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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312
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10answers
336k views

Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
42
votes
1answer
840 views

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, ...
35
votes
4answers
2k views

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 ...
28
votes
3answers
1k views

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 ...
20
votes
2answers
1k views

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 ...
17
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2answers
2k views

LOVE +MATH = can you read this formula?

i don't remember where exactly, i found in internet this image: i tried to replicate the formula with python and i tried this: ...
16
votes
4answers
452 views

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 ...
12
votes
7answers
6k views

Drawing heart in mathematica

It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer ...
11
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1answer
672 views

Folium of Descartes

A colleague came to me with an interesting observation: Consider the folium of Descartes, $$x^3+y^3=3axy$$ which upon implicit differentiation of the latter yields $$\frac{\mathrm dy}{\mathrm ...
10
votes
3answers
797 views

A plane algebraic curve with all four kinds of double points

During my study of plane algebraic curves, I got curious if there is a nontrivial example of a plane algebraic curve that has a node, a cusp (for my purposes I do not care which of the two kinds of ...
10
votes
2answers
297 views

A question about curves in $\mathbb{R}^2$

I need to show this result: Let $\alpha :I\rightarrow \mathbb{R}^2$ a smooth curve, where $I$ is a compact interval of the real line. If $\lVert \alpha (s) - \alpha (t) \rVert$ depends only on ...
10
votes
4answers
305 views

When is the moment of inertia of a smooth plane curve is maximum?

Given a smooth plane curve $(x(s),y(s))$, parameterized in arc length $s$, of fixed finite length $L$, its moment of inertia about its center of mass (axis perpendicular to the plane) is given as $$MI ...
9
votes
5answers
267 views

Parametric equation of $x^y=y^x$ curve

What is the easiest/most natural way to parametrize the following curve? $$\{(x,y)\in\Bbb R^2\mid x^y=y^x, x\neq y\}\cup\{(e,e)\}$$ The best I could do was taking it apart, and for $x>y$ use ...
9
votes
3answers
308 views

Hilbert curve, understanding the original article

I'm trying to read and understand the article in which Hilbert gave an illustration of a space filling curve, namely "Ueber die stetige Abbildung einer Linie auf ein Flächenstück". It's only a short 2 ...
8
votes
2answers
520 views

What type of curve is $\left|\dfrac{x}{a}\right|^{z} + \left|\dfrac{y}{b}\right|^{z} = 1$?

In order to fit experimental data, I want to use a Cartesian equation which looks like: $\left|\dfrac{x}{a}\right|^{z} + \left|\dfrac{y}{b}\right|^{z} = 1$ $a$, $b$, $c$, and $z$ can take any real ...
8
votes
2answers
541 views

Fireworks under inverse-cube gravity

What is the path of a projectile under an inverse-cube gravity law? Imagine that the law of gravity was changed overnight from $F(r) = G m_1 m_2 / r^2$ to $F(r) = G' m_1 m_2 / r^3$. To be ...
8
votes
3answers
989 views

Minimal Ellipse Circumscribing A Right Triangle

Find the equation of the ellipse circumscribing a right triangle whose lengths of it's sides are $3,4,5$ and such that its area is the minimum possible one. You may chose the origin and orientation ...
8
votes
1answer
140 views

Is it possible for a Jordan curve in the plane to enclose a set with area zero?

I read about the Isoperimetric Inequality the other day. It says that for any Jordan curve, $$ \frac{4 \pi A}{L^{2}} \leq 1, $$ where $ L $ is the length of the curve and $ A $ is the area of the ...
8
votes
2answers
210 views

Where do people learn about things like caustics, evolutes, inverse curves, etc.?

When I look up a curve on Wikipedia, I'll often see a lot of properties along the lines of "you can generate curve X by rolling a circle along curve Y and tracing the trajectory of a single point," or ...
8
votes
1answer
602 views

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 ...
8
votes
1answer
2k views

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 ...
8
votes
1answer
510 views

Comprehensive compilation of conic section formulae

My frustration started after hours of searching failed to turn up a formula for the vertex of a parabola in the general form $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ As is already well known, the discriminant ...
7
votes
6answers
2k views

A Math function that draws water droplet shape?

I just need a quick reference. What is the function for this kind of shape? Thanks.
7
votes
1answer
329 views

Circumference of a superellipse?

Could someone help me formulate the circumference of a superellipse? $$\frac{x^n}{a^n} + \frac{y^n}{b^n} = 1$$ If it makes things easier, I'm considering only the cases $n>2$, and ...
7
votes
1answer
155 views

Is there a $C^1$ curve dense in the plane?

Is there a curve $\gamma : \mathbb{R} \to \mathbb{R}^2$ injective and $\mathcal{C}^1$ whose range is dense in $\mathbb{R}^2$?
6
votes
12answers
9k views

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 $x$ ...
6
votes
2answers
572 views

Finding standard ellipse characteristics from specific ellipse parametrisation

I have found the following ellipse representation $(x,y)=(x_0\cos(\theta+d/2),y_0\cos(\theta-d/2))$, $\theta \in [0,2\pi]$. This is a contour of bivariate normal distribution with uneven variances and ...
6
votes
3answers
270 views

Can closed curves have small curvature?

Let $\gamma$ be a smooth curve in Euclidean space of length $2\pi$ whose curvature function satisfies $-1 < k(t) < 1$. Can $\gamma$ be closed? This seems like it should be an easy exercise, at ...
6
votes
2answers
390 views

The Dido problem with an arclength constraint

It is well known that the solution to the classical Dido problem is a semicircle, and that the solution to the classical isoperimetric problem is a circle. It's also reasonably obvious that the ...
6
votes
2answers
494 views

Deriving an implicit Cartesian equation from a polar equation with fractional multiples of the angle

Usually, applying the conversion formulae $r^2=x^2+y^2$ $\cos\;\theta=\frac{x}{r}$ $\sin\;\theta=\frac{y}{r}$ to transform an equation in polar coordinates to an implicit Cartesian equation is ...
6
votes
1answer
217 views

Does every continuous everywhere but differentiable nowhere curve have an infinite length?

Given a curve $\!\,\gamma : [a, b] \rightarrow ℝ^2$ that is continuous everywhere but differentiable nowhere (or almost nowhere), is its length: $$\text{length} (\gamma)=\sup \left\{ \sum_{i=1}^n ...
6
votes
1answer
234 views

Problem with calculating a winding number

I have a problem with calculating the winding number $n\left ( \gamma ,\frac{1}{3} \right )$ of the curve $\gamma :\left [ 0,2\pi \right ]\rightarrow \mathbb{C}, t \mapsto \sin(2t)+i\sin(3t)$. ...
6
votes
1answer
176 views

Examples of stable curves $g\geq 2$?

I'd like to get my hands on some simple examples of families of stable curves. Ideally these would come in the form of a projective curve $C$ over a 1 dimensional base $B$, say $B = \mathbb{A}^1$. ...
6
votes
2answers
267 views

Need help with Curves and parameterizations

I'm having some trouble solving a couple of problems: I know this one must be pretty easy but can't find the way to solve it. I need to find the arc length of a curve described by $ r=1- ...
5
votes
6answers
1k views

Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?

Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
5
votes
4answers
403 views

Curvature of planar implicit curves

I am trying to understand how the curvature equation $$\kappa = -\frac{f_{xx} f_y^2-2f_{xy} f_x f_y + f_x^2 f_{yy}}{(f_x^2+f_y^2)^{3/2}}$$ for implicit curves is derived. These curves arise from ...
5
votes
3answers
3k views

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 ...
5
votes
3answers
288 views

Division of Other curves than circles

The coordinates of an arc of a circle of length $\frac{2pi}{p}$ are an algebraic number, and when $p$ is a Fermat prime you can find it in terms of square roots. Gauss said that the method applied to ...
5
votes
1answer
328 views

What is the path equation that is created with the middle point of a fixed length line segment that touching both ends to an ellipse.

Ellipse equation is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ and the length of line segment is $2k$, if we move the line segment all around of the ellipse while touching both ends to the ellipse. What is ...
5
votes
1answer
938 views

Formula for curve parallel to a parabola

I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
5
votes
2answers
690 views

Tractrix-like curves

Is there a common name for curves, obtained from dragging a point along another curve, similar to how tractrix is obtained by dragging a point along a line? What is a parametric equation of such ...
5
votes
2answers
1k views

Is $r=2\cos(\theta)$ a one-petal polar function?

I'm currently learning about polar functions and their graphs in precalculus, and one of the questions on my homework is to identify the shape of the function $r=2\cos(\theta)$. We were taught that ...
5
votes
2answers
737 views

How do I draw an elliptic curve?

I can draw a circle using a compass. I can draw an ellipse using two focal points and a loop of string. I think that you can draw an arbitrary conic with a "generalized" compass for which the pencil ...
5
votes
3answers
121 views

when is the region bounded by a Jordan curve “skinny”?

How can I formalize and prove the following intuition?: Picture a very skinny rectangle, one with base length 1 and sides length $\epsilon$. Or imagine a very flattened ellipse. The interiors of ...
5
votes
1answer
77 views

Relations between curvature and area of simple closed plane curves.

Let $\gamma$ be a simple closed plane curve. We know that a curve with constant curvature $\kappa$ will trace a circle in the plane. The radius of this circle is the inverse of its curvature. Now, ...
5
votes
2answers
78 views

What is the slope of the tangent at $(0,0)$ on the curve $x^2 y^2 = 4 x^5 + y^3$

This question is arising from the answer to another one: How find this equation integer solution: $x^2y^2=4x^5+y^3$ . For $x < 27$ and $y > -243$ , the basic equation $x^2 y^2 = 4 x^5 + y^3$ is ...
5
votes
3answers
323 views

Level sets of convex functions

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a convex function. For $t\in\mathbb{R}$, consider the corresponding level set $$f^{-1}\{t\}=\{(x,y)\in\mathbb{R}^2: f(x,y)=t\}.$$ For the application I ...
5
votes
2answers
1k views

On deriving the arclength of a hyperbola

In my attempts to derive the closed form for the arclength of the hyperbola, I wound up with the following integral: $$\int\frac{\sqrt{1-m\;\sin^2 u}}{\sin^2 u}\mathrm{d}u$$ I am aware that such ...
5
votes
0answers
114 views

curve which crosses itself at every point

Does there exist a continuous curve $C$ on $\mathbb{R}^2$ that crosses itself at every point on $C$? For two pieces of curve to cross and not just touch, each must have some length either side ...
5
votes
0answers
130 views

What is the parametric equations for the following closed curves?

First case Second case For the sake of simplicity, let the circle of radius $R$ be at the origin, the rectangle width and height be $w$ and $h$, respectively.