Questions tagged [curves]

For questions about or involving curves.

Let $X$ be a topological space and $I$ an interval in $\mathbb{R}$. A continuous curve in $X$ is a continuous map $\gamma : I \to X$.

Let $X$ be a smooth manifold and again, let $I$ be an interval in $\mathbb{R}$. A smooth curve in $X$ is a smooth map $\gamma : I \to X$.

Note, it both cases, a curve is more than its image. That is, given two curves $\gamma_1 : I_1 \to X$ and $\gamma_2 : I_2 \to X$, it may be the case that $\gamma_1(I_1) = \gamma_2(I_2)$. A particular instance of this occurs when there is a map $\sigma : I_2 \to I_1$ which is a homeomorphism in the case of continuous curves or a diffeomorphism in the case of smooth curves, such that $\gamma_2 = \gamma_1\circ\sigma$. In this case, we say that $\gamma_2$ is a reparameterisation of $\gamma_1$.

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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 recently learnt about trigonometric graphs and…
ChrisJWelly
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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:
user50229
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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 finding parametric equations of a curve, a surface,…
qmd
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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 is a perfectly valid approach to calculating arc…
Keshav Srinivasan
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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 escape. (Hint: Draw an imaginary square of sidelength…
user1337
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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 the goal to maximize the number of intersections…
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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 number of infected or deaths as a function of time…
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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 problem: Assume that two equal charges are placed in…
Jack D'Aurizio
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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 the plot of the sine function with amplitude $r$ over…
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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 image $f([0,1]^2)$ always simply connected? The set…
M. Winter
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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 only draw one square that all of its corner points…
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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 in the plane cannot intersect at more than $16$…
Jack D'Aurizio
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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 topologist. Let me repeat the crucial definitions. We…
M. Winter
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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" the line with the curve . They basically said…
Tesla
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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 alternatively due to page 79 in this script)…
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