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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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
30 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
11 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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26 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
34 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 ...
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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 ...
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2answers
137 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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2answers
390 views

Why does an equiangular spiral become logarithmic (intuitively)?

One of the most famous 2D-curves are logarithmic spirals (or Spira mirabilis). They can be constructed by using a machinery that ensures a constant angle between the tangent and the radial lines all ...
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2answers
325 views

Why do definitions of distinct conic sections produce a single equation?

I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
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1answer
295 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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20 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 ...
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36 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 ...
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0answers
33 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 ...
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1answer
46 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
31 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
18 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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19 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
25 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 $n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^*$
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2answers
50 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 ...
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1answer
22 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 ...
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1answer
38 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?
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42 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 ...
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42 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?
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36 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 $\textbf{c}$
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1answer
242 views

fixed point projective geometry

I am thinking about the following: Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
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1answer
43 views

How can we find the arc length of the curve? [closed]

How can I find the length of the curve $$\left(\frac{t^3}{3} - t\right)\mathbf{i}+ t^2 \mathbf{j}, \quad 0≤t≤1?$$
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25 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 ...
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430 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 ...
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557 views

Using equations to draw out complex objects

How do people come up with equations of curves to draw out complex objects? Some popular examples would include: batman curve & PSY curve. This stackexchange link explains the rationale for the ...
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1answer
52 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 ...
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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 $\mathbb R^2$ ...
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267 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 ...
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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 $C^1$ ...
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2answers
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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, ...
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35 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 ...
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How to find the parametric equation of a cycloid?

"A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia In many calculus books I have, the cycloid, in parametric form, is ...
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3answers
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Polar to Parametric Equation?

I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right. Curve C has polar equation ...
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31 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 ...
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1answer
60 views

Osculating circles intersecting a given point

Well, the problem is a question in Montiel's book. How to prove that a planar curve $\alpha$ such that all osculating circles intersects a given point is actually a circle (or a part of it)? I've ...
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4answers
133 views

Proving $\dfrac{dN}{ds}=-\kappa T+\tau B$

Currently revising for a differential geometry exam. The question I am working on is one of those types where the next part of the question follows from the last. I've gotten to the point where I have ...
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2answers
56 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
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106 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 ...
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161 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 ...
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32 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?
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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 $\gamma^{\prime}(t) \neq 0 \;\; \forall t \in [a,b]$ ...
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Adjustable Sigmoid Curve (S-Curve) from (0,0) to (1,1)

I feel like this is such a simple question but I am at such a loss. I currently have a set of values that I would like to weigh by an S Curve. My data ranges from $0$ to $1$ and never leaves those ...
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11 views

Difference between containing point and pass through point?

I do not understand this, What is the difference between the equation of the plane containing the points and the equation of the plane through the point? Is it the same thing or are they different?
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1answer
50 views

injective curve inside curve

I am struggling to prove the following intuitive result: Take $\phi:[a,b]\rightarrow \mathbb{R}^{n}$ a continuous mapping with $\phi(a)\neq\phi(b)$. Then there is a continuous injective mapping ...
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302 views

How to find the length of a curved path.

We have to find a continuous model for a curved path which you then solve. A woman is running in the positive y-direction starting at x=50 (50,0) which is orthogonal to the x axis. At this point a dog ...
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Rolfsen exercise, chord theorem

Here's a problem from Rolfsen's Knots and Links that has me scratching my head: Show that there is always a counterexample to the "chord theorem" if $n$ is not an integer. [Hint: In attempting to ...