What are some cool parametric equations for a butterfly curve?

[Isaac0427]

Hi all! I have recently taught myself parametrics, and I stumbled upon the butterfly curve. So, I was wondering about some cool equations I can plug into a parametric graphing calculator.

 

[ProfuselyQuarky]

Hi all! I have recently taught myself parametrics, and I stumbled upon the butterfly curve. So, I was wondering about some cool equations I can plug into a parametric graphing calculator.

Oh, those are a lot of fun -- even better than polar graphing! My favorite is the Hypotrochoid:

x(t) = (a - b) cos t + c cos ((a/b - 1)t)
x(t) = (a - b) sin t + c sin ((a/b - 1)t)

Sorry, I haven't learned LaTeX, yet . . . Here's a bunch of lovely equations you can try:

https://elepa.files.wordpress.com/2013/11/fifty-famous-curves.pdf

What calculator do you have?

 

[ProfuselyQuarky]

Oh, another good one:

x(t) = sin(7πt)
y(t) = cos(5πt)

Not so sure if it has a name, but it looks way cool. You can always experiment and make up your own, too :)

 

[Ssnow]

The astroid:

$x(t)=a\cos^{3}{t}$
$y(t)=a\sin^{3}{t}$

 

[ProfuselyQuarky]

The astroid:

$x(t)=a\cos^{3}{t}$
$y(t)=a\sin^{3}{t}$

That one is lovely.

 

[pwsnafu]

Special mention should be made of the superellipse
$\left| \frac{x}{a}\right|^n + \left| \frac{y}{b}\right|^n = 1$
which has parametric equations
$x(t) = \pm a \cos^{2/n}(t)$
$y(t) = \pm b \sin^{2/n}(t)$
It contains a number of equations above as special cases.

In turn this is generalized by the superformula.
As I understand it the 3d version is used by No Man's Sky.

 

[Ssnow]

In the Astroid $t\in [0,2\pi]$, there is also the Archimede spiral:

$x=t\cos{t}$
$y=t\sin{t}$

with $t\in [0,+\infty)$. Have good painting ...