# Questions tagged [conic-sections]

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40
questions

**8**

votes

**0**answers

572 views

### A new theorem (discovered in 2013) equivalent to Brianchon theorem (the old theorem) discovered in XIX century?

In 2013, I found a new problem as follows: Let six points $A_1$, $A_2$, ...$A_6$ lie on a circle $(O_1)$, and the six points $B_1$, $B_2$,...,$B_6$ lie on another circle $(O_2)$. If the quadruples $...

**6**

votes

**0**answers

148 views

### What is a geometric construction corresponding to elliptic curve addition for Poncelet's Porism?

Background
At least since Griffiths and Harris [1] we know that the geometric construction "draw the next tangent" appearing in Poncelet's Porism corresponds to addition of a constant in the elliptic ...

**5**

votes

**2**answers

100 views

### On conics curves and increasing unions of ellipses

It is easy to see that the epigraph of a parabola, i.e. the set
$
\\{(x,y)\in \mathbb R^2, y> x^2\\}
$
is a countable increasing union of ellipses in the sense that
$$
\\{(x,y)\in \mathbb R^2, y&...

**6**

votes

**1**answer

292 views

### 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 ...

**4**

votes

**1**answer

278 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, let ...

**1**

vote

**0**answers

58 views

### 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 + ...

**3**

votes

**2**answers

198 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 ...

**2**

votes

**1**answer

186 views

### 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}...

**0**

votes

**0**answers

128 views

### 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'...

**1**

vote

**0**answers

81 views

### 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 ...

**3**

votes

**3**answers

332 views

### 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) (...

**7**

votes

**0**answers

319 views

### 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 ...

**3**

votes

**0**answers

174 views

### 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.
...

**3**

votes

**1**answer

325 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 ...

**3**

votes

**2**answers

648 views

### 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 ...

**2**

votes

**0**answers

403 views

### 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 ...

**9**

votes

**1**answer

841 views

### 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 ...

**1**

vote

**0**answers

78 views

### 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 ...

**18**

votes

**5**answers

1k views

### 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 ...

**6**

votes

**1**answer

285 views

### 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 $...

**2**

votes

**1**answer

202 views

### 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 $P$ ...

**14**

votes

**2**answers

1k views

### 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 ...

**0**

votes

**1**answer

419 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 ...

**1**

vote

**1**answer

119 views

### 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?

**6**

votes

**1**answer

589 views

### 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 $1 \leq p \leq \infty$. ...

**2**

votes

**2**answers

365 views

### 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 ...

**3**

votes

**2**answers

949 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?

**9**

votes

**3**answers

477 views

### 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 ...

**0**

votes

**0**answers

170 views

### 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 ...

**5**

votes

**3**answers

3k views

### 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 "...

**3**

votes

**1**answer

460 views

### 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 ...

**0**

votes

**1**answer

683 views

### 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, $\...

**1**

vote

**0**answers

284 views

### 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 ...

**12**

votes

**4**answers

569 views

### 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 ...

**2**

votes

**2**answers

927 views

### 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 ...

**38**

votes

**3**answers

4k views

### 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 ...

**12**

votes

**2**answers

973 views

### 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-...

**2**

votes

**2**answers

2k views

### 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 ...

**1**

vote

**1**answer

312 views

### 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 ...

**3**

votes

**1**answer

1k views

### 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 ...