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]
195 questions
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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
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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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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
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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…
Money Management Skills
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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…
Myeung Su Kim
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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
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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
$$…
David M.
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Find equation of the cone through the coordinate axes and lines $\frac{x}{1}=\frac{y}{-2}=\frac{z}{3}$ and $\frac{x}{3}=\frac{y}{2}=\frac{z}{-1}$.
Find the equation to the cone which passes through the three coordinate axes and the lines
$$\frac{x}{1}=\frac{y}{-2}=\frac{z}{3}$$ and $$\frac{x}{3}=\frac{y}{2}=\frac{z}{-1}$$
Above is the question from by exercise book, I understand the…
Singh
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Irreducibility of a quadric
I am struggling with a problem in Shafarevich's Basic Algebraic Geometry. First, some context: Fix $k$ an algebraically closed field. Lines in $\mathbb{P}^3$ correspond to planes through the origin in $4$-dimensional space. Thus lines in…
A. S.
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Cohomology of union of quadric surfaces in $\mathbb{C}P^3$
It is known that a degree 4 elliptic curve $E\subset \mathbb{C}P^3$ is the complete intersection of two irreducible quadric surfaces $E=Q_1 \cap Q_2.$ Can one compute the (co)homology groups (over $\mathbb{Q}$ coefficients) of the union of these…
Filip
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Find the sets of lines on a quadric
First of all, let me apologize for my English: I'll be making up all the terms of which I don't know the translation.
This is my issue:
In the real projective space $\mathbb{P}^3$, consider the quadric surface $Q$ defined by the following…
Riccardo Orlando
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Change the variables in $Q(x,y,z)=(x-y+z-1)^2-2z+4$ to have $Q(f(u,v,w))=u^2+v$
I have a problem with this exercise. Initially, they gave me this polynom, and I had to complete the squares:
$$Q(x,y,z)=x^2-2xy+2xz+y^2-2yz+z^2-2x+2y-4z+5.$$
I've done it, and I've checked with maple (so it's correct). We…
Relure
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Is there a way to parametrise general quadrics?
A general quadric is a surface of the form:
$$ 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.
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Quadric surface as a $\mathbb{F}_n$ surface
The minimal models for rational projective smooth surfaces are $\mathbb{P}^2$ or the surfaces $\mathbb{F}_n$ for $n\neq 1$, where
$$\mathbb{F_n}=\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n)).$$
The right…
idioteca
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classification of quadrics
Consider the projective plane $\mathbb{R}P^2$ and a symmetric matrix $B \neq 0$ of a bilinear form that defines a quadric $Q := \{ [v] \in \mathbb{R}P^2 : v^tBv = 0\}$.
Is the following ok? And for the affine part I would need help/tips. I am very…
Jamsss
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