Articles of plane curves

Locus of intersection of two perpendicular normals to an ellipse

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 […]

Rate of change of direction of vector-valued function

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, […]

The function that draws a figure eight

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.

How can I find a curve based on its tangent lines?

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 […]

In how few points can a continuous curve meet all lines?

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? […]

Nontrivial solutions of $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$

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 […]

Deriving an implicit Cartesian equation from a polar equation with fractional multiples of the angle

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 $\frac{p}{q}$ […]

Is this variant of the Jordan Curve Theorem true?

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 […]

A variational strategy for finding a family of curves?

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 […]

find maximum area

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 […]