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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Parabola equation expressed after x

Sorry for the bad title, as English is not my main language. Let me explain better what I mean. I have this equation of parabola: $y = x^2 + 4x $ What I want to do is get the $x$ in one side and ...
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1answer
37 views

Polar correlation and conics in $\Bbb RP^2$

I'm stuck on a small detail in Proposition 1.2.8 in Geiges' Introduction to Contact Topology. Let $C$ be a conic in $\mathbb{R}P^2$ given by $q^tAq=0$, where $A$ is a nonsingular, symmetric 3x3 ...
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39 views

Find the parametric equation of the following parabola?

It doesn't give me $2$ equations this time just $1$ and I have no clue what to do; $y^2 = 4x$ ANSWER IN BOOK: $x = t^2, y = 2t$
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131 views

Ellipsoidal Decomposition: Finding ellipsoids whose sum contains a given ellipsoid

We have a known ellipsoid $E\left(q,Q\right)$ in a 2D space. $q$ represents the center of the ellipsoid and $Q^{-1}$ is the weight matrix. The general equation of the ellipsoid is given as: ...
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1answer
67 views

Parabola - equation from three points

Question: Find the equation of the parabola whose axis is parallel to the y-axis and which passes through the points (0,4) (1,9) and (-2,6) Well as the parabola has its axis parallel to the y-axis ...
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2answers
252 views

Relation of ellipse semi-axes with rotation angle and projection length

In the following setup, assume $w$ (length of the projection of the ellipse) and $\theta$ (the rotation angle) are known. I want to know what equation(s) do I have here that helps me to derive the ...
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1answer
86 views

finding out wheter point is inside ellipse

I'm working on a way to determine if given point is "inside" given ellipse, the problem is I've already forgotten all the related mathematics and don't have time to relearn it and find a way to do it. ...
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3answers
326 views

How to find the outermost points in an ellipse?

If an ellipse is given in the form: $$ A(x − h)^2+ B(x − h)(y − k) + C(y − k)^2 = 1 $$ (where A, B, C, h, and k are given) What would be the simplest way of finding the outermost points, by which I ...
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102 views

Conic section: What is the coordinate matrix of its bilinear form?

Given is the conic section $x^2 + xy + y^2 + 2x +3y - 3 = 0$. I need to find the coordinate matrix $M_\beta(s)$ of the bilinear form $s: \mathbb{R}^2 \times \mathbb{R}^2 -> \mathbb{R}$. I read ...
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91 views

ellipse and segment intersection

I have a rotated ellipse, not centered at the origin, defined by x,y,a,b and angle. Then I have a segment defined by two points x1,y1 and x2,y2 Is there a quick way to find the intersection points?
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117 views

what are some applications of modern algebraic geometry to conic sections?

The simplest non-trivial example of an algebraic curve is probably a conic section (ellipse, parabola and hyperbola). At the same time, we also know that development in advanced theory can provide new ...
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39 views

Conics Confusion

I'm currently reading through a document about the ellipse. I've attached the provided image and working out. From here, it is easy enough to show that $|OP|\sin\gamma=|FP|\sin\alpha$ using say the ...
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1answer
46 views

What is the equation of hyperbola

Given that the equation of asymptotes to the hyperbola be: $y=\pm\frac{3x}{2}$ and $b=4$ How to find the equation of hyperbola? I know that asymtotes have the equation $y=\pm\frac{bx}{a}$, ...
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3answers
213 views

The line is tangent to a parabola

The line $y = 4x-7$ is tangent to a parabola that has a $y$-intercept of $-3$ and the line $x=\frac{1}{2}$ as its axis of symmetry. Find the equation of the parabola. I really need help solving this ...
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1answer
56 views

Locus on a parabola

How could I find the locus of M as P moves of the parabola. P is.(2at, at^2) . M is the midpoint of the x and y intercepts of the normal through P. So far I was able to find the quation of the normal ...
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2answers
64 views

Degenerate conics

I was studying about the discriminant of a conic and got to the case where it equals 0. The book I'm referring to says that such a case means that the equation represents a parabola, a pair of ...
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53 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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176 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
76 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
44 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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$P$ 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
225 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
303 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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56 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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143 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
349 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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112 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
109 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
286 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
111 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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253 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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145 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 ...
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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
67 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
75 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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179 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
196 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
474 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
452 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
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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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347 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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3k 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
55 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
115 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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145 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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54 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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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 $\Sigma$ ...