Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [plane-curves]

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.

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

1368 questions
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Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real? Batman Equation in text form: \begin{align} &\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-3}}+\left(\frac…
a_hardin
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Fractal behavior along the boundary of convergence?

The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, emeritus of Reed College) showed me 15 years ago,…
2'5 9'2
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A circle rolls along a parabola

I'm thinking about a circle rolling along a parabola. Would this be a parametric representation? $(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$ A gives us the radius of the circle, B changes the frequency of the rotations, C, of course, varies the…
futurebird
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Do wheels of a car always travel the same distance?

Consider the left wheel and the right wheel of a car (say, rear wheels). The distances two wheels travel differ when they turn left or right. But, if the car starts traveling towards the north direction and ends traveling towards the same…
govindah
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Derivation of the formula for the vertex of a parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $$y = a x^2 + b x + c$$ My teacher gave me the formula: $$x = -\frac{b}{2a}$$ as the $x$ coordinate of the vertex. I asked her why, and…
Justin L.
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What are curves (generalized ellipses) with more than two focal points called and how do they look like?

An ellipse is usually defined as the locus of points so that sum of the distances to the two foci is constant. But what are curves called which are defined as the locus of points so that the sum of the distances to three foci is constant? Trilipse?…
asmaier
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Is the complement of an injective continuous map $\mathbb{R}\to \mathbb{R}^2$ with closed image necessarily disconnected?

I am interested in the following Jordan curve theorem-esque question: Suppose that you are given a continuous, injective map $\gamma: \mathbb{R}\to \mathbb{R}^2$ such that the image is a closed subset of $\mathbb{R}^2$. The complement of the…
AnonymousCoward
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Why do the roots of $\int_0^x (1-s^2)^n ds$ lie on a lemniscate?

Consider the polynomial $$f_n(x)=\int_0^x (1-s^2)^n ds$$ for some integer $n$. I am interested in these polynomials for reasons unrelated to this question: these are the odd polynomials of which the maximum possible number of derivatives vanish at…
Wouter
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Ambiguous Curve: can you follow the bicycle?

Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance between the two wheels is $1$ then we can describe the…
user52188
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How many cubic curves are there?

It is well-known that there is only one "kind" of line, and that there are three "kinds" of quadratic curves (the nature of which depends on the sign of a so-called "discriminant"). It is noteworthy that many of the named cubic curves look rather…
J. M. ain't a mathematician
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Parametrizing implicit algebraic curves

Back in the day, I was absolutely enthralled by the study of plane curves and their properties (I have Lockwood and Zwikker to thank). I learned early on that for the purposes of generating plots on a computer (and for that matter deducing equations…
J. M. ain't a mathematician
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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 used in examples to find arc length of parametric…
Argon
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Parametric curve resembling a bean.

I am looking for a parametric closed curve that roughly resembles a bean. I am looking for something with an explicit parametrization of the form $C(t) = (X(t), Y(t))$ I tried searching online but "parametric bean" is not yielding much of use.
Makogan
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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 4/7, how would I find it?
Christopher King
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For which $L^p$ is $\pi=3.2$?

This is a silly question that came to mind after watching numberphile video on How Pi was nearly changed to 3.2. For which $p\in(1,+\infty)$ is the ratio of the perimeter of the $L^p$ disc in $\Bbb R^2$ over its diameter equal to $3.2$? EDIT 1 As…
Olivier Bégassat
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