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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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Can you define arc length using a piece of string?

In calculus, how we calculate the arc length of a curve is by approximating the curve with a series of line segments, and then we take the limit as the number of line segments goes to infinity. This ...
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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 ...
865 views

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 ...
561 views

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 ...
230 views

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 ...
133 views

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 ...
541 views

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 ...
189 views

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 ...
245 views

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 ...
7k views

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 ...
303 views

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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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 ...
431 views

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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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 ...
Let $\textbf{$\gamma$}: \mathbb{R} \rightarrow \mathbb{R}^n$ be a smooth curve and let $T \in \mathbb{R}$. We say that $\textbf{$\gamma$}$ is $T$-periodic if \textbf{$\gamma$}(t+T)=\textbf{$\gamma$}...
I think that every lightlike curve in $\mathbb{S}_1^2 \subseteq \mathbb{L}^3$ must be a line. But I'm having trouble concluding it. Let \$\alpha\colon I \subseteq \Bbb R \to \Bbb S^2_1 \subseteq \Bbb ...