# Questions tagged [conic-sections]

The conic-sections tag has no usage guidance.

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### Common conic section between plane/conicoid

A plane
$$ y=m x +n $$
intersects a general second degree conicoid in $\mathbb R^3$ :
$$ ax^2+by^2+cz^2+ 2 dxy+ 2eyz+ 2fzx + 2 gx+ 2hy+ 2 kz + Q=0 ; $$
Eliminating $y$ how do we get the same $z-...

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### A problem of four conics

I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof:
Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...

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259 views

### Is it a new method to construction of a conic, how can prove?

There are some methods to construct a conic, example: Based on Pascal theorem, Steiner construction, .....I propose a method to construct a conic as follows:
Let $L_1, L_2$ be two parallel lines, ...

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### Suppressing some but not all terms of a polynomial equation

(I'll ask the question over $\mathbb{R}$, but feel free to change fields if that makes the answer more straightforward or more interesting.)
Let $Q$ denote a bivariate quadratic:
$$Q(x,y) = Ax^2 + ...

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130 views

### Cone-Torus intersection in 3D

Problem. I have a solid torus and a solid cone in $\mathbb R^3$ and need an efficient algorithm that determines if they intersect or not.
The center of the torus is at a given position $\mathbf p \in ...

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### Definition of a Discriminant in Three Variables

I am studying pell conics and the source I am using (Franz Lemmermeyer: Conics - A Poor Man's Elliptic Curves) defines its discriminant as follows:
For equations of the form $X^2 + XY + \frac{1-d}{4}...

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### Orthogonal Grassmanians: cases where $\text{OG}( \mathbb{P}^1 , Q) \not \simeq \mathbb{P}^3$

Let $Q = \{ q(x_0, \dots, x_4) = 0 \}$ be a quadric-threefold over a field $k$. Are there cases where the orthogonal Grassmanian $\text{OG}( \mathbb{P}^1 , Q)$ is not a copy of $\mathbb{P}^3$?
Here'...

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### Decomposition of conic equation for two intersecting lines [closed]

By using homogeneous coordinates ${\bf x}=[x, y, 1]^T$, a conic section can be expressed by ${\bf x}^T {\bf C} {\bf x} = 0$ with ${\bf C}$ being a 3x3 symmetric matrix. A degenerate form of the conic ...

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### Conics, string art, and Bezier-like curves

It is well documented that certain string-art patterns generate quadratic Bezier curves: let $x, y_1, y_2$ be three points in $\mathbb{E}^2$, consider the family of line segments joining $x + (1-t) (...

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### Can generalization of a generalization Pascal theorem, Pappus theorem to Higher Dimensions? [closed]

Please see a chain of six circles associated with a conic. This is a generalization of Pascal theorem, Pappus theorem. I reformulate as following:
Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a ...

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### Is algebraic geometry related to conical intersection in potential energy surface of molecules?

I post this question here because it seems that the equations describing conical intersections in molecular potential energy surface are similar to what algebraic geometry research concerns.
...

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257 views

### When do two lines and three points determine exactly two conics? Exactly four?

In the real projective plane, I am given three points and two lines. I want to find out how many conic sections there are that are incident to each of the three points and tangent to each of the two ...

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### approaches to Apollonius circle problems

I've been looking for solutions to finding the set of circles tangent to two other circles. one circle can be inverted to a line, but two circles can be mapped to a line and a circle
or equivalently ...

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### Algorithm: Computing the intersection of two conics [closed]

I am looking for a detailed description of an algorithm for the classical problem of computing the intersection of two conics curves. The curves are given by two equations of the form:
$a x^2 + b y^2 ...

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### A chain of six circles associated with a conic

I found this problems three years ago. But I never have been a proof. Recently I posted in math.stackexchange.com. I am looking for a solution of the following problems:
A chain of six circles ...

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### Intersection of a hyper-plane with a hyper-paraboloid of revolution [closed]

I have the equation of a hyper-Paraboloid of revolution:
$2cw=x^2+y^2+z^2$
and the equation of a hyperplane:
$Ax+ By+ Cz+ Dw+E=0$
These do intersect by my construction. How do I find the surface ...

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### differential equation of conics

Function $y(x)$ on the plane defines a conic (strictly speaking, at most second order algebraic curve) if and only if $\frac{d^3}{dx^3} (y'')^{-2/3}=0$. What is intuition behind this? How may I see ...

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### Synthetic projective definition of cubic curves

In a synthetic (Pappian) projective plane, one can define a conic in various clever ways not referring to coordinates. For instance, if $f$ is a projectivity from the pencil of lines through a point $...

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### Reference to parabola lemma

I am looking for a previous reference to the following very simple geometric lemma, which I use in my paper [arXiv:1503.03462]:
Let $P$ be the parabola $y=x^2$. Let $a,b,c,d$ be four points on $P$ ...

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### Discovery and Study of Conic Sections in Ancient Greece

Is there anything known about what drew the attention of ancient greek mathematicians to conic sections and, what were the models they used to study conic sections?
What I would like to know, is ...

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384 views

### the paraboloid model for hyperbolic space [closed]

In Thurston's Three-Dimensional Geometry and Topology, he gives a recipe for a non-standard model for hyperbolic space which he calls the paraboloid model. I'd like to use the model to try out ...

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### Empty real conic containing two pairs of conjugate points in the projective plane?

Given two conjugate pairs of points in general position in $\mathbb{CP}^2$, there is a pencil of real conics containing these four points. Is there a real empty conic in this pencil?

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### Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?

The question is in the title: Let $E$ be an origin-centered ellipse in ${\mathbb R}^2$ and let $S$ be an "$L^p$-circle": $S = \{(x,y) : |x|^p + |y|^p = \text{const}\}$, where $1 \leq p \leq \infty$. ...

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### Reference question: Poncelet theorem

A very famous theorem of Poncelet states that for an elliptic billiard all $n$-periodic trajectories are tangent to some ellipse. As far as I know, Poncelet proved this theorem while sitting in ...

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807 views

### Enclosing a set of ellipses within one ellipse

Is there an algorithm that takes in a set of ellipses and gives back an ellipse that encloses the original set of ellipses?

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### Mutually tangent ellipsoids in 3 space

I recently heard a claim that for any n, it is possible to arrange n ellipsoids in 3 space such that each pair of ellipsoids is kissing. Is this true, and if so, how?
Edit: By kissing, I mean that I ...

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### Asymptotes of hyperbolic sections of a given cone

A book I'm reading (Companion to Concrete Math Vol. I by Melzak) mentions, "...any ellipse occurs as a plane section of any given cone. This is not the case with hyperbolas: for a fixed cone only ...

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### Minimum distance between two arbitrary circles in space?

What is the minimum distance between two arbitrary circles in space?
I am working out the problem with Maxima, but I am surprised by how complicated this rapidly turns out to unfold for such a "...

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### In the classical construction of conic sections, where does the axis of the cone intersect the plane?

Everybody knows that if I take the intersection of a right circular cone with a plane, I get a conic section. My question is, where does the symmetry axis of the cone intersect the plane? Does this ...

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### Can an ellipse's center be determined from a perimeter point's coordinates?

Any arbitrary ellipse in the x-y plane can be described with five parameters -- usually the center’s x and y coordinate positions, x0 and y0; the distance between focal points, d; the eccentricity, $\...

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### Singular conics on certain algebraic surfaces

Let S be an algebraic surface in 3-dimensional complex projective space. Suppose that:
The degree of S is either 5 or 6;
The generic plane section of S is a curve of genus 1.
(Equivalently, the ...

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### Why does the parameterization (F:F':1) happen?

1) To parameterize the conic $x^2+y^2=1$, we can use $(x,y)=(\sin t,\sin't)$ ($\sin'$ meaning the derivative of $\sin$, namely $\cos$).
2) To parameterize an elliptic curve $y^2=4x^3-g_2x-g_3$, we ...

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### Isolated conics on a del Pezzo surface

Is there anything known about isolated conics in a del Pezzo surface: their number, arrangement, and the corresponding elements of the class group of surface's minimal desingularization? (Isolated ...

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### Parabolic envelope of fireworks

The envelope of parabolic trajectories from a common launch point is itself a parabola.
In the U.S. soon many will have a chance to observe this fact directly, as the 4th of July is traditionally ...

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### Geometrically interpreting the answer to a vector calculus question involving tangent line segments to ellipses.

Let E be an ellipse centered at the origin on the x, y plane with major radius b and minor radius a. The length of the shortest line segment tangent to E that begins on the x-axis and ends on the y-...

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### Is ellipse on a sphere convex? (proof)

Is 'small enough' ellipse projected on a surface of a sphere convex? By ellipse I mean a set of points 'C' with a constant sum |AC| + |BC|, A and B are the centers. By 'small enough' I mean that the ...

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### Systems of conics

It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves ...

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### How to find the Fermat Point using the construction of the tangent to ellipse?

Be done the triangle ABC, it is known the method to finding the point Q that minimises the sum QA+QB+QC among all points Q in the plane (The Fermat point).
I want a hint for solving this problem using ...