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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20
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606 views

How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
9
votes
0answers
535 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( ...
5
votes
0answers
579 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 ...
5
votes
0answers
88 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
votes
0answers
65 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 ...
4
votes
0answers
85 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 ...
3
votes
0answers
26 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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0answers
28 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
votes
0answers
49 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
votes
0answers
280 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) ...
3
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0answers
67 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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0answers
72 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
votes
0answers
70 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
votes
0answers
361 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
votes
0answers
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
votes
0answers
621 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 ...
3
votes
0answers
225 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
votes
0answers
19 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
32 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?
2
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47 views

Partial Integral of an ellipse

this is my first question on stack exchange so please bear with me. I am trying to generate a synthetic image of an ellipse in Matlab where each pixel is shaded according to how much of that pixel ...
2
votes
0answers
66 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
41 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
86 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
59 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
30 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
votes
0answers
59 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
votes
0answers
77 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
votes
0answers
422 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
votes
0answers
32 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
votes
0answers
48 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
votes
0answers
296 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 ...
2
votes
0answers
65 views

Path traced by a water drop on an ellipsoid

If we have a smooth football in the shape of an ellipsoid, and that water runs down on its sides, can we trace the path of a water drop on it? For a sphere it seems easy because the force tangential ...
2
votes
0answers
279 views

How to calculate a PHI-ellipse defined by 3 points and its width/length ratio

in the field of technical analysis for stock markets, the usage of so-called Phi-Ellipses is getting popular. One important property of this ellipses is its constant length/width ratio (e.g. 1.618). ...
2
votes
0answers
151 views

Equation of an intersection of two cones when the intersection is an ellipse

The two cones with vertex $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ and generating angle of two cones is $\alpha$ given. I need to write the equation of the intersection of two cones ...
2
votes
0answers
159 views

How to find a point in an ellipse given the angle

I found a couple of formulas but I can't transform them in code. From the answer in Calculating a Point that lies on an Ellipse given an Angle , for instance, I get to: ...
2
votes
0answers
173 views

My solution is right and the book is wrong (parabolas) or did I misunderstand it?

Find the equation of the parabola with the vertex at the origin; directrix 2x = 3 So what I did is, find the equation of the directrix $$x = \frac{3}{2}$$ and then because its the directrix, the ...
2
votes
0answers
396 views

How to find the center of an (scaled) ellipse?

This question is an extension of How to find the center of an ellipse?. The solution there works well, but in Javascript the floating point calculations are not that accurate. The workaround is to ...
2
votes
0answers
423 views

Best fit hyperbola with specific constraint

Given a set of points $(x_i, y_i)$ in $\mathbb{R}^2$, I can find the best fit hyperbola in the least squares sense by using the method given here. But, is there a way to constrict the hyperbola to ...
1
vote
0answers
20 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 = ...
1
vote
0answers
35 views

Conic property pedal length and polar/tangent rotations

From standard Newtonian form for focal conics $ p/r = ( 1- \epsilon \cos \theta), $ I obtained by differentiating with respect to arc: $$ \dfrac{FN}{p} = \dfrac{\cos \phi}{\sin \theta}. $$ ...
1
vote
0answers
28 views

Can I generate a skewed ellipse tangent to two points?

I'm trying to write a python script to generate a trailing edge (TE) for an airfoil with no TE. Basically want to make a smooth round-off nose profile to the right, the closure line should come out ...
1
vote
0answers
15 views

Find the distance between two points on a curve (between two IMU sensors)

I have an elastic belt with six sensors on it. Each sensor contains a gyroscope and an accelerometer. I know the problem of finding the distance between two points on a curved surface has been asked ...
1
vote
0answers
41 views

Find the parabola given two endpoints and the midpoint along the curve

It has arbitrary orientation in 2D. I thought to equate the formulas for the arc lengths (s) between the midpoint and each end point from ...
1
vote
0answers
15 views

How to find foci from size and center?

How can I calculate the foci of an ellipse, given its width, height, and center? Everything I've found uses an equation instead of parameters.
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0answers
55 views

Proof 5 points determine a conic without projective geometry

So I'm trying to prove that any five points, of which no 3 are colinear, there is a single conic that passes through al of them. I don't want to use projective geometry but rather, only analytic ...
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vote
0answers
37 views

What is the intersection between $x + y - z = -2$ and $z^2 = x^2 + y^2$

I got the answer as $4x + 4y + 2xy + 4 = 0$ by substituting $z = x + y + 2$ into the second equation, but I feel as this is wrong since I am missing $z$ in the function. How do I approach this ...
1
vote
0answers
64 views

Finding eccentricity, directrix, foci of diagonal ellipse by rotating it

A problem I'm working on has the equation for a diagonal ellipse $$5x^2 + 5y^2 - 6xy - 8 = 0$$ which can be rotated 45 degrees to get the vertical ellipse $$\frac{x^2}{1}+\frac{y^2}{2^2} = 1$$ The ...
1
vote
0answers
58 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, ...
1
vote
0answers
54 views

Finding the equation of a rational function or a conic section given three points

I have a rational equation derived from 2 points, $(2, 2)$ and $(10, 10)$. Solving for the rational equation gives the equation $$y = \frac{20}{12-x}.$$ What I want to happen right now is that given ...
1
vote
0answers
24 views

Sections of cones in higher dimensions

Everybody knows that when a plane intersects a cone at different angles and positions, we get conic sections. But, I wanted to know that if the same was possible in higher dimensions. If we take the 4 ...