17
votes
Accepted
differential equation of conics
I didn't see this question when it originally appeared, but an edit today brought it to the front page of "new questions" where I saw it. Back around 2005 or 2006 I came across this ...
- 3,157
10
votes
Accepted
A problem of four conics
This is a consequence of the Double Contact Theorem.
If $S_1$, $S_2$, and $S_3$ are three conics having the property that there
is a point $X$, not on any of the conics, lying on a common chord of
...
- 660
7
votes
Accepted
Definition of a Discriminant in Three Variables
Given $n$ homogeneous polynomials $F_i(x_1,\ldots,x_n)$ in $n$ variables with respective degrees $d_i$, there is a unique polynomial ${\rm Res}(F_1,\ldots,F_n)$ in the coefficients of the $F_i$ called ...
- 21.8k
7
votes
Accepted
Geometric construction of the fourth intersection points of two conics
Based on the "Intersection de deux coniques" section of this French language page,
the crux of the construction is this:
let two conics intersect in $A,B,C,D$.
let any line through $A$ ...
- 660
6
votes
Are there any non-elementary functions that are computable?
Obviously the perimeter of an ellipse is a computable function of the parameters (e.g. semi-major and semi-minor axes). What it means for this to be computable is precisely that we can compute ...
- 225k
6
votes
A question on motivic zeta-function
$S^nC$ will equal $\mathbb P^n$ for $n$ even and a Severi-Brauer variety with the same Brauer class as $C$ for $n$ odd. In the $n$ odd case, its class in the Grothendieck group will be $[C] ( 1+ L^2 + ...
- 137k
5
votes
Accepted
When do two lines and three points determine exactly two conics? Exactly four?
In general there will be four solutions, possibly complex. In Conic by three points and two tangent lines I've asked for ways to compute these, and based on a comment there found a way which is in my ...
- 534
5
votes
Accepted
Reference request on a characterization of ellipses
Yes, this characterization is a theorem proven by Blaschke in "Kreis und Kugel" (1916). The theorem has a higher dimensional version, characterizing ellipsoids as the unique strongly convex ...
- 85.1k
5
votes
Does anyone know of an academic reference for this proof that the tangent to a hyperbola at a point bisects the angle between each focus to the point?
This is the "reflection property" of the hyperbola. In the context of "academic references", it is good practice to cite the original source, which is
• Apollonius of Perga, 200 ...
- 178k
4
votes
Inscribed $n$-gons of maximum perimeter for an ellipse
Consider allowing one vertex $B$ to vary on your curve, with neighbouring vertices $A$ and $C$ fixed. You want $B$ to be the point on the curve between $A$ and $C$ that maximizes total distance from $...
- 53.6k
3
votes
On conics curves and increasing unions of ellipses
Here is a proof of this fact. Suppose by contradiction there is such a family of ellipses. Each of them is given by an inequality $P_n(x,y)\le 0$, where $P_n$ is a degree two polynomial, $P_n=Q_n+L_n+...
- 28.8k
3
votes
Accepted
Six conelliptic points
Note that there exists a triangle $\tilde{F}\tilde{D}\tilde{E}$ with side lengths $\tilde{F}\tilde{D}=\sqrt{FD}$, $\tilde{E}\tilde{D}=\sqrt{ED}$, $\tilde{F}\tilde{E}=\sqrt{FE}$. Make an affine ...
- 103k
3
votes
Accepted
Equal products of triangle areas
I am using the following fact given a triangle $ABC$ its area is equal to $\frac{1}{2}AB.BC.\text{sin}(\hat{ABC})$. Now use this formula for the both hand sides. First of all note that the fact that $...
- 5,851
3
votes
Accepted
Is the smallest root of this quartic always the closest point on the Hyperbola?
No.
E.g., take $a=4$ and $b=27/8$. Then the positive roots of $g$ are $x_1\approx0.338$, $x_2\approx0.826$, $x_3\approx3.78$, and $f(x_1)\approx13.6$ and $f(x_3)\approx9.72$, so that $x_1$ is not a ...
- 117k
3
votes
Accepted
On maximum perimeter triangles inscribed in convex regions with one vertex fixed
The maximal variation is $4/3$, achieved by the convex region which is a $5-5-6$ triangle. In that region, inscribed triangles have perimeter of at most $16$, and inscribed triangles with a vertex at ...
- 22.2k
3
votes
Accepted
Is it a new method to construction of a conic, how can prove?
Without loss of generality, let $L_1$ be the x-axis and $t$ be a parameter. Denote $C,D,E,F$ by $$C(x_1,y_1),D(x_2,y_2),E(t,0),F(t+a,b)$$ such that the vector $<a,b>$ is in the same direction as ...
3
votes
Cone-Torus intersection in 3D
The following paper shows how to compute the minimum distance between
a canal surface,
e.g., a torus, and a "simple surface," e.g, a cone.
She reduces the computation to finding the roots of a ...
- 149k
3
votes
differential equation of conics
I think your original 5th order differential equation and the 5th order differential equation in the references cited by Dave Renfro can be partially understood without calculation.
What matters is ...
3
votes
In the classical construction of conic sections, where does the axis of the cone intersect the plane?
This question and accepted answer are almost ten years old, but in case anybody stumbles upon this question, here's some more information on the topic. (For cone terminology and background, see ...
- 660
3
votes
Accepted
A chain of six circles associated with a conic
Since $С_1$, $C_2$, and your conic $\alpha$ pass through $A_2$ and $B_2$ we get that $A_1B_1$ and $A_3B_3$ are parallel (it calls three conic theorem http://mathworld.wolfram.com/ThreeConicsTheorem....
- 1,431
2
votes
differential equation of conics
Let me try to give a semi-conceptual answer, which is still not free of some computations. It is inspired by beautiful book "Projective Differential Geometry Old and New: From the Schwarzian ...
- 4,296
2
votes
approaches to Apollonius circle problems
Perhaps you are looking for a Steiner chain?
(Images (above & below) from Wikipedia.)
- 149k
2
votes
Conics, string art, and Bezier-like curves
Here I show how the description given by bubba connects the parabolic case to the hyperbolic case.
Consider the three triples in $\mathbf{R}^3$:
$$ (x_0, y_0, z_0), (x_0^+, y_0^+, 1), (x_0^-, y_0^-, ...
- 37.6k
2
votes
Accepted
Conics, string art, and Bezier-like curves
You could do the usual string-art construction in 3D (giving a 3D parabola), and then do a central projection $(x,y,z) \mapsto (x/z,y/z,1)$ down onto the plane $z=1$. This will give you any conic ...
- 649
2
votes
Accepted
Reconstructing an ellipse from an arc, synthetically
Five points in general position lie on a unique conic. With the help of Pascal's theorem one can construct arbitrarily many points on the same conic.
EDIT. Answering the question about construction of ...
- 6,257
1
vote
Smallest 3-ellipses that contain triangles
My answer here may help,
especially the citation to
Nie, Jiawang, Pablo A. Parrilo, and Bernd Sturmfels. "Semidefinite representation of the $k$-ellipse." In Algorithms in Algebraic ...
- 149k
1
vote
Accepted
Thirteen-point conic and four-point line, are they new?
There is a systematic method to solve problems of this sort (the $p,q$ method) by reducing them to computations which can be done by hand. Assume the vertices are $(0,0), (1,0), (p,q)$. The three new ...
- 266
1
vote
On conics curves and increasing unions of ellipses
Thanks, that's great. Let me write my comment as an answer since I have only 5 minutes to write down a comment, which is a bit short.
Some minor remarks on your two last paragraphs. Let's call $\...
- 15.2k
Only top scored, non community-wiki answers of a minimum length are eligible