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
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votes
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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
- 44,617
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
- 157
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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…
Leafar
- 1,803
6
votes
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
votes
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
- 3
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
votes
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.
- 19,100
5
votes
0 answers
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…
Myeung Su Kim
- 133
5
votes
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.
- 199
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votes
1 answer
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
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5
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1 answer
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
5
votes
3 answers
Find equation of the cone through the coordinate axes and lines $\frac{x}{1}=\frac{y}{-2}=\frac{z}{3}$ and $\frac{x}{3}=\frac{y}{2}=\frac{z}{-1}$.
Find the equation to the cone which passes through the three coordinate axes and the lines
$$\frac{x}{1}=\frac{y}{-2}=\frac{z}{3}$$ and $$\frac{x}{3}=\frac{y}{2}=\frac{z}{-1}$$
Above is the question from by exercise book, I understand the…
Singh
- 2,168
5
votes
1 answer
Irreducibility of a quadric
I am struggling with a problem in Shafarevich's Basic Algebraic Geometry. First, some context: Fix $k$ an algebraically closed field. Lines in $\mathbb{P}^3$ correspond to planes through the origin in $4$-dimensional space. Thus lines in…
A. S.
- 497