Not to be confused with quadratic equations, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).
Questions tagged [quadrics]
260 questions
11
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Locus of points equidistant to three spheres
Suppose we have three disjoint spheres in plain ordinary 3D space, with three different radii. I want to know the locus $L$ of points that are equidistant from these three spheres. Partial answers: In 2D, the locus of points equidistant from two…
bubba
- 45,007
10
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
- 12,014
9
votes
3 answers
Given four points, determine a condition on a fifth point such that the conic containing all of them is an ellipse
The image of the question if you don't see all the symbols The given points $p_1,p_2,p_3,p_4$ are located at the vertices of a convex quadrilateral on the real affine plane. I am looking for an explicit condition on the point $p_5$ necessary and…
Kelly
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1 answer
Show that any quadric in $\mathbb{P}^3$ is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$
Show that any non-singular irreducible quadric in $\mathbb{P}^3$ is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$ I know that every non-singular and irreducible quadric in $\mathbb{P}^3$ can be written in the form $xy=zw$ after a suitable change…
Leafar
- 1,803
7
votes
3 answers
What is the best definition of a quadric surface?
I'm defining a quadric surface as "a 3D surface which is a graph of an algebraic equation of degree 2, i.e. a quadratic equation". Since more than one variable can be squared, should I change the last part to "i.e. an equation with at least one…
Nate
- 333
6
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2 answers
How to identify all the right circular cones passing through six arbitrary points
I have this interesting question. Given $6$ arbitrary points, I want to identify all the possible circular cones passing through them. The equation of a right circular cone whose vertex is at $\mathbf{r_0}$ and whose axis is along the unit vector…
user948761
6
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1 answer
Number of points determining a quadric
I know that, in ${\Bbb R}^2$ , in general, $5$ points determine a unique (non-degenerate) conic. I was wondering about the higher-dimensional analogue of this. Is it true, for example, that, in general, $9$ points determine a unique (non-degenerate)…
guest1
- 63
6
votes
2 answers
Quadric surfaces and transformation matrices
Currently, I'm facing a problem and I think I have found a solution to it. However, I'd be interested to hear your opinion or hints on this. I have written geometric modeling software and I have users who want to define a number of different bodies…
user68158
- 61
6
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1 answer
Is there a way to parametrise general quadrics?
A general quadric is a surface of the form: $$ Ax^2 + By^2 + Cz^2 + 2Dxy + 2Eyz + 2Fxz + 2Gx + 2Hy + 2Iz + J = 0$$ It can be written as a matrix expression $$ [x, y, z, 1]\begin{bmatrix} A && D && F && G \\ D && B && E && H \\ F && E && C && I \\ G…
Henricus V.
- 19,160
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Property that defines Quadric Surface
The book < Geometry and the Imagination > (written by David Hilbert) introduces a property of a Quadric Surface without a proof. Property : The cone consisting of all the tangents from a fixed point to a quadric cuts every plane in a conic, and the…
Myeung Su Kim
- 133
5
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1 answer
Solve a linear system of three trigonometric equations in three angles
You are given the linear trigonometric system of equations described by: $ a_1 \cos(x) + a_2 \cos(y) + a_3 \cos(z) + a_4 \sin(x) + a_5 \sin(y) + a_6 \sin(z) = d \tag{1} $ $ b_1 \cos(x) + b_2 \cos(y) + b_3 \cos(z) + b_4 \sin(x) + b_5 \sin(y) + b_6…
user948761
5
votes
3 answers
Maximize $xy + 2 yz + 6 x $ subject to $x^2 + y^2 + z^2 = 36 $
Question: Maximize $f(x,y, z) = x y + 2 y z + 6 x $ subject to $ x^2 + y^2 + z^2 = 36 $ . This question is different from a a previous one due to the existence of the linear term $6x$ . Here is my approach: Following the analysis done in the…
user948761
5
votes
2 answers
Intersecting three quadrics in $3D$
A quadric in $3D$ can be expressed as $ r^T Q r = 0 $ where $ r = [x, y, z, 1] $ , and $Q $ is a symmetric $4 \times 4 $ matrix. Suppose I have three quadrics and want to find their intersection points $(x, y, z) $ . For that I'd like to use the…
Gus L.
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5
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Can all quadric surfaces be obtained by cutting a 4-dimensional cone?
In my high school multivariable calculus class, we recently learned of quadric surfaces. Since they appeared to be a generalization of conic sections to 3 dimensions, I wondered if they could be generated by finding the intersection of a…
Anonymous Person
- 203
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Vertex and axis of $n$-dimensional paraboloid
Consider a surface defined by the form $$ x^\text{T}Ax+b^\text{T}x+c=0, $$ where $A\in\mathbb{R}^{n\times{n}}$ is non-zero symmetric positive semi-definite, $b\in\mathbb{R}^n$ and $c\in\mathbb{R}$. Suppose that $\det(A)=0$, and that $$…
David M.
- 2,703