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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Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
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7answers
8k 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 ...
3
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2answers
958 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 ...
7
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12answers
14k 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 $x$ ...
10
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7answers
497 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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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 ...
12
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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 ...
2
votes
2answers
929 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 ...
11
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5answers
483 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 ...
7
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2answers
493 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 ...
3
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1answer
174 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 ...
46
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1answer
1k 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, ...
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6answers
4k 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.
6
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1answer
3k 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?
6
votes
3answers
657 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 ...
4
votes
1answer
496 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 ...
4
votes
2answers
776 views

Reconstructing space curves from its curvature and torsion

I have to write a program which gets two functions (curvature and torsion) and 3 vectors of the Frenet-Serret "frame" at the starting point - and I have to reconstruct the space curve from this given ...
2
votes
2answers
133 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
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0answers
99 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 ...
2
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4answers
806 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 ...
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2answers
85 views

Image of any curve can be parametrized without zero derivative [closed]

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a $C^{1} ([a,b])$ injective application. Prove that there is another continuous parametrization $\rho:[c,d]\to\mathbb{R}^2$ such that the following two sets are ...
0
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1answer
39 views

Curvature and convexity of a plane curve

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2$ be a $C^2([a,b])$ regular curve. Is it true that $\mathbf{r}$ is convex if and only if its curvature $\kappa(t)\leq 0, \forall t\in [a,b]$ or $\kappa(t)\geq 0, ...
4
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1answer
183 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 ...
2
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2answers
959 views

Direction of the second derivative of an arclength parametrized curve

I have a question, it's so simple and stupid ._. If I have a planar curve parametrized by arc length, it's easy to show that the second derivate is orthogonal to the first derivate vector (tangent ...
23
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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 ...
9
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1answer
1k 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
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1answer
210 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, ...
10
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1answer
577 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 ...
4
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1answer
732 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 ...
3
votes
2answers
261 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 ...
3
votes
3answers
2k 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 ...
2
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0answers
71 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 $C^1$ ...
6
votes
1answer
134 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
651 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 ...
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
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1answer
1k 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 ...
4
votes
2answers
148 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 ...
4
votes
0answers
138 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 ...
4
votes
1answer
635 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 ...
4
votes
2answers
348 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 ...
3
votes
1answer
869 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
3answers
974 views

A hyperbola as a constant difference of distances

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, the distance between the two ...
2
votes
2answers
451 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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vote
1answer
136 views

Is each space filling curve everywhere self-intersecting?

Consider a continuous surjection $f:[0,1]\to [0,1]\times[0,1]$. Is $$\{x:\exists(t_1\not=t_2) f(t_1)=f(t_2)=x\}=[0,1]\times[0,1]?$$
0
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0answers
59 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 ...
0
votes
2answers
445 views

Moment of inertia about center of mass of a curve that is the arc of a circle.

Let $(x(s),y(s))$ be a smooth 2-d plane curve which is an arc of a circle of a certain radius $r$. Assume it is represented by an inelastic string $S$ of finite length, lying in a 2-d plane. Let there ...
8
votes
2answers
535 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 ...
5
votes
3answers
150 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 ...
4
votes
2answers
106 views

why is an annulus close to it's boundary when it's boundary curves are close?

This is the motivating question for the rather vague question here: when is the region bounded by a Jordan curve "skinny"? Suppose we are given two Jordan curves in the plane, one inside ...
3
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4answers
254 views

how to find the equation of one curve in XY plan which satisfies such functions?

I want to find the equation of one curve in $X-Y$ plan which satisfies the functions as follows: 1) $A(x_1,y_1)$, $B(x_2,y_2)$ are two known points and $f(x,y)$ (or $y(x)$) is the equation of the ...