12
votes
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
11
votes
Accepted
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 ...
- 16.8k
9
votes
Accepted
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 $\...
- 52.3k
8
votes
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 ...
- 20.6k
5
votes
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.
- 5,493
4
votes
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,...
- 52.3k
4
votes
Accepted
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
$$D=\begin{pmatrix}
x & 0 \\
0 & y \end{pmatrix}, C=\begin{pmatrix}p\\q\end{pmatrix}$$
$$r_1=\begin{pmatrix}a\\b\end{pmatrix},r_2=\begin{pmatrix}c\\d\end{pmatrix},g_1=\begin{pmatrix}e\\f\...
- 145k
3
votes
Golden ratio points in ellipse
Using polar coordinates,
$$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$$
Since $d_1+d_2=2c$,
\begin{align}
\frac{\sqrt{5}+1}{2} &= \frac{...
- 19.1k
3
votes
Line tangent to a parabola
To be tangent, the line $y = l(x)$ and the parabola $y = p(x)$ must intersect at that point, so we have $p(x_i) = l(x_i) \implies p(x_i) - l(x_i) = 0$, i.e. $x = x_i$ is a root of the parabola $p(x) - ...
- 25.6k
3
votes
Accepted
Area swept by the circumference of an ellipse as it slides such that it is always tangent to the $x$ axis at the origin
To find the sliding ellipse, using tangency properties, is quite easy.
Taking an ellipse as
$$
\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^...
- 34.2k
3
votes
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 ...
- 83.9k
3
votes
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 $B$, $C$ ...
- 55k
2
votes
Accepted
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 ...
- 1,940
2
votes
Accepted
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 ...
- 52.3k
2
votes
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
2
votes
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 ...
- 5,726
2
votes
Accepted
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 $\phi=\angle WSP$:
$$
y_P={a(1-...
- 52.3k
2
votes
Approximating an Ellipse with Circular Arcs.
This approximation is of limited use, because it works only if the parallelogram is a rhombus. Try to do that with a generic parallelogram and you'll see that it fails.
In general, if you want to ...
- 52.3k
2
votes
Accepted
Complete specification of the intersection between an elliptical cone and a plane
Eliminate $z$ between the equations of cone and plane. The result is the equation of the projection of their intersection on the $xy$ plane.
You can then find a pair of conjugate diameters of this ...
- 52.3k
1
vote
Construct a cone from independently sampled surface points
I've developed a procedure for recovering the right circular cone from $7$ sample points from its surface. A right circular cone is determined by $6$ geometric parameters, which usually implies that ...
- 24.5k
1
vote
Golden ratio points in ellipse
Given the equation of the ellipse in standard form:
$$
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1
$$
where the foci are at $( \pm c, 0)$ and $c=\sqrt{a^2-b^2}$.
The distances between foci and the points are ...
1
vote
Accepted
Centers of three conical sections located on one line
You made a nice observation, and it is implicit in the theory of the Poncelet Porism and the theory of conic pencils. It's a big topic - I won't offer any proofs - but hopefully I can point you to ...
- 8,836
1
vote
Accepted
Is it possible to find an ellipse with these conditions?
The equation of the ellipse can be written as:
$$
{(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
1
vote
Accepted
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 ...
- 4,993
1
vote
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 ...
- 100k
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