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0answers
48 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 ...
7
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1answer
464 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
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0answers
20 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 ...
7
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2answers
214 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 ...
5
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1answer
178 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
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1answer
172 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 $P$ ...
12
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2answers
776 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
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1answer
192 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
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1answer
110 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
1answer
464 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 $1 \leq p \leq \infty$. ...
9
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3answers
414 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 ...
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0answers
150 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
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3answers
2k 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
1answer
406 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
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1answer
625 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
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0answers
243 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 ...
11
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4answers
476 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
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2answers
760 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 ...
29
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3answers
3k views

Parabolic envelope of fireworks

The envelope of parabolic trajectories from a common launch point is itself a parabola. In the U.S. this weekend many will have a chance to observe this fact directly, as the 4th of July is ...
11
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2answers
854 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 ...
2
votes
2answers
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
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1answer
266 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 ...
2
votes
1answer
959 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 ...