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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5
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1answer
380 views

What is the path equation that is created with the middle point of a fixed length line segment that touching both ends to an ellipse.

Ellipse equation is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ and the length of line segment is $2k$, if we move the line segment all around of the ellipse while touching both ends to the ellipse. What is ...
0
votes
2answers
603 views

Convert ellipse parameter from General parametric form to General polar form

I am facing problem to convert ellipse standard parameters. Everything I say here is refer to http://en.wikipedia.org/wiki/Ellipse I know what are the General parametric form parameter . Lets call ...
4
votes
2answers
180 views

How to decide that a curve segment is not an ellipse line segment?

Let me ask a question , given any short curve segment , how can you decide that it is not an ellipse line segment by a finite calculations? Thank you in advance.
1
vote
1answer
105 views

Are two ellipse arcs always almost identical if they have the same end points and the same center of ellipses?

Edit : This question is better to be ignored until the following related question will be discussed enough. This question relates to I know "almost identical " is not mathematics. But if you have ...
10
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6answers
3k 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 ...
7
votes
1answer
412 views

Conics in $\mathbb{A}^2$; Hartshorne, Exercise 3.1

I'm trying to solve Exercise 3.1 in Hartshorne's Algebraic Geometry: Show that any conic in $\mathbb{A}^2$ is isomorphic to $\mathbb{A}^1$ or $\mathbb{A}^1-\{0\}$. I know from a previous ...
1
vote
1answer
457 views

Calculate perimeter from parametric form with an ellipse?

Suppose I have a thing such as an ellipse: $$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1$$ now we can define it so that $\frac{x}{a}=cos(\theta)$ and $\frac{y}{b}=sin(\theta)$. I ...
5
votes
2answers
12k views

What is the equation of an ellipse that is not aligned with the axis?

I have the an ellipse with its semi-minor axis length $x$, and semi major axis $4x$. However, it is oriented $45$ degrees from the axis (but is still centred at the origin). I want to do some work ...
12
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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 ...
2
votes
1answer
733 views

Find intersection(s) between parametrized parabola and a line

I'm trying to find the value(s) of the parameter $t$ at the intersection point(s) between a 2D general parabola (as a parametric function of $t$) and a line whose equations can be derived from two ...
4
votes
2answers
1k views

Finding Coordinate along Ellipse Perimeter

Given an ellipse at (0, 0), with height "h" and width "w", what's the "x" coordinate along the perimeter for a given "y" coordinate?
0
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1answer
577 views

How to find a generic parabola through 3 arbitrary points in R^2?

Given $(a,b)$, $(c,d)$, and $(e,f)$ (assume non-collinear and $a\neq c$, $c\neq e$, and $a\neq e$), is there a generic way to find a parabolic function between the three?
4
votes
1answer
116 views

Ellipse: Name for the ratio $a/b$?

Given an ellipse with semi-major axis $a$ and semi-minor axis $b$, is there a "common" (or at least standard) name for either $\frac{a}{b}$ or $\frac{b}{a}$? I keep wanting to (informally) call it ...
4
votes
3answers
3k views

Calculate intersection of two ellipses

I used the equations found here to calculate the intersection points of two circles: (P3 is what I'm trying to get) Except, now I want to do the same with two ellipses. Calculating ...
1
vote
2answers
1k views

Finding & Plotting equation of hyperbola given foci, and difference in distances between them.

I have to plot the hyperbola (3 of them actually) in MATLAB, and so it'd be good if I could find some sort of general formula. The foci do not necessarily have to be on the axes (e.g. $(5,3)$ and ...
0
votes
1answer
787 views

How to calculate the X Y coordinates of an ellipse with only the X and Y radius length?

I have an ellipse where the radius of x-axis = 100 and y-axis = 30. I have 3 objects where I want to evenly distribute it along the ellipse. I have already done this for a circle where both axis' ...
2
votes
1answer
295 views

How to describe foсi of en ellipse inscribed in the triangle thru triangles angles points?

I was looking at Marden's theorem and could not help but wonder how foсi of en ellipse inscribed in the triangle can be described thru triangles angles points?
2
votes
3answers
120 views

How to find orignal equations of type $y=ax^2+bx+c$. given 3 coordinate points?

Ok, simple question, having trouble understanding this in school. So given a set of 3 points (xy-plane), such as (40,30) (60,28) (20,25) i have to find the equation of the parabola. I ...
2
votes
2answers
267 views

Conditions for intersection of parabolas?

What are the conditions for the existence of real solutions for the following equations: $$\begin{align} x^2&=a\cdot y+b\\ y^2&=c\cdot x+d\end{align}$$ where $a,b,c,d $ are real numbers. ...
3
votes
2answers
9k views

How do I find the equation of a tangent line to a curve?

I'm given $x^2+2x-4$ at $x=2$ and I have to find the tangent line to this curve at that point...
0
votes
1answer
696 views

Calculating Intersection of an Ellipse and a Line

I found this page which gave me some equations on solving the intersection of a line with an ellipse given a point on the line and the slope of the line: There Isn't much explanation but ...
4
votes
3answers
2k views

How to get the limits of rotated ellipse?

The box that an ellipse fits is easily calculated if there are no rotation, or if the rotation is ${x*90^o}$ (where x is an integer) is easy. For a (major radius) and b (minor radius), it is : ...
9
votes
1answer
4k 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 ...
2
votes
1answer
360 views

Need help with the proof of conic section

Prove that the intersection of a plane and a object consist of one cone and one upside-down cone where the tip of cone meet is either degenerate conic or conic Also, idenify in what situation, the ...
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vote
0answers
72 views

Some question concerning curve of second order

Let $$F(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f,$$ $$\phi(x,y)=ax^2+2bxy+cy^2,$$ $x,y \in \mathbb{R}$. Assume that for some $x_0, y_0 \in \mathbb{R}$ and for some $\alpha, \beta \in \mathbb{R}$ such that ...
0
votes
1answer
146 views

Parabola with a variable starting point

I am trying to build an equation where I could start at (x,y) which are known and create a parabola from that starting point. I have no idea where it intercepts the X or Y. I know where I want the ...
2
votes
1answer
459 views

Properties of Parabola / Optimization

I've been working through some past papers for an exam which I am due to be sitting tomorrow. In the Conic Sections paper from a couple of years ago, the following question came up: The path of a ...
1
vote
1answer
3k views

how to find the parabola of a flying object [closed]

how can you find the parabola of a flying object without testing it? what variables do you need? I want to calculate the maximum hight and distance using a parabola. Is this possible? Any help will be ...
1
vote
4answers
251 views

Difficult equations to rewrite as ellipses

I have this equality that defines an elliptic boundary. I am trying to rewrite it in the form of the equation of an ellipse, but I am having trouble doing that. How would I go about rewriting this ...
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 ...
1
vote
1answer
562 views

Center of gravity of an ellipse

I think the center of gravity of an ellipse is the intersection point of it's two radius. But I didn't see it anywhere, so I'm having some doubt about it. Am I right? Thanks to all.
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4answers
2k views

Is an ellipse a circle transformed by a simple formula?

Does any ellipse $E$ have a circle $C$ such that you can obtain $E$ by transforming $C$ by a simple formula $F$? In details , both $E$ and $C$ have the same center and the axes of $E$ are the XY axes. ...
0
votes
1answer
219 views

How to compute the cross point between an ellipse and a straight line?

Please let me know how to compute the possible cross points between an ellipse and a straight line. In details , I know the basic properties of the two shapes. So if the ellipse had its center at the ...
0
votes
2answers
189 views

Hyperbola property

I am posting the following question under homework category. I hope I will have very good answer from mathematicians about conic sections. I have seen closely the conic sections and their ...
15
votes
3answers
16k 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 ...
8
votes
3answers
940 views

an important property of an ellipse

Good morning everybody. I would like to know the proof of the following observation on the ellipse. A circle is drawn with the right latus rectum as diameter. Another circle is drawn with its ...
3
votes
3answers
299 views

Apostol Section 13.25 #13 - Conic Sections

Question: Prove that a similarity transformation (replacing $x$ by $tx$ and $y$ by $ty$) carries an ellipse with center at the origin into another ellipse with the same eccentricity. (The next ...
4
votes
2answers
345 views

Maximize the distance between a line normal to an ellipse and its center

My friend sent me this problem, which (upon Googling) seems to be from a Cornell class (1220?). Anywho. My advice to him was to parametrize the ellipse (say, in the first quadrant) with $x = a ...
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votes
1answer
828 views

A Hunt for a Mathematical Machine That Gives Points

The central question is : Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ? Explanation: ...
0
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2answers
343 views

why we only have a approximation for every circumference for ellipse but not define a special formula for each ellipse

Why do we only have an approximation for every circumference for ellipse, but we cannot define a special ratio formula for each ellipse? Is it possible for people to use a computer to find the exact ...
1
vote
1answer
326 views

How to find points of tangency on a hyperbola?

If tangent lines to the hyperbola $9x^2-y^2=36 \;$ intersect y-axis at point $(0,6)$, find the points of tangency.
2
votes
4answers
4k views

How to find points of tangency on an ellipse?

The problem I have to solve is: If tangent lines to ellipse $9x^2+4y^2=36$ intersect the y-axis at point $(0,6)$, find the points of tangency.
6
votes
5answers
603 views

Generating coordinates for 'N' points on the circumference of an ellipse with fixed nearest-neighbor spacing

I have an ellipse with semimajor axis $A$ and semiminor axis $B$. I would like to pick $N$ points along the circumference of the ellipse such that the Euclidean distance between any two ...
0
votes
1answer
144 views

How to “transform” $(f(x), g(x))$ to $(x, y(x))$?

I'm currently trying to solve the following problem: Let $L$ be the set of points of $\mathbb{R}^2$ that satisfy the condition $f(x,y) = 7x^2-6 \sqrt{3} xy + 13y^2 = 16$. It is possible to ...
3
votes
2answers
342 views

Why do definitions of distinct conic sections produce a single equation?

I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
6
votes
3answers
966 views

Parametric form of an ellipse given by $ax^2 + by^2 + cxy = d$

If $c = 0$, the parametric form is obviously $x = \sqrt{\frac{d}{a}} \cos(t), y = \sqrt{\frac{d}{b}} \sin(t)$. When $c \neq 0$ the sine and cosine should be phase shifted from each other. How do I ...
14
votes
2answers
849 views

What is the simplest ellipse that goes through exactly 13 lattice points?

The ellipse $-30 x + 3 x^2 - 10 y - 3 x y + 4 y^2$ goes through exactly 11 lattice points. Another such ellipse is $4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$. What is the simplest ellipse that goes ...
0
votes
1answer
330 views

Irreducible conic implies that the underlying matrix is invertible

I guess that it is true that a conic (2nd degree homogeneous equation in complex variables) is irreducible (i.e can't be factorized over polynomials) if and only if the underlying matrix of ...
0
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2answers
1k views

Plot ellipse in cartesian coordinates

I am working with a computer program that needs to draw objects in a 3d field. Over time, these objects will move (at different rates) on an elliptical path around the 0,0,0 point. I've read the ...
3
votes
1answer
170 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 ...