This evening I ran across a paper on an unusual coordinate system that creates interesting graphs based from simple functions. It’s called “circular coordinates,” but this doesn’t mean polar coordinates; it’s more complicated than that. [1]

Here’s a plot reproduced from [1], with some color added (the default colors matplotlib uses for multiple plots).

The plot above was based on a the gamma function. Here are a few plots replacing the gamma function with another function.

Here’s *x*/sin(*x*):

Here’s *x*^{5}:

And here’s tan(*x*):

Here’s how the plots were created. For a given function *f*, plot the parametric curves given by

See [1] for what this has to do with circles and coordinates.

The plots based on a function *g*(*x*) are given by setting *f*(*x*) = *g*(*x*) + *c* where *c* = −10, −9, −8, …, 10.

## Related posts

[1] Elliot Tanis and Lee Kuivinen, Circular Coordinates and Computer Drawn Designs. Mathematics Magazine. Vol 52 No 3. May, 1979.

Is there a typo in your post, or have I misunderstood the maths?

You say at teh end of the post

“The plots based on a function g(x) are given by setting f(x) = g(x) + c…”

Should that not be “f(x) = g(x) * c” – ie it is times c, not plus c

Or have I got the maths wrong?

PS I’m trying to play around with this in R

It is addition, g(x) + c, not multiplication.