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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18
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441 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
417 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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0answers
83 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. $$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$ $a$ ...
5
votes
0answers
354 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 ...
4
votes
0answers
130 views

Focus-Focus Definition for a Parabola

I am looking at the focus-focus definitions of the conics, i.e. defining them as the locus of points with the property that some function of the distance from the point to two foci is a constant. ...
4
votes
0answers
340 views

Calculating equidistant points around an ellipse arc

As an extension to this question on equiangular fisheye distortion, how can I calculate equidistant points around an ellipse (or 1/4 segment of) given it's aspect ratio? When it's circular, I can use ...
3
votes
0answers
50 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
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0answers
58 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
91 views

History behind the choice of letters $h$ and $k$ for the vertex of a parabola?

After failing to find a historical explanation for usage of letters $h$ and $k$ for the vertex of a parabola in most relatively recent textbooks in anglosphere, I turn to math.SE. Is there any ...
3
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0answers
108 views

General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
3
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0answers
55 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
246 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
189 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
29 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
votes
0answers
20 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
54 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
61 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
248 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
30 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
33 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
124 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
150 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
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0answers
130 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
113 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
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0answers
25 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, ...
2
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0answers
431 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 ...
2
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0answers
133 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
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0answers
326 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
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0answers
322 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
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0answers
21 views

find an ellipsoid given its intersection with axes and knowing the lengths of its principal axes

My question is about ellipsoids. I have an ellipsoid in 3D centered at zero so it has an equation: $x^T U \Sigma^2 U^T x = 1$ I know the lengths of it's principal axes (therefore the $\Sigma$ ...
1
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0answers
16 views

Given a pair of conics, construct (synthetically) their shared tangent lines

There are various well-known ways to construct the common tangents to a pair of circles; this is an easy one. I also just learned that we can use Pascal's Theorem to construct a tangent to a conic at ...
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0answers
59 views

Intersection between sphere and ellipsoid

I am failing since two days to compute and to plot the intersection of an ellipsoid in parametric notation ...
1
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0answers
31 views

Problem on co-ordinate geometry

Suppose the circle with equation $x^2 + y^2 + 2fx + 2gy + c = 0$ cuts the parabola $y^2 = 4ax$, ($a > 0$) at four distinct points. If d denotes the sum of ordinates of these four points, then find ...
1
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0answers
37 views

Finding the distance from a parabola (ballistic trajectory) to a point (for use in collision detection)

I need to have some form of collision detection / prevention for an object moving along a ballistic trajectory and a second stationary object on the same plane plane. The ballistic trajectory is ...
1
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0answers
99 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) ...
1
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0answers
18 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 ...
1
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0answers
39 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 ...
1
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0answers
71 views

Finding a positive definite matrix to satisfy the general equation of an ellipse

I am trying to find a matrix A such that $(1)$ can be written as $v^TAv=1$ where $v=(x, y)^T$. $(1)$: $$\left(\frac{x}{a_1}\right)^2 + \left(\frac{y}{a_2}\right)^2 - ...
1
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0answers
89 views

How to see and proof that the hyperbola as a constant difference of distances holds for $\frac{1}{x}$?

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, which is the distance ...
1
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0answers
83 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 ...
1
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0answers
36 views

Equation of ellipsoid given foci and two semi-axes

How does one find the equation of an ellipsoid given two foci, $(a,b,c)$ and $(d,e,f)$, and one semi-axis $l$? $c$ may not be equal to $f$.
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0answers
41 views

How To Mathematically Slice A Parabolic Trough at an Angle?

Given a Parabolic Trough is defined as $y=x^2$ and extending infinitely in the z direction. How may I find the equation of the curve obtained through slicing the parabolic trough using a plane through ...
1
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0answers
91 views

Please help in solving $ax^2 + bxy + cx + dy + e$ = 0

Sometime back when trying to work out how to solve $ax^2 - by^2 + cx - dy + e = 0$ I learned that the way to solve such forms is to 'square the terms' and give it the form $A^2 - B^2 - E = 0$, $A = ax ...
1
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0answers
196 views

Find angle given arc length and radius

I've got a function, $r(\theta)$, of the radius of an ellipse relative to one focus of the ellipse: $$ r(\theta) = \frac{l}{1 - e\cos \theta} $$ where $e$ is the eccentricity and $l$ is the semi-latus ...
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0answers
91 views

Locus Problem .

Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabola is $y^2(2x+a)=a(3x+a)^2$
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47 views

Why loci of conic sections are defined in the way they are?

I understand how conic sections are produced i.e. when a plane cuts a double nappe right circular cone at different angles, we get different types of conic sections like parabola, ellipse etc. But I ...
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0answers
98 views

Plotting an elliptical arc given 3 points, radius ratio and angle

I'm trying to plot an elliptical arc. I know the starting point $P_1$, ending point $P_2$ and a control point $P_3$. I'm also given the ratio of radii $a/b$ and the angle $\theta$ of the ellipse. As ...
1
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0answers
57 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 ...
1
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0answers
75 views

Conic Sections - Points on a cone

On page 80 of Spivak's Calculus, 4th Edition, he writes: One of the simplest subsets of this three-dimensional space is the (infinite) cone illustrated in Figure 2; this cone may be produced by ...
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0answers
70 views

Circle Geometry and Conic Section textbook

I seek a textbook for good conic section and circle geometry questions. Slightly above introductory level. - slightly. But I wouldn't mind introductory level questions to consolidate my knowledge. I ...