the code and image which I tried. ...
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Correct projection of error ellipsoid onto horizontal plane.

While solving some problem, I obtained the error ellipsoid as an uncertainty estimate of point location in 3-D space. In fact, error ellipsoid is given by standard error (SERR), azimuth, and dip of ...
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Tangent basis to Ellipsoid

I have an ellipsoid centered at $0$ (the contour of a Gaussian distribution centered at $0$ with covariance matrix $\Sigma=\Lambda^{-1}$) $$ x^\top \Lambda x = \gamma $$ and I know that the gradient ...
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A general approach to transforming an ellipsoid to an arbitrary sphere

The Problem: I am trying to map a point on an ellipsoid to its corresponding point on a sphere of arbitrary size centered at the origin. I would like to be able to shift any point with the following ...
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What is the elliptical cone that bounds two ellipsoids

I would like to determine the elliptical cone that contains two ellipsoids such that each ellipsoid is tangent to the cone along the corresponding ellipsoid's ellipse (as conic section). See picture ...
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Ellipsoid equation: Converting from implicit form to explicit matrix form

The implicit equation of a general ellipsoid can be written as follows: $a_0x^2 + a_1y^2 + a_2z^2 +a_3xy + a_4yz + a_5xz + a_6x + a_7y + a_8z + 1 = 0$ I can also define the same ellipsoid with a 3x3 ...
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Finding polar set [duplicate]

I am trying to solve this question but am not able to understand how to approach it: What is the polar of an ellipsoid described by the equation: {$(z_1, . . . , z_d) ∈ R^d: a_1z_1^2 + · · · + a_dz_d^...
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Condition for axially symmetric ellipsoid [closed]

Can someone help me understand what the condition is for an ellipsoid to be considered axially symmetric? If I have an ellipsoid with a = 8.00e-07, then what should the values of the parameters b and ...
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