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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637 views

Definition of an ellipsoid based on its focal points

I have a question concerning the formulation of an (3D) ellipsoid. The most common definition for an ellipsoid seems to be: $E = \{ x=\left( x_1, \dots x_n \right)^T \in R^n: \sum_{i=1}^n \left( ...
6
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84 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 ...
6
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908 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 ...
6
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885 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 ...
5
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66 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 ...
5
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66 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 ...
5
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96 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. $$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$ $a$ ...
4
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113 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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109 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 ...
4
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74 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: ...
3
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48 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 ...
3
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91 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 ...
3
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31 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". ...
3
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40 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 ...
3
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53 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 ...
3
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50 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 ...
3
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99 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 ...
3
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458 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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97 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 ...
3
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92 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 ...
3
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0answers
587 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 ...
3
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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, ...
3
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0answers
241 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 $l$ ...
2
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54 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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37 views

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 ...
2
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44 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 ...
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25 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 ...
2
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45 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 $\left|z-v\right|= \left|vz\right|$ ...
2
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0answers
45 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 ...
2
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45 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 ...
2
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0answers
67 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 ...
2
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0answers
148 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 ...
2
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0answers
54 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 ...
2
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0answers
66 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 ...
2
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0answers
119 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 ...
2
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0answers
34 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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188 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, ...
2
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0answers
81 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 ...
2
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0answers
67 views

Find the number of points on ellipse $\frac{x^2}{50}+\frac{y^2}{20}=1$ from which pair of perpendicular…

Find the number of points on ellipse $\frac{x^2}{50}+\frac{y^2}{20}=1$ from which pair of perpendicular tangents are drawn to ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ Normal from a point ...
2
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0answers
147 views

Can one prove Brianchon's theorem using Ceva's theorem?

Can I prove Brianchon's theorem using Ceva's? I am also wondering if parabola and hyperbola can be inscribed in a hexagon?
2
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0answers
85 views

Locus of centre of circle in Lambert theorem

A beautiful theorem, when three tangents to a parabola form a triangle,the focus of the parabola lies on the circumcircle of the triangle. But what is the locus of the centre of the circumcircle of ...
2
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0answers
35 views

Is there a Focal Point/Area/Line of a Parabola for not perpendicular Lines

I'm not sure if this is mathematical enough for this forum, since it's my first post, but please don't be too harsh! So my question is: If the incoming lines of a Parabola come in perpendicular to ...
2
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0answers
33 views

Number of ellipses to uniquely define a co-centered circumscribing ellipse

I have a bit of a tricky problem that has come up in my engineering research, but I haven't quite got the brains to figure it out, though I've gotten pretty far. Suppose that there is an unknown ...
2
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0answers
62 views

An Easier way to solve simple equations of this type

Im currently working with ellipses and I've been given two points on a ellipse whose major axis is along the x-axis, $(4,3)$ and $(-1,4)$. The question asks me to find the length of the major and ...
2
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0answers
90 views

Best fit circle to “planetary” elliptical orbit?

I considered posting this to astronomy.stackexchange.com, but I've bugged them enough for today... Let $p(t)$ be a parametric function that traverses an ellipse such that it sweeps out equal ...
2
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0answers
583 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 ...
2
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0answers
203 views

Finding Quadrants of ellipse from ellipsoid of a Conic section

This is my first post here, hope I won't be giving tough time for you. I will be giving bit non relevant information here to describe my problem as it may help understand the problem better. I will ...
2
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0answers
35 views

find inscribed ellipses in quads

I know in an convex quads,there are a family of inscribed ellipses. what I want to konw is when the semi-axis 'a' and four vertexs are given,how to determine the rotation angle.there may be three ...
2
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0answers
281 views

What are defintions of ellipticity?

Ellipticity seems to have many definitions. So far, I'm aware of three. Are there other definitions of ellipticity that you know of, and if so, where did you encounter them? The three definitions ...
2
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0answers
447 views

Dome with a parabolic shape?

Here is the question: The dome over a town hall has a parabolic shape. The dome measures 48 m across and rises 12 m at its centre. a) Determine the quadratic equation that models the shape of the ...