# Questions tagged [curves]

For questions about or involving curves.

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### Is it possible to plot a graph of any shape?

In school, I have learnt to plot simple graphs such as $y=x^2$ followed by $y=x^3$. A grade or two later, I learnt to plot other interesting graphs such as $y=1/x$, $y=\ln x$, $y=e^x$. I have also ...
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### Helix with a helix as its axis

Does anyone know if there is a name for the curve which is a helix, which itself has a helical axis? I tried to draw what I mean:
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### What is parameterization?

I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up. On Wikipedia it says: Parametrization is... the process of ...
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### Prove that the boy cannot escape the teacher

I'm struggling with the following problem from Terence Tao's "Solving Mathematical Problems": Suppose the teacher can run six times as fast as the boy can swim. Now show that the boy cannot ...
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### Coronavirus growth rate and its (possibly spurious) resemblance to the vapor pressure model

The objective is the model the growth rate of the Coronavirus using avaibale data. As opposed to the standard epidemiology models such as SIR and SEIR, I tried to model a direct relation between the ...
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### Area enclosed by an equipotential curve for an electric dipole on the plane

I am currently teaching Physics in an Italian junior high school. Today, while talking about the electric dipole generated by two equal charges in the plane, I was wondering about the following ...
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### Does a closed curve exist for which a square cannot intersect it 8 or more times?

To phrase my question more clearly: Imagine you have a game with two players, Minnie and Maxime. Minnie starts by defining some closed curve. Then Maxime translates, rotates, and scales a square with ...
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### Strangely but closely related parametrized curves

Compare the following two parametrized curves for $k \in \mathbb{N}^+$: $$x_r(t) = \cos(t)(1 + r\sin(kt))$$ $$y_r(t) = \sin(t)(1 + r\sin(kt))$$ with $0 \leq t < 2\pi$ and $0 \leq r \leq 1$ (being ...
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### Can a smooth curve have a segment of straight line?

Setting: we are given a smooth curve $\gamma: \mathbb{R} \rightarrow \mathbb{R}^n$ Informal Question: Is it possible that $\gamma$ is a straight line on $[a,b]$, but not a straight line on $[a,b]^c$? ...
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### Tracing a curve along itself - can the result have holes?

Let $\varphi:[0,1]\to\Bbb R^2$ be a continuous curve (not necessarily injective) with $\varphi(0)=(0,0)$. Let $f:[0,1]^2\to\Bbb R^2$ be defined as $f(s,t)=\varphi(s)-\varphi(t)$. Question: Is the ...
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### Can the boundaries of two pentagons intersect at $20$ points?

This question is a follow-up to Maximum number of intersections between a quadrilateral and a pentagon, where it is shown that the boundaries $\partial Q,\partial P$ of a quadrilateral and a pentagon ...
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### Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

While reading about the square peg problem, I found this paper of Jerrard, where he described that for the spiral $$r = k\theta \quad 2\pi \leq \theta \leq 4\pi$$ if we join the endpoints, you can ...
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### How does the homogenization of a curve using a given line work?

I am given a curve $$C_1:2x^2 +3y^2 =5$$ and a line $$L_1: 3x-4y=5$$ and I needed to find curve joining the origin and the points of intersection of $C_1$ and $L_1$ so I was told to "homogenize" ...
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### Can every curve be subdivided equichordally?

This question build on top of this other question: Dividing a curve into chords of equal length, for which I wrote an (incomplete) answer. I got the feeling we might need some help from a real ...
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### Exercise 1.24 from Joe Harris' Algebraic Geometry: First Course

I have a question about solvability of Exercise 1.24 (p. 14) from Joe Harris' Algebraic Geometry: A First Course and correctness of following 'synthetic' construction which according to the book (or ...
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### Why is the second derivative of this function a straight line?

This might be an unusual question but I was wondering why the 2nd derivative of this function is a straight line? I kind of have the feeling this is not that easy to answer. But it kind of struck me ...
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### Sigmoid function with fixed bounds and variable steepness [partially solved]

(see edits below with attempts made in the meanwhile after posting the question) Problem I need to modify a sigmoid function for an AI application, but cannot figure out the correct math. Given a ...
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### Constraints on conical coffee cup constructions of cardioids & catacaustics

The Mathologer video Times Tables, Mandelbrot and the Heart of Mathematics discusses several relationships. For the n=2 and 3 cases, the cardiod and catacaustic (or nephroid per @Rahul's comment) ...
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### Geodesic between two points

I have a question about geodesics. So far I know that for any surface $S$ defined by some immersion $f: U \subset\mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3,$ we have that for any point on the ...
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### Fitting a closed curve on the roots of ${x \choose k}-c$

Let $${n \choose k} = \frac1{(n+1) \operatorname{B}(n-k+1, k+1)}.$$ be the generalized binomial coefficient, and here $\operatorname{B}$ is the Beta function. Let $f_{k,c}(x)$ be the following ...
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### Can the boy escape the teacher for a regular $n$-gon?

This is related to Prove that the boy cannot escape the teacher Suppose there is a boy in the center of a regular $n$-gon. The teacher is on the edge of the $n$-gon (but cannot leave the edge) and ...
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### Weighting a cubic hermite spline

I am trying to figure out a function behind the software's curve drawing algorithm. Originally, each node comes with 3 parameters : time, value, and tangent. I have found that it fits cubic Hermite ...
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### How wide can a unit-length planar curve be?

The width of a bounded set in the plane is the minimum distance between two parallel lines bounding the set. Suppose that we have some curve $C: [0,1]\to\mathbb{R}^2$ of unit length. How large can its ...
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### Locus of points such that facing Mecca is the same as facing east

We came to think of this problem: Ali is a good Muslim who happens to travel a lot. On one occasion when Ali is praying, properly oriented towards Mecca, he notices that he is also facing ...
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### Curves with constant curvature and constant torsion

Describe all curves in $\mathbb{R}^3$ which have constant curvature $κ > 0$ and constant torsion $τ$. Any ideas what we can do to describe all such curves? Do we have to use the formulas of the ...
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### On which tangent bundles of $\mathbb R^2$ does position, velocity, acceleration live?

I am studying differential geometry, and am unable to align what I have learnt in classical mechanics with how differential geometers describe the situation. Consider a particle travelling in a circle....
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