Questions tagged [conic-sections]
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63 questions
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Equation of a projective conic over a division ring
The equation of a conic section in homogeneous coordinates in the projective plane $\mathbb{RP}^2$ can be written as
$$
\left[\matrix{x&y&z\cr}\right]
\left[\matrix{a&b/2&d/2\cr b/2&...
- 51
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votes
2
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177
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Radical line of two ellipses
The equation of an ellipse with focal points $(a_1,b_1)$ and $(a_2,b_2)$ which passes through the point $(a_3,b_3)$ is given by the equation
$$\begin{gathered}
\sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x-...
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On the elliptic curve $X^3+6d^2X-7d^3 = Y^2$ and the ellipse $p^2+3q^2-d = 0$?
From the ellipse $p^2+3q^2 - d = 0$ we can find a solution to the equation,
$$a^3+b^3+c^3 = (c+m)^3$$
if we solve the elliptic curve,
$$E:=X^3+6d^2X-7d^3 = Y^2$$
More details can be found in this MSE ...
- 12.6k
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2
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Locus of points for which the sum of the angles subtended there by two different line segments is a constant
Given a line segment AB, the locus of points P such that the angle APB has a constant value is a 'biconvex lens' formed by 2 circular arcs that passes through the points A and B. Special case: if APB ...
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vote
1
answer
135
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Reconstructing an ellipse from an arc, synthetically
Given only an arc of a circle, we can easily reconstruct it fully without any use of analytic geometry - indeed using only compass and straightedge. Note that by "given only an arc", we mean ...
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74
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The closest ellipse and circle to a given triangle - 2
We add a little more to The closest ellipse to a given triangle.
The above linked discussion used the Hausdorff distance to quantify how close two planar convex regions are.
In an earlier post - ...
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1
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Are there any non-elementary functions that are computable?
Does a function $\mathit{f}:\mathbb{R}→\mathbb{R}$ being non-elementary (not expressible as a combination of finitely many elementary operations), imply that it is not computable?
The particular case ...
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182
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The closest ellipse to a given triangle
Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given a general triangle T, to ...
- 5,979
3
votes
1
answer
133
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Is the smallest root of this quartic always the closest point on the Hyperbola? [closed]
Let $a>b>0$.
Suppose we want to minimize
$$
f(x)=(x-a)^2+(1/x-b)^2,
$$
over $x>0$.
Equating $f'(x)=0$ leads to the quartic equation
$$
g(x)=x^4-ax^3+bx-1=0. \tag{1}
$$
Question:
Is the ...
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1
answer
302
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On maximum perimeter triangles inscribed in convex regions with one vertex fixed
Ref: Convex curves with many inscribed triangles maximizing perimeter
Given a planar convex region C. Let P be a variable point on its boundary.
Observations: When C is an ellipse, the variation in ...
- 5,979
2
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0
answers
118
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Ellipse of least perimeter that contains a given triangle
This post is related to Smallest 3-ellipses that contain triangles and tries to clarify a basic issue.
Question: Given a general triangle T, How does one find and characterize the ellipse of least ...
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Tiling with triangles with same Steiner ellipses
We continue from Tiling with triangles of same circumradius and inradius .
Definitions: Given any triangle, its Steiner circumellipse is the unique circumellipse (ellipse that touches the triangle at ...
- 5,979
3
votes
1
answer
323
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Inscribed $n$-gons of maximum perimeter for an ellipse
It appears that the max area inscribed $n$-gon for an ellipse is not unique - if one inscribes a regular $n$-gon in a circle with radius a (there are infinitely many orientations for this inscribed ...
- 5,979
2
votes
1
answer
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Smallest 3-ellipses that contain triangles
Reference: https://en.wikipedia.org/wiki/N-ellipse
Question: How does one find and characterize the smallest 3-ellipses (n-ellipses with n =3) that contain a given triangle? 'Smallest' can mean 'least ...
- 5,979
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1
answer
300
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A question on motivic zeta-function
It's well-known that over $\mathbb F_q$ every smooth projective conic $C$ is isomorphic to a projective line. So the formula for the motivic zeta-function $Z_{mot}(C)$ is evident since $S^n\mathbb P^1 ...
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131
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Largest disk contained in an ellipsoid
It is known that any ellipsoid with principal semi-axes $a$, $b$, $c$ has circular planar sections (https://en.wikipedia.org/wiki/Circular_section).
Is the largest circular disk contained within any ...
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2
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Six conelliptic points
Can you prove the following proposition:
Proposition. Given an arbitrary triangle $\triangle ABC$. Let $D,E,F$ be the points on the sides $AB$,$BC$ and $AC$ respectively , such that $\frac{AB}{DA}=\...
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vote
1
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Equal products of triangle areas
Can you prove the following claim:
Claim. Given hexagon circumscribed about an ellipse. Let $A_1,A_2,A_3,A_4,A_5,A_6$ be the vertices of the hexagon and let $B$ be the intersection point of its ...
- 2,661
5
votes
1
answer
433
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Golden ratio as a property of conic section (is it known?)
I am looking for a proof of a discovery as follows:
Let $ABC$ be arbitrary triangle and $(\Omega)$ be an arbitrary circumconic of $ABC$ let $A'B'C'$ is its tangential triangle of $ABC$ respect to $(\...
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1
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352
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Thirteen-point conic and four-point line, are they new?
We know that Five points determine a conic and Two Points Determine a Line. Here I found a simple construct of a conic through $7$ points (in PS I note that how the conic through thirteen points) and ...
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1
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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?
I am writing a report and as part of it I need to prove the property that for a point $P$ on a hyperbola, the tangent to the hyperbola at $P$ bisects the angle $\angle F_1PF_2$ where $F_1$ and $F_2$ ...
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2
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803
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Geometric construction of the fourth intersection points of two conics
In general, two conics in the plane intersect at most 4 points. Suppose three of those points are given as $A,B,C$. Then let $c_1$ be the conic passing through those three points and $D_1,E_1$. Let $...
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1
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Reference request on a characterization of ellipses
One up-vote, 35 views, and no comments and no answers have resulted from this reference request that I posted on math.stackexchange.com . This was actually inspired by a probability exercise, and at ...
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votes
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answers
910
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 $...
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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 ...
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133
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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
$$
\\{(x,y)\in \mathbb R^2, y&...
- 16.2k
7
votes
1
answer
676
views
A problem of four conics
I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof:
Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...
- 1,776
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votes
1
answer
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views
Is it a new method to construction of a conic, how can prove?
There are some methods to construct a conic, example: Based on Pascal theorem, Steiner construction, .....I propose a method to construct a conic as follows:
Let $L_1, L_2$ be two parallel lines, let ...
- 1,776
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vote
0
answers
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views
Suppressing some but not all terms of a polynomial equation
(I'll ask the question over $\mathbb{R}$, but feel free to change fields if that makes the answer more straightforward or more interesting.)
Let $Q$ denote a bivariate quadratic:
$$Q(x,y) = Ax^2 + ...
- 3,793
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votes
2
answers
381
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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1
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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'...
- 22.8k
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vote
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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 ...
- 11
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votes
3
answers
465
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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) (...
- 39k
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votes
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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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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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0
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778
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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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votes
1
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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 ...
- 1,044
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vote
0
answers
99
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 ...
- 11
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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 ...
- 109k
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votes
1
answer
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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 $...
- 66.7k
2
votes
1
answer
240
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 $P$ ...
- 1,174
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votes
2
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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 ...
- 13.2k
0
votes
1
answer
559
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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 ...
- 2,436
1
vote
1
answer
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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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2
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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 ...
3
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2
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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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