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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Find the algebraic equation of the ellipsoid of known orientation inscribed in a cuboid of known dimensions

Given a cuboid of dimensions $2 a \times 2 b \times 2 c$, and given a $3 \times 3$ rotation matrix $R$, I want to inscribe an ellipsoid whose axes are respectively along the directions specified by ...
Hosam Hajeer's user avatar
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Finding an equation of a plane passing through the origin with cylinder such that the intersection is a circle.

I have the following question here... Find an equation of a plane through the origin such that the intersection between the plane and the elliptical cylinder $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ...
Future Math person's user avatar
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If seven vertices of a hexahedron lie on a sphere, then so does the eighth vertex.

I'm trying to prove https://imomath.com/index.cgi?page=inversion (Problem 11) by projective geometry: If seven vertices of a (quadrilaterally-faced) hexahedron lie on a sphere, then so does the ...
auntyellow's user avatar
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Is this quadratic form B(v, v) a symmetric bilinear?

I'm self-studying the Quadrics by following https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Projective_geometry/Chapter_2_Quadrics.pdf. On page 25, there is an Example Consider the quadratic ...
Rowing0914's user avatar
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Volume of water in a tilted paraboloidal bowl

Suppose that you initially have a container which has the shape of a paraboloid with equation $ z = a x^2 + b y^2 $ where $ 0 \le z \le h $ Now you tilt this paraboloid by rotating it about any point (...
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Determine position of sound source from arrival times of a blip

The setup is a follows: You have three sound receivers (microphones) at known locations $P_1(x_1,y_1), P_2(x_2,y_2), P_3(x_3,y_3)$ that all lie on a plane. A sound source in the same plane produces ...
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A quadric has the origin as centre if and only if its equation has no terms of degree $1$ [closed]

I'd like to know how to prove that if a(n affine) quadric of A^n (regarded as the affine space canonically associated to K^n, being a field) has the origin as centre, then its equation has no terms of ...
Amanda Wealth's user avatar
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Minimum and Maximum distance to the intersection of an ellipsoid with a plane

Given the ellipsoid $ (r - C)^T Q (r - C) = 1$ and the plane $ a^T r = b $ where $ r = [ x,y,z]^T $, $C \in \mathbb{R}^3$ is the center of the ellipsoid, and $Q \in \mathbb{R}^{3 \times 3} $ is a ...
Hosam Hajeer's user avatar
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Closest and Farthest point on curves of intersection between two ellipsoids to the origin

A curve is the intersection of two ellipsoids that are given by $ (r - C_1)^T Q_1 (r - C_1) = 1 $ $ (r - C_2)^T Q_2 (r - C_2) = 1 $ I'd like to find the points on this curve of intersection that are ...
Hosam Hajeer's user avatar
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Minimum and maximum distance of a $3D$ circle from the origin

A $3d$ circle is the intersection of two spheres given by $ (r - C_1)^T (r - C_1) = R_1^2 $ $ (r - C_2)^T (r - C_2) = R_2^2 $ I'd like to find the points on this circle of intersection that are at a ...
Hosam Hajeer's user avatar
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Finding the minimum and maximum distance between the origin and the intersection curve between a cone and a sphere

Suppose you have the cone with its vertex at the origin given by $ r^T Q r = 0 \tag{1}$ where $r=[x,y,z]^T $, and $Q$ is a $3 \times 3$ symmetric indefinite matrix. And you have the sphere centered at ...
Hosam Hajeer's user avatar
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Solving a Lagrange multiplier optimization problem

I have the Lagrange multiplier problem where the objective function is $ f(r) = r^T r $ where $ r \in \mathbb{R}^3 $ subject to $r^T Q r = 0 $ where $Q$ is a $3 \times 3$ symmetric indefinite matrix, ...
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Viewing a paraboloid from a point outside it

Suppose you're given the paraboloid $ z = a x^2 + b y^2 + c $ which you're viewing from the point $A$ that lies outside. What will be the equation of the cone of view of the paraboloid from $A$? My ...
Hosam Hajeer's user avatar
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Determine equation of cone of view of an ellipsoid

Given the ellipsoid $ (p - C)^T Q (p - C) = 1 \tag{1}$ where $ C $ is the center of the sphere, $p $ is a point on the ellipsoid surface, and $Q$ is a $3\times3$ symmetric and positive definite matrix....
Hosam Hajeer's user avatar
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Elliptic cone dimensions such that mutually perpendicular axes can be drawn on its surface

Given the elliptic cone $r^T Q r = 0 $ , where $ r= [x, y, z]^T $ and $$ Q = \begin{bmatrix} \dfrac{1}{a^2} && 0 && 0 \\ 0 && \dfrac{1}{b^2} && 0 \\ 0 && 0 &...
Hosam Hajeer's user avatar
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Locus of point from which three mutually perpendicular lines tangent to an ellipsoid can be drawn

Inspired by this problem, I would like to find the locus of all points from which three mutually perpendicular tangent lines can be drawn to a given ellipsoid. The ellipsoid is given by $$ \dfrac{x^2}{...
Hosam Hajeer's user avatar
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Three mutually perpendicular tangent planes to a paraboloid which intersect at a given point

Inspired by this problem, consider the paraboloid $$ z = \dfrac{1}{4} (x^2 + y^2) $$ And the point $P(2, 3, -2)$. I want to find a set of three mutually perpendicular planes tangent to the paraboloid ...
Hosam Hajeer's user avatar
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Completing square quadric equation

I am trying to find a canonical form for the quadric $x^2+y^2-3z^2-2xy-xz-6yz=0$ by completing squares. For example, if it were $x^2+y^2-3z^2-2xy-6xz-6yz=0$, I would write $$x^2+y^2-3z^2-2xy-6xz-6yz=(...
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Nature of relationship between a cone and a pair of planes

A cone is defined as a surface generated by lines passing through a fixed point and interesecting a given conic, or touching a given surface. A pair of non-parallel planes seems to loosely satisfy ...
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Equation of inscribed ellipsoid in a parallelepiped given $2$ tangency points

Given three vectors $u_1, u_2, u_3 \in \mathbb{R}^3$ that are linearly independent, you build a parallelepiped by specifying a vertex $V_1$, and then the other $7$ vertices follow: $V_2 = V_1 + u_1 $ $...
Hosam Hajeer's user avatar
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3 votes
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Equation of inscribed ellipsoid in a given tetrahedron with three given tangency points

Suppose you're given the four vertices of a tetrahedron $ABCD$. You want to find the equation of the ellipsoid that is inscribed in it and tangent to three of its faces at $r_1, r_2, r_3$. How would ...
Hosam Hajeer's user avatar
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Area of projection of ellipsoid onto the $xy$ plane

Given the ellipsoid $$ (\mathbf{r} - \mathbf{r_0} )^T Q (\mathbf{r} - \mathbf{r_0} ) = 1 $$ with $Q, \mathbf{r_0}$ known. Question: Find the area of its projection onto the $xy$ plane. If $a,b,c$ are ...
Hosam Hajeer's user avatar
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Closed form expression for the area of the shadow of an ellipsoid subject to light rays from a point light source

An ellipsoid with equation $ (\mathbf{r} - \mathbf{r_0} )^T Q (\mathbf{r} - \mathbf{r_0} ) = 1 $ is subject to light rays from an omni-directional light (light source emitting light rays in all ...
Hosam Hajeer's user avatar
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Closed form expression for shadow area of an ellipsoid subject to uniform direction light

An ellipsoid has the equation $$ (\mathbf{r} - \mathbf{r_0} )^T Q_e (\mathbf{r} - \mathbf{r_0} ) = 1 $$ It is positioned such that all of the ellipsoid lies above a projection plane whose equation is $...
Hosam Hajeer's user avatar
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Cutting an elliptical cone

An right circular cone with vertex at the origin, and with it axis pointing along the $z$ axis, is scaled (stretched) along the $y$ axis direction by a factor of $2$. The angle between the surface of ...
Hosam Hajeer's user avatar
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Proving a system of quadratic forms has no (non-zero) solutions

A system of homogeneous linear equations always has the solution $ x=(0,\dots, 0) $. Suppose we have a system of $ n $ homogeneous linear equations in $ k $ variables. If $ k > n $ then there will ...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
222 views

Plane through feet of normals to ellipsoid

The problem If $P,Q,R,P',Q',R'$ be the feet of six normals drawn from a point to the ellipsoid $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} - 1=0$$ and the plane $PQR$ is represented by $lx+...
s_a94248's user avatar
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1 answer
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System of $ n $ simultaneously diagonal real quadrics in $ n+1 $ variables has all solutions real

A single isotropic real quadric in 2 variables always has 2 (projective) real solutions. See Zero set of system of two real quadratic forms for the explicit form. I've noticed that a system of 2 ...
Ian Gershon Teixeira's user avatar
1 vote
2 answers
112 views

Closest point on the surface of an ellipsoid to a given point

Given the ellipsoid $ (r - r_0)^T Q (r - r_0) = 1$ And a point $A$ anywhere in space, I want to find the closest point $r^*$ on the ellipsoid to $A$. My attempt: Define the scalar function: $F(r) = (r ...
Hosam Hajeer's user avatar
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The closest points on a line and an ellipsoid

Inspired by this problem, I've devised the following problem where the cone is replaced with an ellipsoid. Given the line $$ p(t) = p_0 + t \ d $$ where WLOG, $d$ is a unit vector, and the ellipsoid, $...
Hosam Hajeer's user avatar
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Conformal mapping from quadrics to the plane

I am writing a raytracer (a type of 3D engine) to render quadrics, and I am working on rendering a one-sheeted hyperboloid defined by the zero set of $x^2-y^2+z^2=1$. I need 2D coordinates $a \in [0, ...
zenzicubic's user avatar
2 votes
1 answer
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Placing an ellipsoid in a rectangular box

Suppose you have an ellipsoid with semi-axes lengths $a,b,c$, and you a rectangular box (i.e. cuboid) of dimensions $L, W, H$, with its length $L$ along the $x$ axis, its width $W$ along the $y$ axis, ...
Hosam Hajeer's user avatar
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Why is a degenerate quadric well defined?

Let $K$ be a field with $\text{char}(K) \neq 2$ and $Q$ be a quadric in the projective space $P(K^n)$. Let $M$ be a symmetric $n×n$-matrix over $K$ such that $Q \leftrightarrow x^T M x$. In my course ...
Steve's user avatar
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Shadow of a circle

A circle of radius $10$ units is centered at $(0, 0, 50)$. The plane it lies in, has a unit normal vector $(1, 0, 0)$. A uniform direction ray of light falls on the circle, generating a shadow on ...
Hosam Hajeer's user avatar
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5 votes
3 answers
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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 ...
Hosam Hajeer's user avatar
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Intersection of two quadrics using matrix form

Given any quadric can be represented in matrix form, assume I have one quadric $Q_1$ and another quadric $Q_2$. How can I find the equation of the surface that defines intersection of $Q_1$ and $Q_2$ (...
BeginnersMindTruly's user avatar
2 votes
2 answers
123 views

Maximum of $xy + 2 yz$ subject to $x^2 + y^2 + z^2 = 36$

I want to find the maximum of $ f(x, y, z) = xy + 2 y z $ subject to $ x^2 + y^2 + z^2 = 36 $ My Approach: The most direct way is to parameterize $(x, y, z)$ which is easy in this case because $(x,y,z)...
Hosam Hajeer's user avatar
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Volume of water that can fill up a tilted conical cup

A cup takes the shape of a right circular conical frustum. The bottom radius is $r_1$ and the top radius is $r_2$ and its height is $h$. It is tilted from the vertical by an angle $\theta$. Find an ...
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3 votes
2 answers
161 views

Angle of tilt of a cup so that the water surface touches the rim

A cup is in the shape of a right circular conical frustum with bottom diameter $4 cm$ and top diameter $8 cm$, and slant height $10 cm$. It is filled to $\dfrac{2}{3}$ of its height with water, then ...
Hosam Hajeer's user avatar
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4 votes
2 answers
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A planar cut through an oblique cone

An oblique cone has a circular base centered at the origin of radius $5$, and an apex at $(0, 5, 20)$. A plane whose equation is $3x-4y+5z = 40$ cuts through the oblique cone, and the resulting cut ...
Hosam Hajeer's user avatar
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Intersection of a certain ellipsoid with a given plane

An ellipsoid centered at the origin, with semi-axes of lengths $10, 15, 30$ along the $x, y, z$ directions, is cut by a plane whose equation is $x + 3 y + 2 z = 40 $ The intersection of the ellipsoid ...
Hosam Hajeer's user avatar
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hyperplanes as intersection with cones

An affine hyperplane $H$ in $\mathbb{R}^{3}$ is an affine subspace in $\mathbb{R}^{3}$ of dimension 2, i.e. there exists a $h \in \mathbb{R}^{3}$ and a two-dimensional $\mathbb{R}$ vector subspace $U$ ...
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Find the projected ellipse on a plane from a given $3D$ ellipse

Given the ellipse $ E(t) = C + V_1 \cos t + V_2 \sin t $ I want to generate rays from a point $P$ lying outside the plane of $E(t)$, then intersect those rays with the projection plane $n^T (r - r_0) =...
Hosam Hajeer's user avatar
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Determining the distance between a plane and an elliptic paraboloid

Given an elliptic paraboloid whose equation is $ (r - V)^T R D R^T (r - V) + b_0^T R^T (r - V) = 0 $ where $r$ is a point on the surface of the paraboloid, and $V$ is its vertex. $R$ is a rotation ...
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2 votes
1 answer
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Identifying an ellipsoid from $9$ planes that are tangent to it

The algebraic (implicit) equation of an ellipsoid is $ (r - C)^T Q (r - C) = 1 $ where $ r = [x, y, z]^T $ is a point on the ellipsoid surface, and $ C = [C_x, C_y, C_z]^T $ is the center of the ...
Hosam Hajeer's user avatar
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2 votes
2 answers
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Identifying a paraboloid tangent to a number of planes

Context: Recently, I investigated the problem of finding a parabola tangent to $4$ given lines in the Cartesian plane. I was able to write a program that finds this parabola. I wanted to extend ...
Hosam Hajeer's user avatar
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Quadratic surface where variables are functions

Is it possible for a quadratic surface to have variables and coefficients that are functions? For example, an equation such as $$A(T)x(T)^2 + B(T)y(T)^2 + C(T)z(T)^2 + D(T)x(T)y(T)\;...$$ where both ...
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Converting a sphere equation to a homogenous one to obtain the equation of a cone

A variable plane parallel to the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0$ meets the coordinate axes at $A, B, C$. Find the equation of the cone whose vertex is at the origin and the guiding curve ...
Sasikuttan's user avatar
2 votes
0 answers
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How many points are needed to uniquely determine a cone?

The general equation of a cone that passes through the origin is $$ax^{2}+by^{2}+cz^{2}+2fyz+2gzx+2hxy=0$$If I'm given $5$ points on the cone, I should be able to get $5$ equations and be able to ...
Sasikuttan's user avatar
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How can I call "trivial" quadrics, like empty set, single point, straight line, plane, two lines, two planes...?

My temptation is to call it degenerate quadrics. But, in Wikipedia, the list of degenerate quadrics also includes cylinders and conic quadrics. On the other hand, in these notes, the author uses the ...
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