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Questions tagged [conic-sections]

For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.

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Can we define trigonometric function for every conic section?

Just like we can define circular function (or trigonometric function) for circles, hyperbolic function for hyperbola; in a similar way can we also define trigonometric parabolic or elliptic functions?...
Priyansh Agarwal's user avatar
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40 views

polarization ellipse for complex eigenvalues corresponding the phase and eigenstates. [closed]

I want to draw polarization ellipses at 0.0 eV, 0.12 eV, 0.16 eV, and 0.2 eV for my transmission eigen-polarization-values plots using eigenphase data (in radians). I've attached the final result ...
Anshul Bhardwaj's user avatar
0 votes
1 answer
44 views

Least Squares Ellipse with known parameters

Given a set of points in 2D space $$P = \{(x_i, y_i), \text{for } i \text{ in }1 \dots N\}$$ I want to find the least squares fit of an ellipse $$\frac{(x - c_x)^2}{r_x^2} + \frac{(y - c_y)^2}{r_y^2} =...
Dominik Ficek's user avatar
3 votes
0 answers
76 views

A beautiful property of two parabolas that intersect in four points

I have just come up with a very cool property of two parabolas intersecting at four points, I want to know whether this property is already known or not and how to prove it. We have two parabolas ...
زكريا حسناوي's user avatar
0 votes
1 answer
57 views

Locus of a point whose distance from two points is fixed (but not necessarily equal) in 3D geometry

Suppose there are two fixed points $S_1$ and $S_2$ Let the moving point be $P$ $PS_1$ and $PS_2$ are fixed but not necessarily equal. Now I think it is a circle. Obtained by rotation of vertex of ...
Aurelius's user avatar
  • 471
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0 answers
48 views

Elliptical Grid Mapping in Shader

I wanted to make a Elliptical Grid Mapping Shader, but it is not a perfect square and it is rotated. If i multiply the coords by sqrt(2.) and divides them after again, it is an square, but still ...
Taxy's user avatar
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5 votes
1 answer
672 views

The center of gravity of a triangle on a parabola lies on the axis of symmetry

About an hour ago, I discovered a beautiful property of a parabola. If a circle intersects a parabola at four points, one of which is the vertex of the parabola, then the center of the triangle, ...
زكريا حسناوي's user avatar
8 votes
1 answer
134 views

Double Contact Chained Ellipses Problem

A few years ago, when I played around with GeoGebra, I came up with the following conjecture. Conjecture Let $n\in\mathbb{N}, n\geq3$. Let $E$ be an ellipse, and let $E_{1}, E_{2}, \dots, E_{n}$ be ...
K. Miyamoto's user avatar
7 votes
4 answers
202 views

Largest Area Triangle in the Vesica Piscis

I can place any three points in or on a vesica piscis1. I wish to find the triangle of maximum area. I know the area of the vesica piscis is $(\frac{2π}{3}-\frac{\sqrt{3}}{2})d^2$ (where d is the ...
WakkaTrout's user avatar
1 vote
3 answers
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Properties that relate to the chord of a parabola passing from the perpendicular projection of the focus point on the parabola guide [closed]

Now I remembered my previous question about finding the harmonic mean using a parabola and my answer, which included a second method. That second method inspired me to try more in this configuration ...
زكريا حسناوي's user avatar
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1 answer
88 views

Line tangent to a parabola

So I was doing some AoPS Alcumus and came across this problem with a weird solution. A quadratic function $p(x)$ has lines of tangency $y=-11x-37$, $y=x-1$, and $y=9x+3$. These lines are tangent to $p$...
RightOnYourLeft911 gates's user avatar
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1 answer
65 views

Complete specification of the intersection between an elliptical cone and a plane [closed]

Suppose you're given the elliptical cone $ (z - h)^2 = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} $ And the plane $ N \cdot r = d $ where $r = (x,y,z) $. Assume that $N$ is such that the intersection ...
Quadrics's user avatar
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1 vote
3 answers
171 views

Construct a cone from independently sampled surface points

2 points are sufficient to determine a 3D line, 3 points are sufficient to determine a 3D plane and there are well-known formulas to construct lines and planes from such points. I understand that ...
Francesco Solera's user avatar
1 vote
2 answers
105 views

Golden ratio points in ellipse

This is a property of the ellipse. The sum of distances to the foci is constant: In particular, some of these points must satisfy the golden ratio relationship: Given the equation of the ellipse in ...
vallev's user avatar
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Largest Elliptic Cone Intersecting with Sphere

I have a function that can be reasonably approximated with an elliptic cone with a certain excentricity I can calculate, I have the dimensions of the axes so for example a = 1 and b = 0.25. I then ...
redorav's user avatar
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Parametric eqn of an ellipse and the meaning of the angle "t"

I don't understand what the angle "t" is in the parametric equation of an ellipse. The parametric equation from books is given as: $$x = a\cos t$$ $$y = b\sin t$$ Referring to the diagram, ...
rdemo's user avatar
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1 vote
1 answer
213 views

Solving the system $D(r_i-C)=\frac{g_i}{\sqrt{g_i^TD^{-1}g_i}}$, $i\in\{1,2\}$, for $2\times2$ diagonal matrix $D$ and $2\times1$ vector $C$

I want to solve the following equations for the unknown $2 \times 2$ diagonal matrix $D$ and the $2 \times 1$ vector $C$. Given are $2 \times 1$ vectors $r_1$ and $r_2$, and the $2 \times 1$ vectors $...
Quadrics's user avatar
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1 answer
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Help troubleshooting ellipse perimeter calculation algorithm

I'm trying to troubleshoot my implementation of an Infinite Series algorithm to calculate the perimeter of an ellipse. I'm sorry I don't have the expertise to express it in, what appears to be a ...
bielawski's user avatar
  • 209
2 votes
1 answer
63 views

Centers of three conical sections located on one line

Yesterday while playing with the Geogebra application, I discovered a fairly initial theorem in the conic sections, and I don't know if it was already discovered or not, please if there are references ...
زكريا حسناوي's user avatar
3 votes
1 answer
89 views

A question about general equations of Hyperbolas [closed]

We know that the general equation for a hyperbola is $$\sqrt{(x-\alpha)^2+(y-\beta)^2}=e\cdot\frac{|lx+my+n|}{\sqrt{l^2+m^2}}$$ where $(\alpha, \beta)$ is the focus of the hyperbola, $e>1$ is the ...
Kheerii's user avatar
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2 votes
2 answers
191 views

Area swept by the circumference of an ellipse as it slides such that it is always tangent to the $x$ axis at the origin

You're given the ellipse $\frac{x^2}{a^2} + \frac{(y - b)^2}{b^2} = 1,$ for known $a$ and $b$. Now you slide the ellipse and rotate it such that it remains tangent to the $x$ axis at the origin all ...
Quadrics's user avatar
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0 votes
3 answers
89 views

Is it possible to find an ellipse with these conditions?

I am trying to generate an ellipse using two given points in the XY plane. I will call these two points $(x_{start},y_{start})$ and $(x_{end},y_{end})$. The point $(x_{start},y_{start})$ is the ...
ishan_ae's user avatar
9 votes
2 answers
263 views

Relationship between major and minor axis of an ellipse's circumference

I'm not a mathematician. I'm just doing 3D modeling and can't find an equation to solve this problem. In openscad, an ellipse is created by scaling a circle. So this code creates an ellipse with a ...
bielawski's user avatar
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-1 votes
0 answers
40 views

Why only 2 tangents can be drawn to a hyperbola plz try to explain as an 12th grade student [duplicate]

Why only 2 tangents can be drawn to a hyperbola plz try to explain as an 12th grade student by seeing just figure we can say 4 tangents can be drawn to hyperbola
Vinayak Gadag's user avatar
0 votes
1 answer
107 views

How to maximize the area of the triangle? [duplicate]

Let $W$ be a focus of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. $P$ and $Q$ are two points on the ellipse such that for any position of $P$ and $Q$ on the ellipse, perimeter of $\Delta PQW$ is ...
whatamidoing's user avatar
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2 votes
1 answer
88 views

Finding the vertices of the hyperbola $x^2+6xy-7y^2=20$

Given the hyperbola $$x^2+6xy-7y^2=20$$ how do I find its vertices? I have found its asymptotes by factoring it into $$(x+7y)(x-y)=20$$ to obtain its asymptotes: $$y=x$$ and $$y=-\frac{1}{7}x.$$ I do ...
ERROR 404's user avatar
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0 answers
31 views

Can axes $x$ and $y$ be rotated to eliminate the crossed product terms in a cubic form?

I've just learned that it is posible to rotate the axes $x$ and $y$ to obtain the axes $x'$ and $y'$ such that the quadratic form $$ax^2+bxy+cy^2$$ converts to $$\lambda _1x'^2+\lambda _2y'^2$$ So, is ...
Manuel Ocaña's user avatar
0 votes
1 answer
110 views

Find the point $P$ on an ellipse such that $\overline{AP} + \overline{BP}$ is minimum for given points $A$ and $B$

An ellipse in $3D$ space is specified in parametric form as follows: $ E(t) = V_0 + V_1 \cos t + V_2 \sin t $ In addition, two points in space $A$ and $B$ are given. What I would like to find is the ...
Quadrics's user avatar
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-1 votes
0 answers
36 views

Does rotating our axis change the nature of our graph?

If we rotate our axis does it change nature of our graph? For example:-in this image the graph is a parabola. But if use the black arrows as our new axes will it still be parabola? I am confused ...
rth 45's user avatar
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5 votes
4 answers
416 views

Approximating an Ellipse with Circular Arcs.

In engineering drawing class, we were required to draw the projection of a cube with a circle inscribed on a face such that each side of the face is tangent to the circle. It is easy to prove that the ...
Mathematics enjoyer's user avatar
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0 answers
17 views

Does this correctly show that there is an affine transformation that takes a circle in the plane (arbitrarily close) to a parabola?

I believe this is essentially just a way to show that there is an anisotropic scaling of a circle that produces an ellipse arbitrarily close to a parabola. Let $u \in [0,2\pi)$, $R \in \mathbb{R}_{>...
Simon M's user avatar
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3 votes
2 answers
120 views

The shape of parabola in a 1-point perspective drawing with the vanishing point at the parabola's point at infinity

Edit: After a little bit of careful construction, I've worked out that the map I should have been using instead is $$\phi\left(x,y\right)=\left(\frac{x}{ay+1},\frac{ay}{ay+1}\right)$$ where $a\in(0,\...
Simon M's user avatar
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0 votes
0 answers
60 views

Minimum distance between a given ellipse in $3D$ and a given point

An ellipse in $3D$ is given parametrically by $ p(t) = C + v_1 \cos t + v_2 \sin t $ where $ C , v_1, v_2 \in \mathbb{R}^3 $ are known. In addition, a point $q$ in $3D$ is given, where $ q \in \...
Quadrics's user avatar
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1 vote
0 answers
36 views

If the number of intersection of two conics is an odd number, the quadratic forms are not simultaneous diagonalizable

I'm trying to do Exercise 3.6 at the end of this pdf (in $\Bbb CP^2$): Show that the two quadratic forms $$x^2+y^2-z^2, \quad x^2+y^2-y z$$ cannot be simultaneously diagonalized. Attempt 1: Their ...
hbghlyj's user avatar
  • 3,047
1 vote
2 answers
65 views

Determining the Equation of a Parabola in a Three-Dimensional Plane

I have a plane with three points: P1 (x1, y1, z1), P2 (x2, y2, z2) and P3(x3, y3, z3). These points represent a parabola where P3 is the vertex and the other two points are the ends of the parabola. ...
sanjog karki's user avatar
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0 answers
15 views

Isogonal lines and an ellipse

Take two lines $l_1$ and $l_2$ with intersection $A$. We say the lines $AM$ and $AN$ are isogonal with respect to the lines $l_1$ and $l_2$ if $AM$ is the reflection of $AN$ about the bisector of an ...
Numeral's user avatar
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0 votes
1 answer
78 views

Can a parabola be exactly replicated using exponentials?

I came across this interesting artifact of a problem I was helping a friend solve where I used Euler's exponential forms of sine and cosine which later became of the form $e^{ix} + e^{-ix}$ on one ...
person of stuff's user avatar
3 votes
3 answers
168 views

How can I fit a circle into the gap between an ellipse and the coordinate axes?

Consider an ellipse with the equation $\frac{\left(x-ar\right)^{2}}{a^{2}}+\frac{\left(y-br\right)^{2}}{b^{2}}=r^{2}$. How would I fit a circle with an equation $\left(x-k\right)^{2}+\left(y-k\right)^{...
DavidNyan10's user avatar
1 vote
0 answers
21 views

Volume of an elliptical cylinder cut with a curved plane

I am looking to find the volume that is left when a curved plane defined by $z=-y^3+0.5$ and the xy plane cuts an elliptical cylinder defined from five points ($Ax^2+Bxy+Cy^2+Dx+Ey=F$). I plan to ...
Mr. Mister's user avatar
10 votes
1 answer
286 views

Are geometric series related to ellipses in this particular way?

Consider a point $F$ and a line $l$. Let the point $P$ be the foot of the perpendicular to $l$ passing through $F$. Let $E$ be the ellipse with focus $F$, directrix $l$ and eccentricity $r$. Then $E$ ...
Finn Bolton's user avatar
3 votes
2 answers
87 views

Tangent ellipses property

While tinkering with an assignment about orbital transfers I stumbled across a "property" of ellipses in 2D. I would state the property as: If two ellipses are tangent and share a focus, ...
Pietro Deligios's user avatar
0 votes
1 answer
33 views

Finding Transition Point of Two Tangent Parabolas

I have an engineering problem that involves a long cable that is draped (Draped Cable) in a series of reversing parabolas. The high and low points are known, but I need to locate the inflection points;...
dengebre's user avatar
-1 votes
2 answers
65 views

If we spin an ostrich egg along its minor axis will it be oblate shape?

An ostrich egg is classified as an ellipsoid and if we spin it around it's major axis it's classified as a prolate but my fried is arguing that we can not spin that ellipsoid around its minor axis ...
Mathematition_From_Wallmart's user avatar
0 votes
1 answer
51 views

Intersection of parabolas at $y = x$?

Intersection of parabolas at $y = x$? CONJECTURE: if a given point $(x,y)$ such that $x > 0$ and $y > 0$ , generates two parabolas with the $x$ axis and with $y$ axes as directrix, than I ...
Pedro Assumpção's user avatar
2 votes
0 answers
43 views

Infinite Generators of a Cone

Show that the cone ${3yz-2zx-2xy = 0}$ has infinite set of three mutually perpendicular generator Here is what I have tried so far Let three mutually perpendicular line have direction cosine's ($l_{1}$...
Brb's user avatar
  • 21
1 vote
3 answers
137 views

How to find the equation of an ellipse using three points?

I came across this interesting problem yesterday and I am not quite able to find the equation of the ellipse after it has performed that roll. The original problem shows the ellipse to rotate till it ...
whatamidoing's user avatar
  • 2,879
0 votes
1 answer
24 views

stretch a parabolic velocity profile from a circle to an ellipse

The left figure shows a circle of radius a; it is actually the cross section of a liquid jet, which cross section is normal to flow. The velocity profile of liquid within the jet is parabolic and is ...
rdemo's user avatar
  • 341
0 votes
0 answers
33 views

Area of ellipse using Green’s theorem [duplicate]

Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which we want to calculate the area of. Parameterization $\mathbf{r}(t) = (a \cos t, b \sin t), \ t \in [0, 2\pi]$ My book says we can ...
math.lover's user avatar
3 votes
3 answers
243 views

Inscribing two circles and an ellipse in a square

A square of given side length $S$ is to inscribe two circles and an ellipse as shown in the figure below. If the radius of circle $(1)$ is given, determine the center and radius of circle $(2)$, then ...
Quadrics's user avatar
  • 24.5k
6 votes
4 answers
346 views

Rolling an elliptical disc on the $x$ axis

You're given the elliptical disc bounded by $ \dfrac{x^2}{a^2} + \dfrac{(y - b)^2}{b^2} = 1 $ where $a = 5, b = 2 $. You roll this ellipse to the right along the positive $x$ axis, such that it is ...
Quadrics's user avatar
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