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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56 views

Catenary and parabola minimum comparison

Do the catenary and a parabola that approximates the catenary, have the same minimum (maximum sag)? IF plotted, it looks to me they do, and that they only difer somewhere on the "slope". (sorry for ...
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2answers
195 views

Ellipse as projection of a disk - function describing ellipse diameter with disk rotation?

Say I have got a disk of radius $r$ and a plane $p$ in $3D$ space. The disk is "aligned" to $p$ and lies at an arbitrary distance, so that its orthogonal projection on $p$ is an identical disk of ...
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3answers
77 views

Solving $\frac{\mathrm d^2\mathbf{q}}{\mathrm dt^2} = -\frac{\mathbf{q}}{|\mathbf{q}|^3}$

I am reading a set of course notes and it has this example of a system of differential equations given by $$\frac{\mathrm d^2\bf{q}}{\mathrm dt^2} = -\frac{\bf{q}}{|\bf{q}|^3}$$ All it says is that ...
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1answer
45 views

Determining if two general conic sections are tangent to each other

Given two conics in general form $A_ix^2 + B_ixy + C_iy^2 + D_ix + E_iy + F_i = 0$ for $i = 1, 2$, I want to determine if they are tangent to one another, and I'm looking for a method that wouldn't be ...
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1answer
26 views

$P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ on the line that bisects $\angle F_1PF_2$. Prove $|PF_1-PF_2|>|QF_1-QF_2|$.

$\require{cancel}$ Sorry for the grammatical mistake in the title; it was needed to keep the title under 150 characters. $P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ is ...
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2answers
229 views

Ellipse definition

Spivak defines an ellipse as the set of points, the sum of whose distances from two fixed points is a constant. He takes these two points to be $(-c,0)$ and $(c,0)$ and the sum of the distances to be $...
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1answer
315 views

Finding equation of directrix when the parametric equation of parabola is given.

If the parametric equation of the parabola is $( x = t^2 + 1 , y = 2t + 1 )$, then find the equation of the directrix. This was the question in my last test in which I got stuck and wasted much of my ...
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1answer
60 views

finding $\lambda$ when equation of parabola is given

If the equation $\lambda x^2 + 4xy + y^2 + \lambda x + 3y + 2 = 0$ represents a parabola. Then find $\lambda$. I got stuck in this question while solving parabola. Is here anybody who can help me ???...
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1answer
150 views

Prove that the locus is a parabola

The point P(x,y) moves in XY plane such as that its distance from a fixed point (0,-1) is equal to its distance from the line Y=1. Prove that the locus is a parabola. Find it's focus, directrix, ...
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1answer
362 views

Finding the asymptotes of a general hyperbola

I'm looking to find the asymptotes of a general hyperbola in $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ form, assuming I know the center of the hyperbola $(h, k)$. I came up with a solution, but it's too ...
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1answer
47 views

General “Conics” of higher degrees?

A general conic has the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. I understand that there are certain properties of this equation that make it special and allow us to classify the different types of ...
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0answers
117 views

Non linear least square ellipse fitting

I am trying to find a Non linear leasts squares ellipse fit for a set of 100 data points data points $(x,y)$. Now i have found the values of $A,B,C,D,E,F$ according to the conical equation of the ...
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1answer
35 views

Movable “light” in 3d enviroment

A light-emitting object is suspended in a 3 dimensional environment at a known position (eg: X=0, Y=0, Z=10). The object emits light with a certain beam pattern; it is not omnidirectional. The ...
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2answers
113 views

“Conic sections” that are really just two straight lines

My teacher was teaching co-ordinate geometry and today he said that the following equation will always represent a conic section:$$ax^2+by^2+2hxy+2gx+2fy+c=0$$ Then he said that if the determinant of ...
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1answer
302 views

Projective and affine conic classification

I have a doubt on the classification of non-degenerate conics (parabola, ellipse, hyperbola) in projective geometry (my textbook is "Multiple View Geometry in Computer Vision", which, as the title ...
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1answer
116 views

Conic Sections Parameter set constraints

Given the general equation $Ax^2 + Cy^2 + Dx + Ey + F = 0$, what constraints on the set $\{A,C,D,E,F\}$ will apply if the equation represents a (a) parabola? (b) ellipse? (c) hyperbola? Firstly, I ...
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3answers
262 views

What is the cone of the conic section?

Given the general (real valued) equation of a conic section: $$ A x^2 + B xy + C y^2 + D x + E y + F = 0 $$ Then what is the circular cone associated with it ? Is it unique ? And is there a way to ...
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3answers
44 views

How to go from this equation to the equation of an hyperbola

I've seen that $x*y=1$ graphs an hyperbola, but I am struggling to get that equation to the form $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$. How can I do this? Ultimately, what I want is to be able to ...
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0answers
147 views

Axis aligned and non aligned ellipses and semi definite programming

Let's say I have a equation $$X_1^T \Omega X_1 =1 $$ $X_1$ is a $2\times 1$ matrix. $\Omega$ is a $2\times 2$ matrix. This defines an ellipse. $\Omega$ is a positive, semi definite, symmetric matrix....
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1answer
39 views

Edited parabola question

I am confused with the equation of a parabola. My teacher told me that it is in the form $$\text{(axis of parabola)}^2=4\text{(vertex of parabola)}$$ I feel that $\text{(the axis on which the vertex ...
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2answers
70 views

Equation of parabola confusion

I am having a confusion regarding the equation of a parabola. My teacher told me that it is in the form (axis of parabola)^2=4(vertex tangent). I feel that (vertex tangent)^2 should be 4(axis of ...
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1answer
81 views

Finding the area of an equilateral triangle on an ellipse

The question is as follows: Let $E$ be an ellipse with major axis length $4$ and minor axis length $2$. Inscribed an equilateral triangle $ABC$ in $E$ such that $A$ lies on the minor axis and $BC$ is ...
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0answers
200 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?
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1answer
221 views

Why does the focus point distances of an ellipse sum up to the length of the major axis diameter

Why does the distances from the focus points of an ellipse to arbitrary point in the ellipse sum up to the length of the diameter of the ellipse in the major axis? In other words, how to prove: I ...
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1answer
499 views

Properties of the positive definite Hessian matrix of a convex function

I'm reading about nonlinear programming and I'm having trouble understanding the cool properties that a positive definite Hessian matrix $Q$ of $n$-dimensional function $f: \mathbf{R}^n\rightarrow \...
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4answers
487 views

Find depth of a half-filled parabolic cross-section

Given a cross-section of an object that is parabolic in shape, how do you find the depth of the object when it is "half full". A full example given in an exam: A long trough whose cross-section ...
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2answers
92 views

How to determine the point at a set length along a given function (parabola)?

Given a specific function, a parabola in this instance, I can calculate the length of a segment using integrals to sum infinite right angled triangles hypotenuse lengths. My question is, can I reverse ...
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1answer
389 views

Minimal number of points to define a rotated ellipse?

What is the minimal number of points $N$ to uniquely define the semi-major axis $a$, the semi-minor axis $b$ and the rotation angle $\omega$ of an ellipse whose the center is known/fixed (this is ...
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2answers
4k views

Locus of a point - sum of distance from two points

Find the locus of the point $P$ such that the sum of its distances from $(0,2)$ and $(0,-2)$ is $6$. What I did: I tried using the distance formula, but I think that's too much of a task. There has ...
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2answers
56 views

equation for parabola --> Equation for parabolic basin

I have a parabolic basin which i am trying to find the equation for so I can reproduce it. I have taken $3$ points along one line of it to find the equation of the parabola, and I'm wondering if there ...
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1answer
142 views

Equation for the length of a chord parallel to either the minor or major axis in an ellipse

I am looking for a way to compute the length of any chord parallel to the minor (or major) axis of an ellipse. In all cases I know the lengths of both axes, and the distance between the chord and axis ...
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1answer
153 views

Best fit circular arc to an elliptical arc?

Is there a standard procedure or algorithm for finding the best fit circular arc to an elliptical arc ? Where the ellipse arc is: symmetrical about the minor axis, subtending $[+\theta, -\theta]$ ...
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1answer
62 views

intersection of cone axis with plane

So when the plane intersect the cone, the intersection is a conic. Is (or when is) the axis (of the cone) intersection with the plane the focus of the conic?
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48 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 $\Sigma$ ...
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1answer
105 views

Help with finding the arithmetic mean of all the radii from the center to the edge of an ellipse?

So far I approached this problem computationally, I decided to take all the radii add them up, by distance formula, then divide by the number of radaii. To make the distribution even, I rotated the ...
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2answers
229 views

How to find the determinant of this matrix

I'd like to find the determinant of following matrix $$ \begin{pmatrix} {x_1}^2 & x_1y_1 & {y_1}^2 & x_1 & y_1 \\ {x_2}^2 & x_2y_2 & {y_2}^...
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1answer
353 views

How to determine if two ellipse have at least one intersection point

All of the question are in sequence and related. 1.Given 2 ellipse with the position x1,y1, x2,y2 and the radius a1,b1, a2,b2, construct an equation to determine if both of them has at least one ...
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2answers
74 views

Determine whether the intersection of surfaces is a parabola

Let $C$ be the curve of intersection between the cone $z=\sqrt{x^2 + y^2}$ and the plane $z=1-x$. Is $C$ a parabola? I can see that letting $x=t$ we have parametric equations $x=t\\y=\pm\sqrt{1-2t}\\...
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1answer
150 views

Why $b^2-4ac$ as determinant?

I am curious why $b^2-4ac$ is used as a determinant of a conic section? Like why this specific expression is chosen, why the value is always greater, lesser or equal to zero for hyperbola, ellipse ...
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1answer
166 views

Central angle of an ellipse

If I have an ellipse centered at the origin and know the length of $a$ and $b$ and was given the length of an arc, how can I find the angle that is between the two radius from the center of the ...
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1answer
749 views

Equation of parabola, tangent at vertex [closed]

Two tangents on a parabola are $x-y=0$ and $x+y=0$. If $(2,3)$ is the focus of the parabola, then find the equation of tangent at the vertex. Thanks. My thoughts: Can't figure out anything :(
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1answer
44 views

Area of an ellipse proportional to integral of cross-ellipse distances?

I am curious if the area of an ellipse can be shown to be proportional to the integral of all cross-ellipse distances. Before I define cross-ellipse distance, I will give a motivating example from a ...
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2answers
476 views

Find normal to ellipse through arbitrary points

I want to find the normal to ellipse through an arbitrary point. There is an array of points located arround a given ellipse (but not on ellipse curve). What I want to find is the normal of each of ...
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3answers
54 views

Find the point on a parabola that is closest to a given point

As you can see I found the equation but I don't know how to find the points. As far as I tried was $(7, 49)$ but it was wrong.
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1answer
188 views

Find intersections of two ellipses who share one fixed point

Given two ellipses $e_1$ and $e_2$ with $$ e_1 = \{x: \lVert{x - F_1}\rVert + \lVert{x - F_2}\rVert = R \} $$ $$ e_2 = \{ x : \lVert{x - F_1}\rVert + \lVert{x - F_3}\rVert = R \} $$ where $F_1$ is ...
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1answer
492 views

Show that the ellipse and the hyperbola are convex

In Spivak's chapter on differentiation, he asks the reader to prove that the tangent line to an ellipse or a hyperbola intersect the figure at exactly one point. How is this done most elegantly? I ...
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1answer
248 views

how to find the foci, directrix, center of a polar conic section. ($r=\frac{4}{5-4sin\theta} $)

I've been trying to figure this out for a bit and haven't found an answer. the equation is this: $r=\frac{4}{5-4sin\theta} $ I know I need to match this up to a conic graph so I divide top and ...
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2answers
115 views

Is equation for ellipse in polar coordinates correct?

Wikipedia gives the following equation for the conic sections in the polar coordinate system: $r = \frac{l}{1+e\cos\varphi}$. According to the article on conic sections, in case of an ellipse $e = \...
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1answer
751 views

Equation of normal to an ellipse

Show that the equation of the normal at the point $x = a\cos(t)$, $y = b\sin(t)$ of the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ is $$\frac{2a^2 - b^2}{a}$$ Hi, I am not sure how to ...
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1answer
70 views

The image of a conic section under the $z^2$ map

My question in short: In some cases, the image of a conic section under the $z^2$ map is still a conic section. Is there an elegant argument to show that? Let $\Gamma$ be a conic section in the xy-...