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Questions tagged [quadrics]

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).

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General form of quadric surfaces

The general form of quadric surfaces is $$Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0$$ I want to classify all of the possibilities including degenerate cases with the help of ...
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Quadric surface S tangent to plane

Suppose that the quadric surface S is given by $z = x^2 + x + 2y^2 + 3y$ and the plane is given by $x + y + z = k$, where k is a constant. Find the vector equation for the tangent line to the curve ...
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How to express a quadric equation from canonical form to a different basis.

I have the quadric $3X^2-Y^2-Z^2=0$ expressed in the canonical form, and the matrix of change of basis from a basis B to the canonical form is $$P=\begin{bmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\...
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Compute projected radii of a rotated elliptic paraboloid

I'm working of a set of datapoints known to be an elliptic paraboloid on which I best fit the general quadric $$ax²+bxy+cy²+dx+ey+f=0$$ Then I work with what I call radii projected on x an y defined ...
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Find the solution of an outer product induced system

Sorry if the question is lame, but I'm struggling to find the answer to the following problem: Given a matrix $A\in \mathbb{R}^{n,n}$ and a column vector $b\in \mathbb{R}^{n}$, how can one find the ...
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Can quadric surfaces be made 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 ...
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Reduced equation of quadric surfaces

Given the following quadric surfaces: Classify the quadric surface. Find its reduced equation. Find the equation of the axes on which it takes its reduced form. The quadric surfaces are: (1) $3x^2 +...
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Uniqueness of the quadratic form associated with a quadric

Let two quadratic forms $\Phi_1:E\to\mathbb{K}$ and $\Phi_2:E\to\mathbb{K}$ be, such that $\{u\in E:\Phi_1(u)=0\}=\{u\in E:\Phi_2(u)=0\}$. Then, it is known that $\Phi_1=\lambda\Phi_2$ for some $\...
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Intersection of projective quadric and affine plane.

I'm stuck in trying to understand the graphical part of the following problem. Let $\mathcal C = \{ [x:y:w:z] \in \mathbb P ^3: x^2 +xy +yw -w^2 = 0 \}$. Graph the intersection $\mathcal C \cap \...
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Quadratic forms, change of variables

If one has a symmetric matrix $A$, one can diagonalize it with an orthonormal change of basis vectors, e.g. $S^TAS$ is diagonal. Now lets consider the following matrix $$A=\begin{bmatrix} 1&1\\ 1&...
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Rotation of ellipsoid(quadric)

Consider $$φ(x, y , z) = x^2 + 2y^2 + 4z^2 −xy −2xz −3yz$$ find the coordinate transformation (translation or rotation) to eliminate $xy$, $xz$ and $yz$. In $\mathbb R²$, with conic sections, I would ...
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How to convert from a quadratic form to canonical, when there's no squared unknown?

I already know that this is hyperbolic paraboloid, but I can't turn it into a canonical equation: $ z = xy $
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How to find distance between a point and 3D surface, solution to general quadric equation, and visualizing such a surface?

Goal: I am writing software to visualize 3-D objects in Python, using libraries such as sympy, numpy, and ...
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Proof that quadrics of signature $(2, 2, 0)$ in $\mathbb{R}\mathbb{P}^3$ is homeomorphic to torus

I know that a quadrics of signature $(2, 2, 0)$ in $\mathbb{R}\mathbb{P}^3$ is homeomorphic to a torus. However, I know of no simple proof of this fact. I have heard that one proof uses the fact that ...
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Determine conics by four points and a tangent line.

I have small troubles determining conics that go through four points and have a given tangent line. More specifically $P_1 = (0:1:0), P_2 = (0:0:1), P_3 = (1:0:1), P_4 = (1:-1:0) \in \mathbb{RP}^2$ ...
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Canonic form of a quadric

I have a question, how can determine the canonic form of the quadric associated to $F$, where $F$ is a bilinear form $\Bbb R^3\times\Bbb R^3\to\Bbb R $ that its associated matrice is $A=\left(\begin{...
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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 $c\in\mathbb{R}$. ...
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Intersection of two smooth projective quadrics in general position

1) I know what it means for a finite set of points in a projective space to be in general position, but what does it mean for two quadrics in $\mathbb{P}^3(\mathbb{C})$ to be in general position? 2) ...
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Normal form for quadrics

In the classification theorem of the wiki's page on quadric, the normal form is obtained by action of rigid body transformations (called also as change of Cartesian coordinates). If homotheties are ...
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Cone $x^2+y^2-z^2=0$ as a degenerate quadric

We can read here quadric wikipedia that the cone $x^2+y^2-z^2=0$ is a degenerate quadric but they don't define what is a degenerate quadric. I know hat a quadric is non degenerate if the matrix ...
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Volume without Multivariable Calculus

The tangent plane to the ellipsoid at a point $(x_0, y_0, z_0)$ $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ and the co-ordinate axis planes form a tetrahedron. Only with the use of single ...
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Quadric represented in matrix form

I just want to make sure I understand this right. In the book our teacher describes matrix representation of quadric as (sorry for the 3x3 matrix it should be 2x2) $$[ 1\,\,\, x^T]\begin{bmatrix} c ...
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Calculate centres of a quadric

I recently studied quadrics at university, in linear algebra, but professor has skipped parts of proofs which I think are important. We have defined a function $\phi$: $\mathbb{R}^n\to\mathbb{R}$ by $$...
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Project 3D point onto cylinder

I have a set of homogeneous 3D points $X_i = [ x_i, y_i, z_i, 1 ] ^T$ such that $$ X^T C X \le \epsilon $$ where $C$ is a $4 \times 4$ matrix representing a general cylinder (i.e. not necessarily ...
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Prove that three paraboloids have a common tangent plane

Prove that the paraboloids $\frac{x^2}{a_{1}^2} + \frac{y^2}{b_{1}^2} = \frac{2z}{c_{1}}$, $\frac{x^2}{a_{2}^2} + \frac{y^2}{b_{2}^2} = \frac{2z}{c_{2}}$ and $\frac{x^2}{a_{3}^2} + \frac{y^2}{b_{3}^2}...
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Finding values for a and b that satisfy necessary conditions for a function to be convex

In the above problem, I have some thoughts but wanted to make sure I'm on the right track. I know that if $f(x)$ is convex and $g(t)$ is convex and increasing, then $g(f(x))$ is also convex. So in ...
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181 views

Conditions for a plane and an ellipsoid NOT to intersect

I know that the intersection of an ellipsoid and a plane is an ellipse. However, how can I derive the conditions for an ellipsoid and a plane not to intersect? Suppose I have a plane $$Ax+By+Cz+D=0$$ ...
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Two questions on three quadrics in $P^5$ whose intersection is a genus $5$ K3 surface.

It is well known that the intersection of three quadrics in $P^5$ yields a genus $5$ K3 surface. (See this link: https://en.wikipedia.org/wiki/K3_surface ). Question I: Does anyone have an example (...
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Covering map from paraboloid to one-sheeted hyperboloid

I need to construct an explicit covering map between the following two sets: The paraboloid $X = \{(x,y,z) \in \mathbb{R}^3:z = x^2 + y^2\}$ as covering space The one-sheeted hyperboloid $\hat X = \...
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Find all values of a parameter for which some quadrics have 4 common points of intersection

$2|xy-3y-4x+12| = a^2+2a-z-30$ and $3a^2-a-z-32=0$ and $z-x^2-y^2+6x+8y=0$ What I've got: $3a^2-a-32=x^2+y^2-6x-8y$ so $(x-3)^2 + (y-4)^2 = 3a^2-a-7$ and this is circle $2|xy-3y-4x+12|=-2a^2+3a+2$. ...
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Hyperboloids of one sheet, hyperbolic paraboloids, and Hilbert's famous “three skew lines”

One of Hilbert's many deceptively simple observations involved generators of hyperboloids of one sheet and hyperbolic paraboloids: Three skew lines always define a one-sheeted hyperboloid, except in ...
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Suppose it can be shown that any right(left)-regular finite-state grammar implicitly generates a “hypar”" surface

Background: The question I'm about to ask is a follow-on question resulting from a discussion with Paul Sinclair and Henning Makholm in this thread here: Permutation of 0,,,n-1 as two vectors with n ...
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Quadric fitting

Let's have two quadrics (as input): $Q_1$ and $Q_2$. These are symmetric 4x4 matrices of reals. I want to find a transformation (9 degrees of liberty: 3 for rotation, 3 for scale and 3 for ...
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Lines lying in a projective algebraic set

I want to find all lines lying entirely in the projective algebraic set $XY-ZW=0$ in $\mathbb{P}^3$, where $X, Y, Z, W$ are homogeneous coordinates. How can I do this?
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Why is $\frac{x^2}{4} + \frac{y^2}{9} = 1$ the equation of an elliptical cylinder?

I'm attempting to graph the function: $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$ I can tell that at constant $z$ the graph is elliptical, but I found, at constant $x$ and $y$, let's say $x=0$, $y=0$: Let ...
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connection and fundamental group of real and complex quadrics

I would like to calculate the fundamental group of real and complex quadrics. I've seen it in the projective space, but not in $\mathbb{R}^n$ and $\mathbb{C}^n$. Also, I would like to prove that they ...
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Automorphism fixing quadrics.

i'm trying to solve these problems: (1) Suppose that $X\subset\mathbb{P}^2_\mathbb{C}$ is a smooth conic and $A,B\in X$ are two points on it. Find automorphisms of $\mathbb{P}^2_\mathbb{C}$ fixing $...
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Does an Implicit equation for an infinite Cylinder exist ? i.e f(x,y,z)=0

I wanted to know if the surface for any arbitrary cylinder(infinte or restricted does not matter) can be expressed with an implicit equation, like that for a sphere: $$(X-x)^2 + (Y-y)^2 + (Z-z)^2 = R^...
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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: ...
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Determining a quadric surface.

I've been having great difficulty in determining the type of surface that this thing is: $9x^2 - 16xy - 5y^2 + 16xz + 23z^2 = 20x$. Thoughts. So, if we took the RHS to be zero, then I know that ...
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Manifold of Quadric Surface Coefficients

Being a non-mathematician, excuses if I make errors in formulating this question. Consider a general quadric surface of the implicit form: $$ f(x,y,z)=Ax^2+ By^2 + Cz^2 + 2Dxy + 2Exz + 2Fyz + 2Gx + ...
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Space of quadrics

What does one mean when they say that a certain "subspace" of quadrics has dimension $d$? Specifically, let $\mathbb{P}^1(\mathbb{C})\rightarrow \mathbb{P}^3(\mathbb{C})$ be given by $[X_0:X_1]\...
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Where is the parabolic paraboloid?

In theory, any quadric surface can, through a series of translations and linear transforms, be converted to $f(X)+g(Y)+h(Z)=C$, where $f(X)=X^2$ (except in the case of the imaginary ellipse, where $f(...
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Hessian of an Ellipsoid seems invariate

An ellipsoid is the hypersurface of the ellipse in $\mathbb{R}^3$ which belongs to the family of quadrics. It is defined by a quadratic form equating to 1 in its canonical (unrotated) form. $\vec{x}^...
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Classification of a quadric

Is it possible to classify the following quadric along with its centre without using the principal axis theorem? $$x^2 = 1 + x +xy$$ For instance, a simple equation like $x^2 + 2x = 3$ turns out ...
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Question about how to identify a surface just completing squares

Question: Identify the type of surface given by $x^2 + 2xy + z^2 = 1$. Solution: By completing the square, one obtains $x^2 + 2xy + z^2 = 1$, so $(x + y )^2 - y^2 + z^2 = 1$, and then $(x + y)^2 + z^...
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Quadric obtained by rotation of a line

I have a line of equations: \begin{align}r: \begin{cases} x = 2+2t \\ y = 2 - t \\ z=t \end{cases} \end{align} and another line of equations: \begin{align}s: \begin{cases} x = 3 \\ y = 1 \\ z=k \end{...
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Graphing a hyperboloid of one sheet when the right side of the equation is zero

I am attempting to graph a hyperboloid of one sheet. Right now, I am sketching the traces of the hyperboloid in the $(y, z)$ plane. Here is the hyperboloid equation: $$(x-5)^2+(y-5)^2-(z-4)^2 = 1.$$ ...
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Explanation behind a behavior observed for certain quadrics

Having two quadrics, given as polynomials with real coefficients $f (x) = 0$, $g (x) = 0$; $x \in R^3$; $f (x) < 0$ marks the interior, $f (x) > 0$ marks the exterior, $f (x) = 0$ marks the ...
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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 ...