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
16 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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1answer
26 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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1answer
7 views

Calculating rotated half ellipse axis aligned bounding box

I want to find the height and width of a rotated half ellipse axis aligned bounding box. For that, I have the minor and major axes of the ellipse and its rotation angle. I have no idea how to form the ...
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0answers
38 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. We would like to allocate this space to two ellipsoids $X\left(X_q,X_Q\right)$ and $Y\left(Y_q,Y_Q\right)$. The area of these ellipsoids ...
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1answer
22 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
55 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
20 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
37 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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Finding Equation of Hyperbola with given parameters

The Question Goes As:- Q.) Find the equation of Hyperbola whose eccentricity is at the origin, transverse axis along $x$-axis, length of conjugate axis is $5$ and passing through the points $(1,-2)$. ...
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53 views
+100

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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29 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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1answer
33 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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1answer
31 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
29 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
40 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
38 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
26 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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0answers
9 views

How to draw the reciprocal polar of one ellipse w.r.t. another one

I know that the reciprocal polar of one ellipse w.r.t. another one is a hyperbola obtained as the envelope of all polars from points lying in one of the ellipses w.r.t. the other. My question is: how ...
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0answers
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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
21 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
59 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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How to draw a border at a specific distance from a cylinder outline

I have a small cylinder (Cylinder A) with its minor radius of A. Minor radius measures the minor radius of the cylinder ellipse. I need to draw a border at a distance of X from the border of ...
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1answer
22 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
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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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1answer
27 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
39 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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2answers
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Equation of parabola with focus and tangent [closed]

What is the equation of parabola whose focus is $(1,1)$ and the tangent at vertex is $x+y=1$?
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43 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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Curvature of track of focal point

Prove: A ellipse pure roll on a line l,show that the centripetal acceleration of F1 is 0 if F1F2∥l.
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1answer
62 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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0answers
28 views

Predicting Bending of Plywood (Conical Shapes)

I'm building a NASCAR-style banked track for 1/27 scale RC cars, and I'm trying to predict the bends I need to cut to produce the banked track I want. I've already built some simple banked pieces, ...
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1answer
21 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
22 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
29 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
70 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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How to find the tangent vector for intersection of two cones?

Let two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given vertex of two cones and the generating angle of two cones are $m$. Intersection of two cones could be either an ellipse or ...
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1answer
34 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
42 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
110 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
34 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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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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1answer
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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
53 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
40 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
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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
27 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
53 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
59 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
56 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 ...