Plane curves are continuous (or smooth) functions $\gamma\colon I\to\mathbb R^2$ from a real interval to the plane. Sometimes also the image $\gamma(I)$ is called curve.

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Curve fitting using a graph extracted from an article, is it possible?

I want to curve fit a graph from an article which I can only extract from the pdf file as a screenshot. Therefore, I do not have the coordinates of the data points explicitly, yet I know that the ...
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1answer
34 views

$dx$-notation in analysis

In the context of integrals and differential equations, often the symbol $df$ or $dy$ appears, where in some previous steps $f$ and $y$ were functions. What do these symbols mean $df$ and $dy$? ...
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45 views

Showing that a specific curve is regular.

Define the curve by $c(t):=(sin(pt)+r)(cos(qt),sin(qt))$ for $p,q \in \mathbb{Q}$ and $r\in \mathbb{R}$. Determine for which $p,q$ is the curve regular, i.e. $c'(t) \neq (0,0)$ for any $t\in ...
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40 views

Total curvature of a piecewise smooth planar curve

I am trying to show that the total curvature of a piecewise smooth planar curve $\alpha$ has the property that $\displaystyle \int_{C}\kappa $ $\text{d}s + \displaystyle \sum_{j=1}^{l} \epsilon_{j} ...
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21 views

Is there a unique solution?

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a given continuous function and $t_0\in (a,b)$ a fixed point. Is it true that the following problem has a unique continuous solution ...
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21 views

Relative motion between line/circle

By rolling a (line,circle) on a fixed (circle, line) in $\mathbb{R}^2$, the locus of a point on the former is an (involute, cycloid). By what procedure can we inter-convert their parametric ...
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1answer
32 views

Focus of a rolling parabola traces a catenary - geometric explanation

It is known that the focus of a rolling parabola along the x-axis traces a catenary. I'm interested in a geometric explanation. I found this derivation in The surfaces of Delaunay by J. Eells in The ...
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27 views

Another type of primitive

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $\Vert\mathbf{v}(t)\Vert=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}\colon (a,b)\to\mathbb{R}^2$ so ...
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1answer
25 views

Another type of derivative, another type of differential equation

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Is it possible to find a continuous function $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so ...
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21 views

A curve is contained in a circle [duplicate]

I need to prove that if $\alpha: I\rightarrow \mathbb{R}^{2}$ is regular with curvature $\kappa$, then $\alpha$ is on a circle with radius $r>0$ if and only if $|\kappa(t)|=\dfrac{1}{r}$.
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3answers
38 views

Chess Board Coloring of a Paper using an Arbitrary Curve

Pick a piece of paper and a pen. Put the pen on a starting point and begin to draw an arbitrary curve and don't withdraw your hand until you reached the starting point. You can meet your curve during ...
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44 views

Tangent vector for a curve defined by a discrete set of points

I have a curve defined by a discrete set of points (x,y). How can I approximate the tangent vector at a point for such a curve?
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1answer
30 views

How can I get a smooth distortion on a circle with a function g(x,y)

Let's say, $$f(x,y)=x^2+y^2=1$$ gives the unit circle. Now I would like to get a smooth distortion on the circle with a function $g(x,y)$. my guess is to consider the perimeter as one dimension, so ...
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34 views

Cusp singularity

What is the most general way of defining cusps singularities for a continuous curve $\mathbf{r}:(a,b)\to\mathbb{R}^2$ and how can we define them? What are the minimum requirements for the curve ...
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1answer
13 views

Image of a locus via stereographic projections

Yesterday evening I was playing around in my head with stereographic projections and I've come up with this idea. Let $\gamma(t)=(x(t),y(t))$ be a certain curve on a plane. Define a new curve ...
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1answer
25 views

Maximum of the length of a curve and its curvature

I'm trying to solve this problem but I can't find any way to do it. Some hints or helps will be very useful and I'll be very thankful. The problem says: Let $\alpha : (a,b) \rightarrow \mathbb{R}^2$ ...
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1answer
20 views

troublesome area problem

Calculate the area of the region of the graph bounded by: $$\begin{eqnarray} y &=& x \\ y &=& x^2 + 1 \\ y &=& 2 \\ x &=& 0 \end{eqnarray}$$ My final result is ...
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10 views

Equation for a Vinyl curve

This video seems to show an explicit map from the torus to $\mathbb{R}^2$. Does it factorize through the projection $\mathbb{R}^3 \to \mathbb{R}^2$? What is the equation of the curve?
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91 views

Differential geometry: Conformal map

Let $f:\mathbb{R}_{>0} \times (0,2\pi) \rightarrow \mathbb{R}^3$ $$f(t,\xi) := (r(t) \cos( \xi) , r(t) \sin(\xi),z(t))$$ be a surface of revolution, where we assume that $r>0$ and ...
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2answers
22 views

Show that if $v_1$ and $v_2$ are any two vectors in this plane, then for any two scalars, $c_1v_1 + c_2v_2$ is also a vector in the plane

Let $a,\,b$ and $c$ be constants (not all zero) and consider the equation $ax + by + cz = 0$, which has a graph that is a plane that passes through the origin in $\mathbb{R}^3$. Show that if $v_1$ and ...
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1answer
13 views

Distance between two skew lines

I have 2 skew lines $L_A$ and $L_B$ and 2 parallel planes $H_A$ and $H_B$. The line $L_A$ lies in $H_A$ and $L_B$ in $H_B$. If the equations of $H_A$ and $H_B$ are given like this: $x+y+z = 0$ (for ...
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1answer
41 views

Torsion vs Curvature of a curve

I was wondering if there is a simple explanation of the torsion and curvature in $\mathbb{R}^3$ of a curve. The curvature measure somehow the acceleration perpendicular to the tangent vector, but ...
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1answer
37 views

Does every plane curve contain a rational point?

Does every plane curve contain a rational point? I think the answer is yes, but I can not prove this. Please help. However, if it is possible to build a pathological curve - without rational points, ...
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1answer
18 views

Question on a function defined on some plane curve.

Let $C$ be the plane curve defined by $y^d =f(x)$, where $f(x) \in \mathbb{C}[x]$ is a polynomial. Let $g(x,y)$ and $h(x,y)$ are polynomials relatively prime with each other. We consider $r(x,y) ...
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1answer
36 views

Prove that $z = tx + (1 − t)y$ if $d(x, y)= d(x, z) + d(z, y)$

Let $x,y,z$ be elements of $\mathbb{R}^2$ Prove that $z = tx + (1 − t)y$ if $d(x, y)= d(x, z) + d(z, y)$ d is usual euclidean metric.
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3answers
68 views

finding the closest distance between a point a curve

consider the curve $y=x^2$ what are the points on the curve that are the closest to the point $(1,0)$ using calculus I got the two points but what is the connection between normals and the closest ...
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1answer
30 views

What kind of a curve can represent a physical trajectory

It is very well known that conics, spirals, etc. can represent a realistic trajectories of point particles. However, a physical trajectory can also intersect itself, have a cusp, and other kinds of ...
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14 views

How to find a spirograph that approximates another curve?

A spirograph is a curve of the form $$\vec r(t)= \sum_n (a_n \cos b_n t\ i + a_n \sin b_n t\ j)$$. This class of curves includes things like epi/hypocycloids and trochoids, but also a number of other ...
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2answers
129 views

Trains describing a parabola

From the train station – the point S – originante two tracks, i.e. rays, which do not lie on a common straight line. Along these move two trains, which are line segments. On the first track a train is ...
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1k views

Why not use two vectors to define a plane instead of a point and a normal vector?

In Multivariable calculus, it seems planes are defined by a point and a vector normal to the plane. That makes sense, but whereas a single vector clearly isn't enough to define a plane, aren't two ...
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65 views

What is this kind of geometry called?

I want to get Cartesian coordinates of the points of a curve (e.g. a bezier curve) based on the distance (e.g length of the arc) from the start point on the curve. To make this more clear, suppose I ...
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64 views

Newton's Investigation of Cubics: Generalization?

I recently read about Newton's investigation of cubic curves, and how, like for quadratic curves we can classify them into parabolas, ellipses and hyperbolas, Newton was able to classify cubic curves ...
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25 views

Finding the formula of a curve

Is it possible to find the simple formula for curves? For example, you want to plot the letter "A" , is it possible to find a formula to graph letter "A" on the coordinate system? To the best of my ...
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1answer
58 views

Find two planes which are parallel to each other, given that each plane contains a line (just a summary of the question)

Let L be the line through (1, 2, 3) and (3, 1, 2) and let L' be the line through (1, −1, 1) and (0, 2, 1). a) find Find the equation of a plane π containing L, and parallel to a plane containing L'. ...
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1answer
55 views

Why does the arc-length formula have form $\int_a^b\left|\left|\frac{d\vec{f}(t)}{dt}\right|\right|_2dt$ for C1 curves?

This discussion focuses on $\mathcal{C}^1$ curve on $\mathbb{R}^n$. But feel free to talk about the case where we only have a continuous curve or the scenario with a manifold with a metric in general. ...
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24 views

Determine the number of real roots of the system.

Determine the number of real roots of the system,$1.$$x^3y - y^4 =a^2$ $2.$$x^2y+2xy^2+y^3=b^2$ where $a$ and $b$ are real parameters.
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1answer
75 views

Give a closed plane curve C with k (curvature) > 0 that is not convex, Draw closed plane curves with rotation indices 0, 2, -2, and 3

1.) Give a closed plane curve C with k (curvature) > 0 that is not convex. can someone please explain these concepts to me. How can you have a closed plane curve like this? Do you used signed ...
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1answer
18 views

Finding family of curve for given asymptotes

I need to find possible curves, with asymptotes given as $x=0 (x \to -\infty)$ and $y=mx \hspace{0.5cm} m>0$. it is easy to find curves for individual lines, $y= \exp(-\lambda_1 x) + mx$ for $y=mx$ ...
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28 views

About the conditions in Jordan's Curve Theorem

In the original formulation of the theorem, it was stated that a Jordan curve separate the plane in two sets that is not path-connected. The formulation in Wikipedia is that the Jordan curve separate ...
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1answer
36 views

What is the Hilbert curve's equation?!

The Hilbert curve has always bugged me because it had no closed equation or function that I could find. What is its equation or function? For example, if I wanted to find the Hilbert's curve point at ...
2
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2answers
40 views

Is there a thing named a “spiral plane” which is a plane but it's spiral?

Hello, I'm wondering if there is such thing like this. Is there a plane which is not flat but spiral and extending for infinity? I have drawn a representation for what I mean but it's not thorough ...
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32 views

How to translate the polar curves up/down and right/left without referring to Cartesian equations?

If I have a polar equation $r(\theta)$, how can I translate it up/down and right/left? We can do this by converting the equations into Cartesian equations and do the translations we want and then ...
2
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0answers
37 views

Tractable indefinite integral of the exponentiation of some function

Consider the function $z(s)\in\mathbb{C}$ defined as $z(s)=\int_0^s \exp\left[i(q u+\lambda(u))\right]du$ for some $q\in\mathbb{Q}-\mathbb{Z}$ and $\lambda(s)$ a $2\pi$-periodic real differentiable ...
4
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2answers
146 views

Nontrivial solutions of $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$

Let $a=(a_n)$ with $a_n\in\mathbb{C}$ be a vector indexed over all $n\in\mathbb{Z}$, and consider the system of equations $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$ for all ...
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38 views

Cycles non-homologous but with the same winding number at each point outside?

Problem (Fulton's Algebraic Topology: A First Course, Problem 6.28) Can we find a closed set $X\subseteq\mathbb R^2$, such that there's a cycle $\gamma$ in $X$ not homologous to zero but for each ...
3
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1answer
50 views

A generalization of Jordan curve theorem to connected open sets in the plane

Problem (Fulton's Algebraic Topology: A First Course, Problem 5.23) Let $U\subseteq\mathbb R^2$ be any connected open set in the plane. If $X\subseteq U$ is homeomorphic to $[0,1]$, then ...
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1answer
35 views

Tractrix exercise

Exercise: Let $$\begin{align*}\gamma:(0,\pi) &\to \mathbb R^2\\ t &\mapsto \gamma(t)=(\sin t,\ \cos t+\log \left(\tan(\frac{t}{2}) \right), \end{align*}$$ be the parametrized curve of the ...
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1answer
29 views

Cycloid arc lenght question

I've parametrized the cycloid with the function $$\gamma(t)=(\cos(\frac{3}{2}\pi-t)+t,\sin(\frac{3}{2}\pi-t)+1) ; \space t \in [0,+\infty)$$ I am asked to find the arc lenght of the curve which ...
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46 views

Problem with calculating winding number in sum of curves

Let $$ \begin{align} \gamma= \gamma_1 +\gamma_2+\gamma_3,\\ \gamma_1(t)=e^{it}, t\in[0,2\pi] \\ \gamma_2(t)=-1+2e^{-2it}, t\in [0,2\pi]\\ \gamma_3(t)=1-i+e^{it},t\in [\frac{\pi}{2},\frac{9\pi}{2}] ...
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1answer
30 views

winding number of $\gamma$ and point exterior to $\gamma$

$$ \begin{align} n(\gamma,z_0)=\frac{1}{2\pi i}\int_\gamma\frac{1}{z-z_0}dz . \end{align} $$ Is it safe to say that $n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^*$