The Stack Overflow podcast is back! Listen to an interview with our new CEO.

# Questions tagged [curves]

For questions about or involving curves.

1,972 questions
Filter by
Sorted by
Tagged with
12k views

### 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 ...
4k views

### 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:
765 views

### 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 ...
40k views

### 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 ...
252 views

### 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 ...
2k views

4k views

### 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 ...