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).

236 questions
Filter by
Sorted by
Tagged with
1 vote
56 views

### 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 ...
• 22.3k
328 views

### 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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ...
• 2,887
92 views

### 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 ...
• 579
1 vote
29 views

### 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 ...
• 696
135 views

### 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 (...
• 22.3k
34 views

### 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 ...
• 22.3k
93 views

### 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 ...
1 vote
21 views

### 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 ...
• 22.3k
23 views

### 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 ...
• 22.3k
48 views

### 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 ...
• 22.3k
57 views

### 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 ...
• 22.3k
1 vote
86 views

### 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, ...
• 22.3k
48 views

### 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 ...
• 22.3k
1 vote
56 views

### 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....
• 22.3k
61 views

• 22.3k
1 vote
107 views

### 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 ...
• 22.3k