I came up with the following question while playing around with Geogebra, and rather than do it myself I figured I’d offer it here. For a given ellipse, the director circle is defined to be the set of all points where two perpendicular tangents to the ellipse cross each other. Suppose we instead consider the […]

This should be easy, but I cannot figure out what I’m doing wrong, and it’s killing me. Let $\mathbf{f}(t)$ be a function from $\mathbb{R}$ to $\mathbb{R}^3$. I want to find the “rate of change of the angle between $\mathbf{f}(t)$ and nearby vectors (from $\mathbf{f}$)”. I’m going to assume $|\mathbf{f}(t)|=1$, because it makes the writing easier, […]

I’m trying to describe a counterexample for a theorem which includes the figure eight or “infinity” symbol, but I’m having trouble finding a good piecewise function to draw it. I need it to be the symbol, except at the “crossing point” the function jumps (not continuous) so that we still have a manifold.

Let’s say for some curve its tangent lines at every point have a property that the length of a segment within the first quarter $[0;+\infty)^2$ is exactly $C>0$. How can such a curve be defined analytically? Maybe even in terms of $y=f(x)$. Now about the tangent lines themselves. Noticeably for all $k<0$ the tangent line […]

Inspired by this (currently closed) question, I’m wondering about a related topic: given that we have a continuous parametric curve (that is, a curve of the form $(x=x(t), y=y(t))$ for $t\in\mathbb{R}$ and $x(t), y(t)$ continuous functions) which meets every line in the plane, what is the minimum number of times it must meet some line? […]

Let $a=(a_n)$ with $a_n\in\mathbb{C}$ be a vector indexed over all $n\in\mathbb{Z}$, and consider the system of equations $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$ for all $k\in\mathbb{Z}$. One may verify this has a family of trivial solutions given by $a_n=\delta_{nm}$ for some nonzero integer $m$. Assuming $a_0=0$, are there any other solutions? This problem is inspired by (unsuccessful) attempts to find […]

Usually, applying the conversion formulae $r^2=x^2+y^2$ $\cos\;\theta=\frac{x}{r}$ $\sin\;\theta=\frac{y}{r}$ to transform an equation in polar coordinates to an implicit Cartesian equation is quite straightforward for an equation of the form $r=f(\cos(n\theta),\sin(n\theta))$ with $n$ an integer, thanks to multiple angle formulae. Polar equations of the form $r=f\left(\cos\left(\frac{p}{q}\theta\right),\sin\left(\frac{r}{s}\theta\right)\right)$ where $p$, $q$, $r$ and $s$ are integers, and $\frac{p}{q}$ […]

This feels as though it should be falsifiable, but it’s not immediately obvious to me. The informal version of the statement is ‘for every non-intersecting curve between two opposite corners of a square, there’s a curve between the other two corners that only intersects it once.’ Formally: Let $f(): [0, 1]\mapsto [0,1]^2$ be a non-self-intersecting […]

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ examples and there are several piecewise-smooth examples of such in that question. However, my real hope was to see $C^\infty$ examples; as of […]

Consider a problem here : There is a wall in your backyard. It is so long that you can’t see its endpoints. You want to build a fence of length L such that the area enclosed between the wall and the fence is maximized. The fence can be of arbitrary shape, but only its two […]

Intereting Posts

Sum of all real solutions for $x$ to the equation $\displaystyle (x^2+4x+6)^{{(x^2+4x+6)}^{\left(x^2+4x+6\right)}}=2014.$
Is Minimax equals to Maximin?
Gaussian integral evaluation
What is the probability that $X<Y$?
Topology on the general linear group of a topological vector space
Finite Index of Subgroup of Subgroup
If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$?
“Fair” game in Williams
Rotate an area around a diagonal line.
Series Question: $\sum_{n=1}^{\infty}\frac{n+1}{2^nn^2}$
Prove variant of triangle inequality containing p-th power for 0 < p < 1
What can we say about the kernel of $\phi: F_n \rightarrow S_k$
A particular set of generators for $S_n$
Calculate the Wronskian of $f(t)=t|t|$ and $g(t)=t^2$ on the following intervals: $(0,+\infty)$, $(-\infty, 0)$ and $0$?
Why do $x^{x^{x^{\dots}}}=2$ and $x^{x^{x^{\dots}}}=4$ have the same positive root $\sqrt 2$?