Questions on conic sections and their properties; the curves formed by the intersection of a plane and a cone. Circles, ellipses, hyperbolas, and parabolas are examples of conic sections.

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Why does the Ellipsograph/Trammel of Archimedes draw an ellipse, really?

Here's a diagram of the device I mean, hard at work drawing an ellipse. I find this quite surprising, and would like to get to the bottom of things. Essentially, a rod (black line in animation) is ...
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130 views

On the Visual Manifestation of Curves in Nature

All sorts of curves are useful in modelling and describing phenomena we observe. Trig functions, logarithms, exponentials, polynomials, hyperbolas, circles, and so forth are all very useful in this ...
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1k views

To construct an ellipse, being a projection of a great circle, given two points on it

I'm looking for a geometric construction which would allow me to draw an ellipse, which is supposed to be an orthographic projection of a great circle of a sphere, given two points on it. The ...
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1k views

Decomposition of a degenerate conic

As it has been done for the Intersection of conics using matrix representation the aim of this page is providing an exaustive and clear numerical example that describe the math behind the ...
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77 views

Keplerian orbits and closest approaches to Earth.

This question arose out of a discussion on Space.SE, but I think it will appeal to mathematicians more than astronomers: Let's consider a small astronomical object following an ideal elliptic ...
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69 views

Visualising 3rd degree equations

I know that general second degree curve, i.e. $ax^2 + by^2 + 2hxy + 2gx + 2fy + c=0$ gives us the equation of different cross sections of a cone. Similarly, what does a third degree* curve actually ...
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117 views

A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
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99 views

What property does this equation calculate?

It's pretty difficult to Google for the meaning of a formula. This is the equation, it has to do with ellipses and GIS coordinates. $$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$ $a$ ...
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125 views

Polar representation of conic sections $r(\theta)=\frac1{1 + e \cos\theta}$

Consider a curve given in polar coordinates by $r(\theta) = \dfrac1{1 + e \cos\theta}$, where $e\ge0$. a) Show that the distance of each point on this curve to the line $x=\frac1e$ is a constant ...
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523 views

Equation of intersection of two cones

The equations of two cones are given; $(x-x_{0})^2+(y-y_{0})^2=\frac {(z-z_{0})^2}{m^2}$ and $(x-x_{1})^2+(y-y_{1})^2=\frac {(z-z_{1})^2}{m^2}$ How to find the equations of intersections 1) ...
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78 views

Quadric question

I'm trying to prove that given 3 disjoint lines in $\mathbb{P}^{3}$ there exists a non-singular quadric containing them. The exercise is from the following link: http://www.uam.es/personal_pdi/...
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31 views

Find the radius of the circle for given conditions

A circle with center at origin passes through three points $P$, $Q$ and $R$ with the line segment $PQ$ as its diameter along $x$-axis. A line passes through $P$ intersects the chord $QR$ at point $D$. ...
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55 views

Ellipse Area (Trouble understanding answer)

Question: An elipse with equation $$ {x^2\over a^2} + {y^2\over b^2} = 1 $$ is enclosed by the hyperbolas given by $xy=1$ and $xy=-1$. , Determine the largest area of an ellipse enclosed by the ...
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51 views

Construction new ellipse

Using a pencil the thread was pulled on the ellipse. Then the pencil started to rotate around the ellipse. How to prove that a new geometric figure which the pencil drew is also an ellipse (with the ...
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162 views

How to convert general equation of ellipse to a form analogous to standard form?

Sorry for the bad title , please edit it to something better if you can. I need a procedure to convert the general equation of ellipse - $$Ax^2 + By^2 + 2hxy + 2gx + 2fy + c = 0$$ into $$\frac{\...
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36 views

Hyperbola crossing, # of solutions

We typically see hyperbolas drawn the "nice" way. Namely, they are oriented with the arms "opening up" straight up or down, or at 45 degrees. But, in general, they can be at any "angle". Question:...
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43 views

Curios relation between parabola, circumcircle and circumellipse

When playing around with conics in GeoGebra, I have found out that the following relation seems to hold: Let parabola $p$ be tangent to sides/extensions of sides $BC,CA,AB$ of triangle $ABC$ at ...
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56 views

What's an elegant expression for a general conic using complex numbers?

A 'nice' form of writing a line through $2$ points $z_1,z_2\in\mathbb{C}$ is $$z\overline{z_1-z_2}+z_1\overline{z_2-z}+z_2\overline{z-z_1}=0$$ Now I'm asked to give a similar expression for a general ...
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65 views

Probability Density Function for Randomly Oriented Ellipse

I have an ellipse with a long aspect of a and a short aspect of b. The equation for this ellipse is found on this post: What is the general equation of the ellipse that is not in the origin and ...
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121 views

Find the minimum radius of the circle which is orthogonal to two given circles

Problem : Find the minimum radius of the circle which is orthogonal to both the circles $x^2+y^2-12x+35=0$ and $x^2+y^2+4x+3=0$ . Solution : Let the equations : $x^2+y^2-12x+35=0.....(i)$ and $x^2+...
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105 views

any conic in $\mathbb{A}^2$

Exercise 3.1 in Hartshorne's Algebraic Geometry: Show that any conic in $\mathbb{A}^2$ is isomorphic to $\mathbb{A}^1$ or $\mathbb{A}^1-\{0\}$. when the conic given by $x^{2}+y^{2}-1$, what the ...
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661 views

Relation between ellipse general and parametric equation

I am familiar with the fact that one can relate the eigenvectors and corresponding eigenvalues of an ellipse's quadratic equation matrix, to the pose of a circle in 3-space. When say quadratic ...
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101 views

Biggest ellipse included in a convex polygon

Considering a N edges convex 2D polygon called P. Let's name its vertices $\{p_1, p_2, ..., p_N\}$ described in a counter-clockwise order, with $p_i = (x_i, y_i)$ What would be, and how would one ...
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694 views

Volume of the intersection of two solid cones

I want to calculate the volume of the intersection (overlap) of two solid cones, as shown below. I imagine this must be a known problem but I'm having a hard time finding anything online, so any ...
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28 views

Do both these ellipses satisfy the same conditions?

I was solving a problem that asked for the equation of the ellipse with the following properties: vertex at $(-10,5)$, focus at $(-2,5)$, eccentricity $\frac{1}{2}$. I think I found two such ellipses, ...
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246 views

Ellipse and circles

Playing with an ellipse I discovered the following properties and am now looking for a nice proof or references. Let's consider an ellipse with foci $A$ and $B$ and let $C$ be a point on it. Let $l$ ...
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32 views

Points at infinity of a conic section and its eccentricity, foci, and directrix?

Background on projective geometry and conic sections; you might want to skip to the actual question A conic section is analytically described as the zero-locus of points $(x,y)$ in the affine plane ...
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34 views

Fourier transform of function defining half an ellipse

I'm trying to determine the expression for the Fourier transform of a function defining half an ellipse. It's been awhile since I've done Fourier transforms by hand. Obviously I can plug the ...
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30 views

Finding a curve that is orthogonal at $(1,1)$ to the set of given parabolas

I am given a DE like the following: $$\frac{dy}{dx} = \frac{2xy}{x^2-1}$$ When one solves it: $$y = A(x^2-1)$$ Then we obtain an equation for a family of parabolas all intersecting $(-1,0)$ and $(...
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26 views

Area covered by fixed perimeter around polygon.

Suppose I have a polygonal field with a post at each vertex and a non-extensible rope threaded through each post around the perimeter but with some slack. How can I determine the perimeter of the area ...
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26 views

The difference of the focal semi axes of an ellipse and a hyperbola is equal to $4$.If the ratio of their eccentricities is $\frac{3}{7}$.

An ellipse and a hyperbola have their principal axes along the coordinate axes and have a common foci separated by a distance $2\sqrt{13}$,the difference of their focal semi axes is equal to $4$.If ...
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59 views

Proof of equation of ellipse

The ellipse can be defined as a conic section with eccentricity lesser than unity. How can you derive the equation of the ellipse using this definition? I can't find proof of the formula using this ...
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28 views

Polar form of generalized superellipse

I am looking for the polar form of the generalized superellipse: $$ \left|\frac{x}{a}\right|^{n_2}+\left|\frac{y}{b}\right|^{n_3}=1 $$ where $a$ and $b$ are the semi major and semi-minor axes. I have ...
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Prove an ellipse is unique if the foci and a tangent are given.

Given 2 points $F_1$ and $F_2$ and a straight line $l$ which does not cross $[F_1F_2]$. Prove that there exists an unique ellips with $F_1, F_2$ as foci, and tangent $l$. What if $l$ crosses $]F_1,...
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54 views

Inversion across an ellipse

Let's take an ellipse with the standard equation $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ And I am trying to invert the following ellipse across that ellipse $$\frac{4x^2}{3}+4\left(y-\frac{3}{2}\right)^...
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34 views

Coordinates of a plane with a handle

I am trying to find the appropriate coordinates for a plane with a handle (of topology $\mathbb{R}^2 \# \mathbb{T}^2$), without having to use several coordinate patches. My current intuition is to ...
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60 views

Show that a complex equation represents a circle

I'm having troubling understanding the answer to a question. The question is: If $\ v=1+i$ and $\ z=x+iy$, for any real numbers x and y: Show that the equation $\left|z-v\right|= \left|vz\right|$ ...
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49 views

If $p(x,y)$ is a irreducible quadratic polynomial, then there is a line not intersecting $p(x,y)=0$

Let $p(x,y)\in \mathbb{R}[x,y]$ be an irreducible polynomial and $deg(p)=2$. Then $p$ defines a conic $Q$ in $\mathbb{P}^2(\mathbb{R})$ as $$Q=\{[x,y,t]\in \mathbb{P}^2(\mathbb{R})|\hat{p}(x,y,t)=0\}$...
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49 views

How to find smallest enclosing ellipse when multiple lines are given ? This ellipse needs to intersect all those lines

I'm trying to find the smallest ellipse in terms of circumference. I suspect the smallest enclosing ellipse will intersect some lines in one single point. Given a line l: px + qy + r = 0 , L : {p q ...
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82 views

Parabola Terminology

In Danish we call the two halves of a parabola that goes out to each side from the vertex branches like branches on a tree. Is there a name for them in English? Are they just called halves or maybe ...
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62 views

Equation of the ellipse for musical notation (the quarter note/crotchet and shorter)

Note heads are often represented with a slightly rotated ellipse, as shown here for instance (first image). Does anyone happen to know the equation of the ellipsis, and the rotation that's applied to ...
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203 views

Rotating a 3d-ellipse equation?

So I have very limited Linear Algebra knowledge, and I'm trying to program a computer graphics application in Android using OpenGL. I understand my design is not great, so if you have questions as to ...
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57 views

Is the following a conic section

All vectors are in $\mathbb{R}^3$ and only $\mathbf{r} = \left[ x; y; z \right]$ is unknown. My question is does the following system define a conic section in the $x-y$ plane and, if so, how can I ...
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71 views

Solving Kepler's Equation

I've been working on simulating orbits. I've found that, when solving Kepler's equation, $M = E - \varepsilon\sin{E}$, I'm unsure about the solution to use. For a true anomaly $< \pi$, using the ...
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46 views

When is the existence of rational points on an ellipse equivalent to the existence of integral points?

This question is a follow-up to my previous question. For what square-free values of $d$ is the following statement true? For all $n\geq 1$, the equation $x^2+dy^2=n$ has a rational solution if ...
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175 views

Singular Conics and Intersection of Line with a Conic

I've been working through Silverman and Tate's book Rational Points on Elliptic Curves. They use conic equations as an introduction to singular/nonsingular curves. I've reproduced the problem with my ...
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70 views

Finding point on ellipse given an arc length

Given a parametric representation of an ellipse: $$ x = a\cos t \\ y = b\sin t $$ Say I have a known point $P_0$ at $t = t_0$. Given also a known arc length $d$ on the ellipse: $$ d = \int_{t_0}^{...
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36 views

Extremal points relative to origin for an ellipsoid

Suppose I have an ellipsoid of the form $ax^2 + by^2 + az^2 - cxy -cyz = d$ How would I find the points nearest to, and furthest from, the origin?
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240 views

Finding equation for conic section given five points

Problem: Given the points $$(0,1),(0,-1),(2,0),(-2,0),(1,1)$$ find the equation for the conic section that passes through these points. My attempt: Using the general equation for a conic section, $$...
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85 views

Maximum product of lengths involving secant drawn to a parabola.

A chord is drawn from a point $P(1,t)$ to the parabola $y^2=4x$, which cuts the parabola at $A$ and $B$. If $PA\cdot PB=3|t|$, what is the maximum possible value of $|t|$? All I can infer is that the ...