# Cool Parametric Equations

• I
Gold Member
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
Gold Member
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?

Last edited:
Isaac0427
ProfuselyQuarky
Gold Member
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 :)

Isaac0427
Ssnow
Gold Member
The astroid:

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

Isaac0427
ProfuselyQuarky
Gold Member
The astroid:

##x(t)=a\cos^{3}{t}##
##y(t)=a\sin^{3}{t}##
That one is lovely.

Ssnow
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.

Isaac0427 and Ssnow
Ssnow
Gold Member
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 ...

Isaac0427, jim mcnamara and ProfuselyQuarky