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Questions tagged [curves]

For questions about or involving curves.

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51 votes
5 answers
19k 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 ...
ChrisJWelly's user avatar
39 votes
5 answers
5k 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:
user50229's user avatar
  • 3,092
39 votes
2 answers
74k 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 ...
qmd's user avatar
  • 4,285
22 votes
4 answers
1k 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 ...
user1337's user avatar
  • 24.5k
20 votes
3 answers
5k views

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 ...
Nilotpal Sinha's user avatar
20 votes
5 answers
730 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 ...
Jack D'Aurizio's user avatar
20 votes
3 answers
496 views

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 ...
user3635700's user avatar
20 votes
1 answer
280 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 ...
Hans-Peter Stricker's user avatar
18 votes
3 answers
2k views

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 $[a,b]^c$? ...
John Frank's user avatar
18 votes
1 answer
167 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 ...
M. Winter's user avatar
  • 30.1k
17 votes
1 answer
318 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 ...
Jack D'Aurizio's user avatar
17 votes
1 answer
566 views

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 ...
rgvalenciaalbornoz's user avatar
16 votes
1 answer
16k views

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" ...
Tesla's user avatar
  • 2,136
16 votes
4 answers
299 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 ...
M. Winter's user avatar
  • 30.1k
16 votes
1 answer
935 views

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 ...
user267839's user avatar
  • 7,139
15 votes
8 answers
13k 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 ...
Vera Marya's user avatar
15 votes
3 answers
10k views

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 ...
user3554004's user avatar
15 votes
1 answer
1k views

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) ...
uhoh's user avatar
  • 1,862
14 votes
3 answers
5k views

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 ...
user avatar
14 votes
1 answer
340 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 ...
user153012's user avatar
  • 12.3k
14 votes
4 answers
1k 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 ...
Ant's user avatar
  • 21.2k
14 votes
2 answers
2k 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 ...
5argon's user avatar
  • 315
13 votes
1 answer
256 views

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 ...
RavenclawPrefect's user avatar
13 votes
1 answer
323 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 ...
Jeppe Stig Nielsen's user avatar
12 votes
3 answers
11k 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 ...
user avatar
12 votes
1 answer
602 views

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....
Siddharth Bhat's user avatar
11 votes
2 answers
4k views

Curvature of a regular curve is a smooth function of parameter if it does not vanish

We have that the curvature of a curve $\gamma (t)$ is given by $K(t)=\|\gamma ''(t)\|$ iff $\|\gamma '(t)\|=1$. If $\|\gamma '(t)\| \neq 1$, then we find the arclength $s(t)=\int_0^t \|\gamma '(u)\|...
Mary Star's user avatar
  • 14k
11 votes
1 answer
2k views

How does the focus-directrix definition of a conic section apply to a circle?

One of the definitions of a conic section is that the conic section is a locus of points whose distance to the focus $F$ is a constant multiple of distance between them and the directrix $D$, i.e. $e ...
Aleksandar Stefanović's user avatar
11 votes
3 answers
667 views

Is there a generalization of the helix from $\mathbb{R}^3$ to $\mathbb{R}^4$?

The helix is a curve $x(t) \in \mathbb{R}^3$ defined by: $$ x(t) = \begin{bmatrix} \sin(t) \\ \cos(t) \\ t \end{bmatrix} $$ and it takes the classic shape: Does this have a natural extension from $\...
kdbanman's user avatar
  • 446
10 votes
1 answer
1k views

What mathematical shape is the surface of waves on water?

What is the shape of the surface of the water in the animation below? Clearly, the dots that compose the surface are following a sinusoidal path. The curve isn't a simple sine wave, since the peaks ...
Duncan C's user avatar
  • 235
10 votes
1 answer
253 views

Can any curve in 3D space be described by an intersection of two surfaces?

Can any curve in 3D space be described by an intersection of two surfaces? If not, what assumptions I need to let it be true? If this is too general, what if I restrict the scenarios to twice ...
winston's user avatar
  • 1,274
9 votes
2 answers
755 views

What formula could mimic the following curve?

For the purpose of deforming a 3D mesh, I am looking for a formula to generate a curve I could evaluate like the following: Its shape would be more or less a simplified version of wind waves over an ...
aybe's user avatar
  • 297
9 votes
4 answers
7k views

What is the difference between a function and a curve?

Are all curves functions? To my current knowledge a function gives only one output for a given input so x->f(x) , but for example a circle which is a curve is "made" out of two functions, so generally ...
zemara's user avatar
  • 145
9 votes
4 answers
468 views

The smallest distance between any point on a curve and the parabola $y=x^2$ is 1. What is the equation of the curve?

Find the equation of the function $f(x)$, where: the optimal distance between any one point $P$ on the curve $y=f(x)$ and the parabola $y=x^2$ is always equal to $1$ $f(x)>x^2$ for all $x$ (...
H. Luo's user avatar
  • 131
9 votes
1 answer
780 views

Parametric equation for a space curve

With reference to the following image: the blue curve has trivially a parametrization: $$(x, y, z) = (\cos\theta, \, \sin\theta, \, 0) \; \; \; \text{with} \; \theta \in [0,\,2\pi)$$ I would like to ...
πρόσεχε's user avatar
9 votes
2 answers
221 views

Epitrochoids and adjacent loop touching

Consider the pair of parametric equations which describe a (simplified) epitrochoid: \begin{align} x(t) &= \cos (t) - a \cos (\alpha t)\\ y(t) &= \sin (t) - a \sin (\alpha t). \end{align} Here ...
omegadot's user avatar
  • 11.8k
9 votes
2 answers
6k views

Why don't parabolas have asymptotes?

As given on the Wikipedia page, an asymptote is a line which becomes the tangent of the curve as the $x$ or $y$ cordinates of the curve tends to infinity. Hyperbola has asymptotes but parabolas ( ...
Ankit's user avatar
  • 700
9 votes
1 answer
1k views

Is there a neat way to write the parameterization of this tennis-ball-seam-like curve on the sphere?

I have been experimenting with drawing curves on the surface of a sphere (of radius 1). In order for it to lie on the sphere, every point $(x,y,z)$ must satisfy: $$ \begin{equation} \tag{1} \label{Eq1}...
Harry's user avatar
  • 541
9 votes
4 answers
2k views

Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
user239753's user avatar
9 votes
1 answer
330 views

Helices in Lorentz-Minkowski space $\Bbb L^3$.

Context: Consider the Lorentz-Minkowski space $\Bbb L^3$, with the metric: $$ds^2 = dx^2 + dy^2 - dz^2$$ (which I'll denote just by $\langle \cdot, \cdot \rangle$) If $\alpha$ is spacelike and ${\bf ...
Ivo Terek's user avatar
  • 78.1k
9 votes
0 answers
6k views

Is there a function that is the envelope of the sum of ceilings of reciprocal functions

TL;DR: Given a sum of ceilings of reciprocal functions $$y_1 = T = \sum^{n-1}_i \Big\lceil \frac{p_i}{k} \Big\rceil$$ is there a corresponding form for a function that envelopes the $T$ on the left? ...
joseville's user avatar
  • 1,497
8 votes
4 answers
1k views

how to get a nice "cosine looking" curve following the y=x direction?

My aim is to get a "cosine looking" curve rotated 45° counterclockwise. When I graph : g(x)= x + cos(x) , I get a curve that has lost the nice and regular wavering of the ordinary f(x)= cos(x) ...
user avatar
8 votes
3 answers
1k views

How to parametrize a "SpongeBob flower"

I am trying to figure out a way of generating a 5 lobed shaped like the flower sin the sky of spongebob: I.e. a parametric curve that is at least twice differentiable, closed and that transitions ...
Makogan's user avatar
  • 3,399
8 votes
3 answers
1k views

"Ordering" of Complex Plane

I have heard that the Complex Numbers do not form an ordered field, so you can't say that one number is larger or smaller than another. I've always been fine with this, but I recently wondered what ...
Brevan Ellefsen's user avatar
8 votes
6 answers
2k views

What equation can produce these curves?

Does an equation exist that can produce the curves shown in the attached image, by varying a single variable?
user120911's user avatar
8 votes
2 answers
1k views

Isn't a differential curve with corners contradictory?

While reading "Riemannian Geometry" by M. Do Carmo, I've ran into a confusing remark following the definition of a curve. The author presents the following definition (2.8, chapter 1) for a ...
NG_'s user avatar
  • 726
8 votes
1 answer
2k views

How can I construct genus 2 curves

One of the first big formula's you learn in algebraic geometry is the genus-degree formula which states that an irreducible homogeneous polynomial in $f \in \mathbb{C}[x,y,z]$ of degree $d$ gives a ...
54321user's user avatar
  • 3,243
8 votes
1 answer
876 views

A map is injective if it is nonzero at the generic point?

Proposition IV 2.1 in Hartshorne's states that if $f: X\to Y$ is a finite separable morphism of curves. Then $f^{*}\Omega_Y\to \Omega_X$ is injective. And he proves this by saying that it will be ...
Yuyi Zhang's user avatar
  • 1,452
8 votes
3 answers
377 views

A continuous curve intersects its 90 degrees rotated copy?

This is almost the same problem as in this question. However, the OP there was looking for a solution where we could assume any number of things, while I want to stick with just the given assumption (...
YiFan Tey's user avatar
  • 17.5k
8 votes
1 answer
2k views

Finding an example of "A reparametrization of a closed curve need not be closed"$\text{}$

I look at the following exercise of the book "Elementary Differential Geometry" of Andrew Pressley: "Give an example to show that a reparametrization of a closed curve need not be closed." Any ...
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