New answers tagged conic-sections
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Find the point $P$ on an ellipse such that $\overline{AP} + \overline{BP}$ is minimum for given points $A$ and $B$
Given a point $P = E(t)$ for some $t$, it lies on a unique ellipsoid (of revolution) whose foci are $A$ and $B$. It was shown here that the equation of this ellipsoid is
$ (p - C)^T (a^2 I - {UU}^T ) ...
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Largest Area Triangle in the Vesica Piscis
Here is a solution in the "17th century spirit" where extremal solutions were found based on the computation of infinitesimal quantities.
I assume that we look for an optimal solution under ...
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Least Squares Ellipse with known parameters
As your $c_x$ and $r_x$ are known, your problem can be recast as
$$\left|\frac{y}{r_y}-\frac{c_y}{r_y}\right|=\sqrt{1-\left(\frac{x-x_c}{r_c}\right)^2}.$$
This is equivalent to a linear model, but for ...
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Locus of a point whose distance from two points is fixed (but not necessarily equal) in 3D geometry
Just to give you an answer:
Indeed if $d(P,S_1)$ is constant, then $P$ lies on the spherical surface $\odot(S_1,S_1P)$, on the same manner $P \in \odot (S_2,S_2P)$. Because $P$ is on the meeting of ...
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Finding the vertices of the hyperbola $x^2+6xy-7y^2=20$
The conic is central, so the vertices are at least distance from center i.e. origin
With $x=r \cos \theta, y = r \sin \theta$ we get $r^2 = \dfrac{20}{3 \sin 2 \theta + 4 \cos 2 \theta -3} \ge \dfrac{...
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Largest Area Triangle in the Vesica Piscis
Consider a bounded and closed (i.e. compact) region in the plane, not contained in a line. There exists a triangle with largest area with vertices in the figure, $\Delta ABC$. Now if we keep $B$, $C$ ...
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The center of gravity of a triangle on a parabola lies on the axis of symmetry
We can prove a stronger claim, namely that this works for any axis-aligned ellipse which passes through the parabola vertex. First, we choose coordinates such that the parabola's vertex is $(0,0)$ and ...
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Double Contact Chained Ellipses Problem
Not an answer, but a generalization to OP's circle result, Lemma 2.
Consider an ellipse with center $O$ that has an internally doubly-tangent circle of radius $r$ (with touch-points $T$ and $T'$) and ...
- 78.1k
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Largest Area Triangle in the Vesica Piscis
EDIT.
I'm inserting here a purely geometrical solution, the original reasoning can be seen at the end.
I'll repeatedly make use of the following result: if we have a line $r$ and an arc of circle $\...
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Largest Area Triangle in the Vesica Piscis
For triangles with an edge parallel to the line connecting the centers of the circles, the largest is shown in the image below. I expect this to be the largest in general.
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Construct a cone from independently sampled surface points
I've developed a procedure for recovering the right circular cone from $7$ sample points from its surface. A right circular cone is determined by $6$ geometric parameters, which usually implies that ...
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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
And this is another feature I've just come up with about this configuration.
The two straight lines AF, BF are made with parabola guide isosceles triangle
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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
And this is another feature that I have come up with now related to this configuration
The line passing through M and perpendicular to AB must pass from a fixed point which is the image of point F' ...
- 2,654
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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
Now I discovered another property related to this configuration
The center of the circle passing through the points A, B, M belongs to the axis of symmetry of the parabola
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Construct a cone from independently sampled surface points
A point $P=(x,y,z)$ belongs to the surface of a cone with vertex $V=(x_0,y_0,z_0)$ if vector $P-V$ forms a fixed angle $\theta$ with the direction $\vec a=(x_a,y_a,z_a)$ of the axis.
Hence the ...
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Locus of intersection of a pair of variable straight lines $x^2+4y^2+\alpha xy=0$ with ellipse $x^2+4y^2=4$
We can assume the locus point as (X1,y1)and write it's chord of contact equation and homogenize that with the ellipse equation. Finally we can compare this equation with the equation given in question
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Complete specification of the intersection between an elliptical cone and a plane
Eliminate $z$ between the equations of cone and plane. The result is the equation of the projection of their intersection on the $xy$ plane.
You can then find a pair of conjugate diameters of this ...
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Construct a cone from independently sampled surface points
Suppose you have $4$ non-coplanar points. Let's call them $A,B,C,D$. Pick one of them as your cone's apex. The equation of a right circular cone with apex $A$ is
$ (r - A)^T Q (r - A) = 0 $
where $...
- 24.5k
3
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Line tangent to a parabola
To be tangent, the line $y = l(x)$ and the parabola $y = p(x)$ must intersect at that point, so we have $p(x_i) = l(x_i) \implies p(x_i) - l(x_i) = 0$, i.e. $x = x_i$ is a root of the parabola $p(x) - ...
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Centers of three conical sections located on one line
You made a nice observation, and it is implicit in the theory of the Poncelet Porism and the theory of conic pencils. It's a big topic - I won't offer any proofs - but hopefully I can point you to ...
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Area swept by the circumference of an ellipse as it slides such that it is always tangent to the $x$ axis at the origin
To find the sliding ellipse, using tangency properties, is quite easy.
Taking an ellipse as
$$
\mathscr{E}(x,y,x_0,y_0,a,b,t) = ((y-y_0)\cos t+(x-x_0)\sin t)^2a^2+((x-x_0)\cos t-(y-y_0)\sin t)^2-a^2b^...
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Area swept by the circumference of an ellipse as it slides such that it is always tangent to the $x$ axis at the origin
Not a full answer, but I'm posting it in the hope it can be of help to find a complete solution.
We want to find the equation of an ellipse, with semi-axes $a$ and $b$, tangent to the $x$-axis at $(0,...
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4
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Solving the system $D(r_i-C)=\frac{g_i}{\sqrt{g_i^TD^{-1}g_i}}$, $i\in\{1,2\}$, for $2\times2$ diagonal matrix $D$ and $2\times1$ vector $C$
Let
$$D=\begin{pmatrix}
x & 0 \\
0 & y \end{pmatrix}, C=\begin{pmatrix}p\\q\end{pmatrix}$$
$$r_1=\begin{pmatrix}a\\b\end{pmatrix},r_2=\begin{pmatrix}c\\d\end{pmatrix},g_1=\begin{pmatrix}e\\f\...
- 145k
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Golden ratio points in ellipse
Using polar coordinates,
$$r=\frac{b^2}{a-c\cos \theta}$$
which is focal-origin, namely
$$\frac{(x-c)^2}{a^2}+\frac{y^2}{b^2}=1$$
Since $d_1+d_2=2c$,
\begin{align}
\frac{\sqrt{5}+1}{2} &= \frac{...
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Golden ratio points in ellipse
Given the equation of the ellipse in standard form:
$$
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1
$$
where the foci are at $( \pm c, 0)$ and $c=\sqrt{a^2-b^2}$.
The distances between foci and the points are ...
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Help troubleshooting ellipse perimeter calculation algorithm
If you aren't able to figure it out and it's a problem with the code and formula you may want to try one of these instead; they are highly accurate, with maximum errors of .04%, .015% and .02%:
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A question about general equations of Hyperbolas
Given the foci $F_1,F_2$and a point $P$ on the hyperbola, the center is the midpoint $M:(h,k)$ of $F_1,F_2$. Letting the axes be centered at the center of the hyperbola and along and perpendicular to ...
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Relationship between major and minor axis of an ellipse's circumference
As mentioned in the comments and other answers, the perimeter of an ellipse is given by an elliptic integral of the second kind. There are several different notation conventions used for these ...
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Angle between two tangent of the ellipse when parametric line equation is used
The given equation of the ellipse in vector-matrix form is
$ r^T Q r = 6 $
where
$ r = [x, y]^T $ and $ Q = \begin{bmatrix} 1 && 0 \\ 0 && 2 \end{bmatrix} $
The given line can also be ...
- 24.5k
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Angle between two tangent of the ellipse when parametric line equation is used
We have:
$$\frac{x}{2}\cos\theta+y\sin\theta=1\tag1$$
$$\frac{x^2}{6}+\frac{y^2}{3}=1\tag2$$
Comparing $(1)$ with the parametric form of tangent equation,
$$\frac{x\cos\theta}{a}+\frac{y\sin\theta}{b}=...
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Is it possible to find an ellipse with these conditions?
The equation of the ellipse that has its major and minor axes parallel to the $x$ and $y$ axes is given by
$ \dfrac{ (x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1 \tag{1} $
This equation can be written ...
- 24.5k
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Is it possible to find an ellipse with these conditions?
I have use Geogebra for doing such a thing: I have created two points, $A$ and $B$, and from that, I have derived two new points, $C$ and $D$, making a simple rectangle like that ($x_C=x_A$, $y_C=y_B$ ...
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Is it possible to find an ellipse with these conditions?
The equation of the ellipse can be written as:
$$
{(x-x_0)^2\over a^2}+{(y-y_{end})^2\over b^2}=1.
$$
Substitute here the coordinates of $(x_{start},y_{start})$ and $(x_{end},y_{end})$, and add the ...
- 52.3k
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Relationship between major and minor axis of an ellipse's circumference
I don't know if what you seek can be accomplished without invoking calculus and special functions.
Assume everything is centered at the origin in the $x,y$ plane. The circle with radius $\dfrac12$ has ...
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If we spin an ostrich egg along its minor axis will it be oblate shape?
Forget about spinning. An ellipsoid has three perpendicular axes; it can be constructed from a sphere, by stretching it along these axes. The stretch factors may or may not be equal.
It's called a ...
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How to maximize the area of the triangle?
The area can be written as
$$
A_{WPQ}=A_{WPS}+A_{WQS}={1\over2}2ae(y_P-y_Q),
$$
hence we need to find the maximum of $y_P-y_Q$. Taking a convenient parametrization with $\phi=\angle WSP$:
$$
y_P={a(1-...
- 52.3k
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Are geometric series related to ellipses in this particular way?
Here's a fairly nifty interpretation ...
Define $p:=|PF|$, and write $e$ for the ellipse's eccentricity. We can locate points $P_0 (=P)$, $P_1 (=F)$, $P_2$, $P_3$, $\ldots$ on axis-line $PF$ such that ...
- 78.1k
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Approximating an Ellipse with Circular Arcs.
A property of the mechanical-engineering shortcut construction
is that as you change from one arc to the other, the length of the radius changes but the direction of the radius does not.
If you start ...
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Approximating an Ellipse with Circular Arcs.
This approximation is of limited use, because it works only if the parallelogram is a rhombus. Try to do that with a generic parallelogram and you'll see that it fails.
In general, if you want to ...
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