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.

learn more… | top users | synonyms

0
votes
0answers
15 views

curve of class C1

Let $\mathbf{r}:(a,b)\to\mathbb{R}^2,\ r(t)=(x(t),y(t)), x,y\in C^{1}((a,b))$ be a non-constant function on any interval contained in $(a,b)$. Is it true that, if the following two limits exist: ...
2
votes
1answer
39 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 ...
0
votes
1answer
32 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, ...
1
vote
1answer
17 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) ...
1
vote
1answer
33 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.
0
votes
3answers
54 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 ...
1
vote
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 ...
0
votes
0answers
6 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 ...
2
votes
2answers
126 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 ...
9
votes
2answers
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 ...
0
votes
2answers
63 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 ...
4
votes
0answers
54 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 ...
0
votes
0answers
21 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 ...
0
votes
1answer
46 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'. ...
1
vote
1answer
54 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. ...
0
votes
0answers
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.
-1
votes
1answer
65 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 ...
0
votes
1answer
14 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 $y=mx$ ...
0
votes
0answers
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 ...
0
votes
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
votes
2answers
39 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 ...
0
votes
0answers
26 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
votes
0answers
35 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
votes
2answers
144 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 ...
3
votes
0answers
37 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
votes
1answer
49 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 ...
0
votes
1answer
32 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 ...
1
vote
1answer
23 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 ...
0
votes
0answers
33 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}] ...
0
votes
1answer
29 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 $n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^*$
1
vote
1answer
24 views

Problem in showing that contours $\gamma_2$ is equivalent to $ \gamma $

Let $\gamma_2(t)= e^{-it^2}, t\in[0,\sqrt{2\pi}]$ and $\gamma(t)=e^{2\pi it}, t\in[0,1]$ Show that $\gamma_2 \sim \gamma $. I think that for the latter to be true $\gamma_2$ should be ...
0
votes
1answer
39 views

Does $\gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}]$ represent the unit circle?

$$ \begin{align} \gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}] \end{align} $$ Is my thinking correct that $\gamma$ represents the unit circle correct?
0
votes
1answer
46 views

Image of a parametrised curve and its geometrical meaning

$\gamma_1(t)= t^2 + i\, t^4 , t\in [ -1, 1]$ which is the Image of this curve and / or what is the geometrical description of the set of points $ z : z=\gamma_1(t)$ I am helping a friend study ...
2
votes
1answer
45 views

There is at least one straight line that can bisect both the perimeter and the area of the curve. [duplicate]

Consider a closed curve of finite length. There is at least one straight line that can bisect both the perimeter and the area of the curve. Why is this statment true?
0
votes
1answer
37 views

The motion of the particle satisfies $\textbf{v} = \textbf{c}\times \textbf{r}$

Why is the path is contained in a circle that lies in a plane perpendicular to $\textbf{c}$ with centre on a line through the origin in the direction of $\textbf{c}$
0
votes
2answers
83 views

Finding the equation for a (inverted) cycloid given two points

If I have two points on a Cartesian plane, and I know that they are connected by a cycloid, then how do I find the equation for that cycloid? For background information, I have been playing around ...
0
votes
0answers
33 views

How to find the tangent vector to a curve at a special point

The two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given. The line segments $AC$ and $BC$ make equal angle $\alpha$ with the horizontal plane through $C$. The angle ...
3
votes
1answer
63 views

How is equation art created [duplicate]

A friend of mine sent me a Wofram Alpha link to a parametric curve that creates a detailed drawing of a game character. The parametric equation that draws it is about ~9 screens tall on my ...
1
vote
0answers
19 views

How to derive Jordan curve theorem for three arcs from the two arc version?

The Jordan curve theorem can be stated as follows: Let $p,q\in\mathbb R^2$, $p\ne q$ and $a,b$ be arcs between $p$ and $q$ intersecting only in the endpoints. Then $a\cup b$ divides $\mathbb R^2$ ...
3
votes
2answers
293 views

What is a French curve, as mentioned by Feynman?

I'm reading "Surely You're Joking, Mr. Feynman!", he says: I often liked to play tricks on people when I was at MIT. One time, in mechanical drawing class, some joker picked up a French curve (a ...
2
votes
0answers
60 views

A variational strategy for finding a family of curves?

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ ...
0
votes
0answers
38 views

Curve avoiding semi-rational points

A rational point is a point in $\mathbb{R}^d$ all of whose $d$ coordinates are rational. Let me define a semi-rational point as one that has at least one rational coordinate (but whose other ...
1
vote
2answers
83 views

Retrieve the initial cubic Bézier curve subdivided in two Bézier curves

I have a cubic Bezier curve subdivided to two cubic Bezier: Assuming that "t_cut" is the t value where this initial Bezier is cut: example of function subdivision(BezierCurve initialCurve, ...
9
votes
7answers
440 views

Constructing a family of distinct curves with identical area and perimeter

Two recent questions were posed by Arjuba [1] [2] asking for counter-examples regarding whether two different figures could have the same perimeter and area. Responders quickly raised a number of such ...
0
votes
0answers
47 views

Maximum area enclosed by a string attached at fixed points

Two fixed points A and B have a string of length L attached between them. Supposing that the string does not intersect the line segment AB, then the string and AB will form a closed figure. What ...
0
votes
2answers
59 views

Suppose $(x,y,z)$, $(1,1,0)$, and $(1,2,1)$ lie on a plane through the origin.

What determinant is zero? What equation does this give for the plane? I need some help here, am pretty stuck
3
votes
0answers
113 views

Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
0
votes
2answers
36 views

Lattice Points in x-y plane

What are Lattice Points? Which points in x-y planes are Lattice Points? Is (m,n) a lattice point where m,n are any integers?
3
votes
0answers
176 views

Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question. Consider two smooth plane curves $C \equiv ...
0
votes
1answer
35 views

Two definitions for a smooth curve equal.

I've encountered these two definitions: 1. $\gamma\colon [a,b]\longrightarrow\mathbb{R^3}$ is smooth if all three derivatives exist and $\gamma^{\prime}(t) \neq 0 \;\; \forall t \in [a,b]$ ...