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Questions tagged [curves]

For questions about or involving curves.

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50
votes
5answers
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 ...
37
votes
5answers
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:
24
votes
2answers
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 ...
23
votes
2answers
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 ...
23
votes
4answers
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 ...
20
votes
5answers
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 ...
19
votes
1answer
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 ...
17
votes
1answer
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 ...
16
votes
4answers
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 ...
16
votes
1answer
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 ...
15
votes
4answers
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 ...
14
votes
8answers
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 ...
14
votes
1answer
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 ...
12
votes
1answer
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 ...
11
votes
1answer
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 ...
10
votes
2answers
2k 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)\|...
10
votes
1answer
474 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 ...
10
votes
1answer
474 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 ...
9
votes
3answers
2k 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 ...
9
votes
1answer
223 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}...
8
votes
2answers
574 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 ...
8
votes
6answers
1k 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?
8
votes
2answers
542 views

Curvature of curve: equivalence between tangent vector and angle definitions

I know that curvature for some curve $C$ defined parametrically is: $$\kappa=\left\|{d\vec{T}\over ds}\right\|$$ Which basically is the rate at which the tangent vector to the curve changes, as the ...
8
votes
2answers
111 views

How to find the area of a region bounded by a simple closed curve?

I have the following equation: $$ \frac{p}{(a-x)^2+y^2}+\frac{1-p}{(b-x)^2+y^2}=1 \text{ where } 0\leq p\leq 1 $$ Which represent a simple close curve. Obviously, when $p=0,p=1$ or $a=b$ we recover a ...
8
votes
3answers
211 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 (...
8
votes
1answer
138 views

'Spiky Periodic Things' - Do these objects have a name, and is there a method for finding the boundary curves?

This question was originally about evaluating the sum $\sum_{n=0}^\infty e^{nix}$, but I figured out the answer about half way through writing it. So instead, I decided to ask a slightly different ...
8
votes
1answer
266 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 ...
8
votes
0answers
202 views

Variational gradient descent on the space of curves

I want to find a curve $\gamma : [0,T] \to \mathbb{R}^n$ that minimizes the following functional $$J[\gamma]=\int\limits_0^T F(\gamma(t),\gamma'(t))\,dt$$ Suppose I know the endpoints, $a, b \in \...
7
votes
2answers
961 views

Is friction necessary for a Tractrix Curve?

Is friction necessary for a Tractrix Curve? (ANSWERED) 1.If friction is necessary, *what curve will the particle trace if friction is not present?* (NOT ANSWERED) 2.If friction is not necessary, *...
7
votes
3answers
778 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 ...
7
votes
6answers
3k views

How to know when a curve has maximum curvature and why?

This is a question about understanding the concept of curvature. Firstly, what exactly is curvature of a curve (not the formula, what does it actually mean conceptually)? Secondly, I am confused ...
7
votes
4answers
140 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 $x$ (...
7
votes
1answer
9k 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" ...
7
votes
1answer
579 views

How to understand this example in Do Carmo?

I'm reading the book $Riemannian$ $Geometry$ written by Do Carmo. Here is an example in which I cannot understand the explanation he gave. I really don't understand what he said about why $\alpha$ is ...
7
votes
2answers
2k views

How to parametrize the curve $y^2 = x^2(x+3)$

I am emberassed to ask this, but I couldn't find a way. I want to write the curve $y^2 = x^2(x+3)$ as $$y=f(t) \quad \quad x=g(t) \quad \quad t \in \mathbb R$$ I guess I have to do something like $...
7
votes
1answer
307 views

Questions about torsion of a curve in $\mathbb{R}^3$ and analogues of torsion in higher dimensions

Suppose we have a curve $\alpha(s) : I \to \mathbb{R}^3$ parametrized by arc-length that has nowhere-vanishing second derivative, so that we are able to define the torsion $\tau(s)$ for every $s \in I$...
7
votes
1answer
828 views

Show that $\textbf{$\gamma$}$ lies on a sphere of radius $r$

Let $\textbf{$\gamma$ }(t)$ be a unit-speed curve with $\kappa (t) > 0$ and $\tau (t) \neq 0$ for all $t$. Show that, if $\textbf{$\gamma$}$ is spherical, i.e., if it lies on the surface of a ...
7
votes
1answer
248 views

Proof regarding the minimum of a function

I want to show that the function $$f(x)= \frac{2x}{1+(\frac{1}{k-x})^a} + \frac{k-2x}{1+2\cdot(\frac{1}{k-x})^a+ (\frac{1}{k-2x})^a}$$ with $$ x \in [0,\frac{k}{2}], a \in [0,\infty], k \in [0,1]...
7
votes
1answer
722 views

Orthogonal differentiable family of curves

This problem is out of section 4-4 in M. do Carmos' Differential Geometry of Curves and Surfaces: We say that a set of regular curves on a surface $S$ is a differentiable family of curves on $S$ if ...
7
votes
1answer
118 views

Equivalence of Continuous Monotonic Functions $[0, 1] \to [0, 1]$

A couple of later clarifications to my original post ... the monotonic non-decreasing functions mapping $[0, 1] \to [0, 1]$ are surjective - it appears from the comments that this was misunderstood. ...
7
votes
1answer
199 views

The topology of $y^2=(x-1)(x-2)(x-3)(x-4)$

Andreas Gathmann's lecture notes on algebraic geometry start by considering the curve $C_n=\{(x,y): y^2 = (x-1)(x-2)\cdots(x-2n)\} \subset \mathbb{C}^2$. He claims that the topology of this curve is ...
7
votes
0answers
82 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 ...
7
votes
4answers
1k 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 ...
7
votes
0answers
113 views

Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
6
votes
3answers
1k views

Is the metric on the circle, induced from the plane, not a flat one?

My question concerns the highlighted part posted below, from Wikipedia article. (Link to the revision at the time of this post.) I'd say I can't detect the curvature of the unit circle if I go along ...
6
votes
3answers
2k views

Find the exact values of m.

I'm in an American high school but my teacher gave me some IBDP questions and I'm struggling to solve one of them: Find the exact values of $m$ for which the line $y= mx+3$ is tangent to the curve ...
6
votes
1answer
721 views

Show a curve is not a regular surface

A set $S\subseteq\mathbb{R}^3$ is said to be a regular surface, if for every $p\in S$ there exists an open ball $B_p(r)$ around $p$, an open set $A\subseteq\mathbb{R}^2$ and a function $f:A\to\mathbb{...
6
votes
3answers
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 ...
6
votes
3answers
969 views

Closed periodic curve

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{$\gamma$}...
6
votes
1answer
223 views

Are lightlike curves in the De Sitter space straight lines?

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 ...