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

392
votes
11answers
406k views

Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
3
votes
2answers
2k 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 ...
46
votes
4answers
3k 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 ...
13
votes
7answers
11k 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
votes
1answer
4k views

Why does the focus of a rolling parabola trace a catenary?

I keep hearing that when one rolls a parabola on a straight line, the focus traces a catenary. I kept trying to find a proof on the Internet, but no dice. How does one prove this to be true?
10
votes
7answers
543 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 ...
7
votes
1answer
1k views

Calculating equidistant points around an ellipse arc

As an extension to this question on equiangular fisheye distortion, how can I calculate equidistant points around an ellipse (or 1/4 segment of) given it's aspect ratio? When it's circular, I can use ...
13
votes
13answers
19k 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 ...
2
votes
2answers
1k views

Parallel functions.

In 2 dimensions, we can draw 2 parallel lines that have the same distance from a line. I wanted to find parallel functions of a function and their distance is $d$ to the function for all inputs and ...
15
votes
1answer
3k 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 ...
7
votes
3answers
1k 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 ...
7
votes
1answer
905 views

How to place objects equidistantly on an Archimedean spiral?

To place objects equidistantly on an Archimedean (arithmetic) spiral, the arc length of the spiral has to increase linearly between the objects. This is what I have so far: The length of a spiral is ...
8
votes
2answers
584 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 ...
60
votes
2answers
5k 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, ...
11
votes
5answers
537 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
1answer
226 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 $\mathbb{R}^2$?
8
votes
1answer
2k 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 ...
9
votes
4answers
9k 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 ...
2
votes
4answers
912 views

Sketch a curve given parametrically by $x=2t-4t^3$ and $t^2-3t^4$

I am unable to see how to eliminate $t$. Wolfram Alpha fails at it too. $$x=2t-4t^3$$ $$y=t^2-3t^4$$ I can guess that the curve is a polynomial equation so in principle I can write this as $$w_1 ...
2
votes
2answers
156 views

What is the limit distance to the base function if offset curve is a function too?

I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that ...
2
votes
0answers
109 views

Generalizations of equi-oscillation criterion

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
4
votes
1answer
213 views

Assigning alternate crossings to closed curves

This is a minor curiosity that I've been wondering about. Suppose that we draw a closed curve in the plane and that this curve intersects itself several times, but never twice in one spot. We can knot ...
3
votes
1answer
216 views

Does using an ellipse as a template still produce an ellipse?

Suppose I have a (physical) template, consisting of a piece of stiff sheet plastic with a hole cut in the middle. Suppose the hole is in the shape of an ellipse, say, 8 x 12 inches. Suppose I then ...
12
votes
1answer
2k 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 ...
25
votes
2answers
2k 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 ...
11
votes
1answer
656 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 $\...
8
votes
1answer
476 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, let'...
8
votes
6answers
6k 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.
5
votes
3answers
297 views

How to find center of a conic section from the equation?

If we are given a curve in the form $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ and the following determinant $$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$ is non-zero, then this is either a curve ...
5
votes
1answer
96 views

Prove this : $\left(a\cos\alpha\right)^n + \left(b\sin\alpha\right)^n = p^n$

I have this question: If the line $x\cos\alpha + y\sin\alpha = p$ touches the curve $\left(\frac{x}{a}\right)^\frac{n}{n - 1} + \left(\frac{y}{b}\right)^\frac{n}{n - 1} = 1$ then prove that $\left(a\...
3
votes
3answers
3k views

Writing a Polar Equation for the Graph of an Implicit Cartesian Equation

If $(x^2+y^2)^3=4x^2y^2,$ then $r=\sin 2\theta$ for some $\theta$. Using $r^2=x^2+y^2, x=r\cos\theta,y=r\sin\theta$, it's easy to get $r^2=\sin^22\theta$. But I don't know what to do next, since $r$ ...
8
votes
2answers
209 views

Determine the Winding Numbers of the Chinese Unicom Symbol

I'm practicing with Winding Numbers, and encountered an interesting example. You might be familiar with this liantong symbol, the logo of China Unicom: Suppose we make this into a fully closed and ...
6
votes
1answer
152 views

How to calculate the area of a region with a closed plane curve boundary?

Under the conditions of Green’s Theorem, the area of a region $R$ enclosed by a curve $C$ is $$\oint_C x \, dy=-\oint_C y \, dx=\frac{1}{2}\oint_C (x \, dy - y \, dx)$$ I tried to use the result to ...
6
votes
1answer
44 views

How to prove this inequality about the arc-lenght of convex functions?

Let be $f,F:[0,1] \to \mathbb{R}$ with $f,F \in C^2([0,1])$ two convex functions such that $f \le F$ in all points and $f(0)=F(0)$, $f(1)=F(1)$. Considering the plane-curves $\gamma(t)=(t,f(t))$ and $...
4
votes
1answer
885 views

The area of the superellipse

I'm watching this video, where D. Knuth explains the connection of $\pi$ and factorials, and other matters (it is very interesting). Almost at the end of the talk he says the area of the superellipse ...
4
votes
1answer
1k views

Can you write a non-piecewise equation that describes an arbitrary shape?

This batman equation thing got me thinking: for an arbitrary curve drawn on the Cartesian plane, can you write a corresponding equation which is not piecewise? What about closed shapes, a la the ...
3
votes
2answers
478 views

find maximum area

Consider a problem here : There is a wall in your backyard. It is so long that you can’t see its endpoints. You want to build a fence of length L such that the area enclosed between the wall and the ...
2
votes
0answers
81 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 $C^1$ ...
1
vote
1answer
197 views

Is each space filling curve everywhere self-intersecting?

Consider a continuous surjection $f:[0,1]\to [0,1]\times[0,1]$. Is $$\{x:\exists(t_1\not=t_2) f(t_1)=f(t_2)=x\}=[0,1]\times[0,1]?$$
6
votes
1answer
164 views

Is this variant of the Jordan Curve Theorem true?

This feels as though it should be falsifiable, but it's not immediately obvious to me. The informal version of the statement is 'for every non-intersecting curve between two opposite corners of a ...
6
votes
2answers
918 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
0answers
274 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 of ...
6
votes
2answers
1k 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
1answer
2k views

Arc length formula for the lemniscate

This question can be homework for elementary calculus. The lemniscate of Bernoulli $C$ is a plane curve defined as follows. Let $a > 0$ be a real number. Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be ...
5
votes
2answers
402 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 $(c,0)...
4
votes
1answer
80 views

How can I find a curve based on its tangent lines?

Let's say for some curve its tangent lines at every point have a property that the length of a segment within the first quarter $[0;+\infty)^2$ is exactly $C>0$. How can such a curve be defined ...
4
votes
2answers
161 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 $k\in\...
4
votes
4answers
4k views

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 ...
4
votes
0answers
160 views

In how few points can a continuous curve meet all lines?

Inspired by this (currently closed) question, I'm wondering about a related topic: given that we have a continuous parametric curve (that is, a curve of the form $(x=x(t), y=y(t))$ for $t\in\mathbb{R}$...
3
votes
2answers
512 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 ...