# 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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### 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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### 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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### 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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### 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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### 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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