cool function curves?

May 10, 2018 1:34 PM

I'm working on some AI music algorithms and need mathematical curves on which to base certain aspects. What are the equations/names of some beautiful/interesting/unusual 2D function curves?
posted by Ain to Science & Nature (10 answers total)

I'm going to guess that by beautiful/interesting/unusual 2D function curves you DON'T mean the conic section curves (line, parabola, ellipse,...) or the graphs of standard (polynomial, trig., exponential, ...) functions.

Here's a link to Fifty Famous Curves, some of which I find beautiful, interesting, and unusual.
posted by El_Marto at 1:55 PM on May 10, 2018 [5 favorites]

How about a space filling curve? There was a post a few months ago about using them to generate music.
posted by dilaudid at 2:47 PM on May 10, 2018

Lines are very unusual; almost every smooth function is not a line. Likewise any smooth function is unusual, because almost every continuous function is not smooth.

You might get better answers if you explain a little bit about what role you expect these functions to play, or what criteria should they satisfy. Or for that matter if you mean curves or functions or something else, because those are not the same thing, and ‘function curve’ isn’t really a standard term in math.

What’s ‘unusual’ is hard to pin down. Because literally any function that has a name or can be written down at all is unusual in terms of the vast infinite dimensional space of functions from the real line to the real line.

Making lots of guesses: check out the Bessel functions and the family of Weibull distributions. These span a fairly large space on their own, and although they are common in many fields of science and engineering, a lot of people haven’t heard of them. They are beautiful in terms of their elegant properties, their utility in application, and the aesthetics of their graphs. You can manipulate their maxima, minima, and lots of oher properties, and they are implemented in many software packages.
posted by SaltySalticid at 2:53 PM on May 10, 2018

The sinc function is very beautiful. It is defined as sinc(x) = sin(x)/x;
posted by hz37 at 3:07 PM on May 10, 2018

Do you want your functions one-at-a-time or in families. If families of functions are of any interest, you might be interested in orthogonal functions. The best known is the Fourier Series and it represents exactly what music is made of: oscillations at different intervals. If, for example, you have computer-based tuning app, it's taking a snapshot of the sound, computing the Fourier Series using an algorithm called a Fast Fourier Transform. The result shows, among other things, the frequency of the loudest frequency (which is the pitch) of the tone.

There are lots of families of orthogonal functions. Some are based on Trig functions, some are based on polynomials, etc. Wikipedia explains. I like Chebyshev polynomials, myself.
posted by SemiSalt at 4:18 PM on May 10, 2018

Consider our friends the polar coordinate functions.
posted by Huffy Puffy at 7:06 PM on May 10, 2018

In addition to the excellent suggestions above, maybe look at the Jacobi elliptic functions, which can describe the exact motion of a pendulum, certain types of water waves (cnoidal waves), and the representation of a number as the sum of four squares.
posted by egregious theorem at 9:35 PM on May 10, 2018

the aesthetic of tan and cotan - always graceful
posted by Barbara Spitzer at 10:17 PM on May 10, 2018

Many of the "cool" curves look nice because they mimic some pattern in the nature. The catenary looks like suspended cords. It is related to the tractrix, which is the curve produced by dragging a thing along a straight track, by a process called the evolute (the inverse of involute). The logarithmic spiral can be found in plants and shelled molluscs, and the parabolic spiral in the arrangement of sunflower seeds on the disk.

I think one fun way to find more curves is to think something we encounter in the nature and search for topics from analytic geometry using such keywords. Such as "curve of fern leaf" etc.

From work: I find the curves of the Beta distribution pretty cool to behold. The Cauchy distribution gives arise to a curve (mis)identified as the "Witch of Agnesi" but Maria Gaetana Agnesi was a cool woman mathematician.
posted by runcifex at 4:55 AM on May 11, 2018

Lissajous Curves
posted by wittgenstein at 1:31 PM on May 11, 2018