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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6
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6answers
18k views

Find the area of largest rectangle that can be inscribed in an ellipse

The actual problem reads: Find the area of the largest rectangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$ I got as far as coming up with the equation ...
9
votes
6answers
5k views

How to find an ellipse , given 2 passing points and the tangents at them?

Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ...
21
votes
4answers
6k views

The locus of two perpendicular tangents to a given ellipse

For a given ellipse, find the locus of all points P for which the two tangents are perpendicular. I have a trigonometric proof that the locus is a circle, but I'd like a pure (synthetic) geometry ...
4
votes
3answers
3k views

Finding Intersection of an ellipse with another ellipse when both are rotated

Equation of first ellipse=> $$\dfrac {((x-xFirstEllipseCenterPoint)\cdot \cos(A)+(y-yFirstEllipseCenterPoint)\cdot \sin(A))^2}{(a_1^2)}+\dfrac{((x-xFirstEllipseCenterPoint)\cdot \sin(A)-(y-...
5
votes
2answers
2k views

Determining the angle degree of an arc in ellipse?

Is it possible to determine the angle in degree of an arc in ellipse by knowing the arc length, ellipse semi-major and semi-minor axis ? If I have an arc length at the first quarter of an ellipse and ...
9
votes
1answer
5k views

Intersection of conics using matrix representation

I came across a very interesting section of a wikipedia article on conics: http://en.wikipedia.org/wiki/Conic_section#Intersecting_two_conics I am trying to work out a couple of examples to add to ...
13
votes
3answers
27k views

How to determine the arc length of ellipse?

I want to determine the arc length of a ellipse. So what data should I know ? And what law should I use ? For example I have this ellipse on picture below: How can I determine the $d$ length of ...
7
votes
1answer
1k 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 ...
8
votes
4answers
1k views

How to find the center of an ellipse?

I have the following data:- I have two points ($P_1$, $P_2$) that lie somewhere on the ellipse's circumference. I know the angle ($\alpha$) that the major-axis subtends on x-axis. I have both the ...
7
votes
1answer
2k views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
13
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13answers
20k views

Derivation of the formula for the vertex of a parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $$y = a x^2 + b x + c$$ My teacher gave me the formula: $$x = -\frac{b}{2a}$$ as the ...
8
votes
3answers
18k views

Finding the angle of rotation of an ellipse from its general equation and the other way around

The general equation for an ellipse is $Ax^2+Bxy+Cy^2+D=0$. How do I find the angle of rotation, the dimensions, and the coordinates of the center of the ellipse from the general equation and vice ...
13
votes
1answer
6k views

Calculating Distance of a Point from an Ellipse Border

I'm thinking about using oriented ellipses to represent curves (dents/bumps etc.) in my physics engine, and have a few questions about working with them: What methods are there to finding the ...
5
votes
3answers
5k views

How to find an ellipse, given five points?

Is there a way to find the parameters $$A, B, \alpha, x_0, y_0$$ for the ellipse formula $$\frac{(x \cos\alpha+y\sin\alpha-x_0\cos\alpha-y_0\sin\alpha)^2}{A^2}+\frac{(-x \sin\alpha+y\cos\alpha+x_0\...
2
votes
1answer
387 views

Compute center, axes and rotation from equation of ellipse

Suppose I have the equation of an ellipse, in its implicit form $$Ax^2 + By^2 + Cxy + Dx + Ey + F = 0$$ For example the following: $$4.36\,x^2 + 2.89\,y^2 - 5.04\,xy + 30.8\,x - 0.6\,y + 81 = 0$$ ...
48
votes
5answers
4k views

Do “Parabolic Trigonometric Functions” exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) &...
18
votes
4answers
680 views

A geometric reason why the square of the focal length of a hyperbola is equal to the sum of the squares of the axes.

This may be a phenomenally stupid question, so apologies in advance. But when I teach conics, I show why $c^2=a^2-b^2$ for ellipses geometrically, just by drawing the obvious isosceles triangle from ...
5
votes
4answers
2k views

Area of an ellipse

An ellipse has equation : $$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F =0$$ Can you provide an optimum method to find it's area?
6
votes
0answers
1k 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 ...
5
votes
4answers
2k views

How to partition area of an ellipse into odd number of regions?

Is it possible to divide an ellipse into 3,5 or 7 etc. parts of equal area? If yes then how? Describe a circle around the ellipse and the circle of an equilateral triangle we construct. Projection ...
3
votes
1answer
1k views

Canonical form of conic section

I have $x^2+2xy-2y^2+x-4y=0$ and I have to find its canonical form, but I'm a little confused.. I'd like to understand very well what I have to do.. Can you help me, please? Thanks!
1
vote
4answers
785 views

Finding Eccentricity from the rotating ellipse formula

I see that from a normal ellipse formula, we can acquire the eccentricity via this formula here. However, for this formula (1): $A(x − h)^2 + B(x − h)(y − k) + C(y − k)^2 = 1$ When parameter $B ...
5
votes
1answer
505 views

Ellipses given focus and two points

I would like to find all ellipses which contain 2 given points and has one focus at origin (zero). All in 2D plane. There are several possible approaches but I'm not sure which is the best - both ...
0
votes
1answer
121 views

Matrix representations of parabola.

Continuing the epic quest on finding matrix representations from here: Representation of hyperbolas. with a last part, the only conic section left: the parabola. I will present one idea of how to ...
0
votes
2answers
599 views

A circle touches the parabola $y^2=4ax$ at P. It also passes through the focus S of the parabola and int…

Problem : A circle touches the parabola $y^2=4ax$ at P. It also passes through the focus S of the parabola and intersects its axis at Q. If angle SPQ is $\frac{\pi}{2}$ find the equation of the ...
24
votes
4answers
27k views

Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by ...
13
votes
1answer
1k views

The Ellipse Problem - finding an ellipse inside a triangle

The problem statement is as follows: A triangle is dissected into six smaller triangles by its angle bisectors. Prove that the intersections of the angle bisectors of each of these smaller triangles ...
6
votes
6answers
5k views

Plot $|z - i| + |z + i| = 16$ on the complex plane

Plot $|z - i| + |z + i| = 16$ on the complex plane Conceptually I can see what is going on. I am going to be drawing the set of points who's combine distance between $i$ and $-i = 16$, which will ...
12
votes
4answers
4k views

What Does Homogenisation Of An Equation Actually Mean?

For example, if we have a conic; ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 What does homogenising this equation with another line (say ax + by + c = 0 ) actually mean? As in, what are the graphical ...
3
votes
2answers
3k views

Determining the major/minor axes of an ellipse from general form

I'm implementing a system that uses a least squares algorithm to fit an ellipse to a set of data points. I've successfully managed to obtain approximate locations for the centre of the ellipse but I ...
3
votes
1answer
707 views

Enlarging an ellipses along normal direction

Given an ellipses, enlarge it along normal direction a fixed length say 1cm. Do we get another ellipses? If so, how to prove ?
8
votes
1answer
2k views

Formula for curve parallel to a parabola

I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
7
votes
1answer
38k views

Ellipse in polar coordinates

I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants $a$ and $b$ in this format - Where $2a$ is the total height of the ellipse and $2b$ ...
5
votes
3answers
4k views

Calculate the intersection points of two ellipses

I used the equations found on Paul Bourke's "Circles and spheres" page to calculate the intersection points of two circles: $P_3$ is what I'm trying to get, except now I want to do the same with ...
3
votes
1answer
418 views

Why are two definitions of ellipses equivalent?

In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum. When we speak of an ellipse ...
2
votes
1answer
2k views

Calculate Ellipse From Points?

How can I calculate an ellipse from a group of points ? Result: center point, x-radius, y-radius ? I'm not mathematician so I don't really know the best parameter style for ellipses. This ellipse ...
2
votes
3answers
317 views

How to construct the point of intersection of a line and a parabola whose focus and directrix are known?

I found this problem in Polya's "How to solve it". It goes as follows Using only a straight edge and a compass, construct the point(s) of intersection of a given line and a parabola whose focus ...
1
vote
2answers
2k views

How to convert the general form of ellipse equation to the standard form?

How to convert the general form of ellipse equation to the standard form? $$-x+2y+x^2+xy+y^2=0$$
0
votes
3answers
144 views

Convert this equation into the standard form of an ellipse

$$\frac{\left(\frac{xa^2}{a^2y^2+\ x^2}-p\right)^2}{a^2}+\left(\frac{ya^2}{a^2y^2+\ x^2}-q\right)^2=k^2$$ Could someone please convert this into standard form of equation $$\frac{\left(X-H\right)^2}{...
0
votes
2answers
64 views

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose center is $C$ be such that $CP$ is perpendicular to $CQ$

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose center is $C$ be such that $CP$ is perpendicular to $CQ$ and $a<b$,then prove that $\frac{1}{CP^2}+\frac{1}{CQ^...
6
votes
2answers
1k views

Relationship between ellipsoid radii and eigenvalues

I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup. I went through a theorem in the book stating the ...
4
votes
2answers
389 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 ...
4
votes
3answers
709 views

Is it possible to find out $x^2$ parabola and function from 3 given points?

I am programming a ball falling down from a cliff and bouncing back. The physics can be ignored and I want to use a simple $y = ax^2$ parabola to draw the falling ball. I have given two points, the ...
2
votes
1answer
442 views

Given area of sector and a starting angle from focus of an ellipse, finding angle needed to get area.

Problem Background: I'm trying to make a rough simulation of Kepler's second law (equal areas over equal time) and to do this I've divided the area of the ellipse into some number of pieces. I want ...
2
votes
3answers
2k views

Proving a property of an ellipse and a tangent line of the ellipse

Suppose that there is line $l$ that is tangent to an ellipse $A$ at point $\,P\,$. The ellipse has the foci $F'$ and $F$. One then creates two lines - each from each focus to the tangency point $\,...
1
vote
2answers
58 views

Prove that the directrix-focus and focus-focus definitions are equivalent

(NOTE: This is my attempt at answering this question and this question, but I rewrote it in order to make it easier for me to solve. Also, I've made a YouTube video explaining this whole proof. Note ...
1
vote
1answer
330 views

Homography between ellipses

This is a spin-off from a comment on Stack Overflow. How can I find a homography between two ellipses in the plane?
0
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1answer
471 views

How to find centre,vertics,foci,focal radii,letus rectum… when exists of a general quadratic equation in x and y

Is there a generalized way( a particular conic section of any shape,for instance an ellipse without determining its major/minor axis) to find the centre,vertics,focus,focal radii,letus rectum,...
3
votes
1answer
551 views

Locus of vertex of parabolas through three points

Consider all parabolas through three given points $A,B,C$. What is the locus of the vertex? Qualitatively it traces three branches with the lines through the mid-points of the sides of the triangle $...
3
votes
1answer
220 views

Does using an ellipse as a template still produce an ellipse?

Suppose I have a (physical) template, consisting of a piece of stiff sheet plastic with a hole cut in the middle. Suppose the hole is in the shape of an ellipse, say, 8 x 12 inches. Suppose I then ...