I want a picture book of what zillions of different mathematical curves look like.
OK, OK, everybody get in line and take turns explaining to me why this is a dumb thing to want. Yes, of course it depends on too many considerations.
Really, there’s a legitimate need. I often have to model data empirically, fitting smooth simple curves to groups of points. If there’s an a priori reason to expect a particular form of curve, that’s sweet, but if there isn’t, I start wondering if an exponential, or a rational polynomial, or some hyperbolic or trig function, or what else, would work. And I maintain that many of us have at least a little ability to distinguish between for example exponential looking curves versus polynomial looking curves.
There are broad families. Sometimes I definitely want a function that increases monotonically from zero to one as its argument increases from zero to one, but what function?
Or other times, I want something that in some transform is asymptotic to two different straight lines in the extremes, and yet is a nice fit in that ticklish transitional region between them.
People who can recognize kinds of functions by their curves are at a great advantage in this. I’m looking for something that helps do this.
However, I’m not really sure that this is going to be of fundamental use to you. The shape of many curves, and especially anything more complex than the conic and hyperbolic curves are fairly arbitrary, depending on parameters. For instance, although a real polynomial will have as many reversals as it has exponents minus one. For any given set of n points there are an infinite number of polynomials of n-1 exponents which can be fit to them, notwithstanding higher order polynomials or other functions. A fit should represent some anticipated behavior in a system, e.g. a linear or logarithmic change, or orbital motion in a conic, or somesuch. Overfitting by assigning arbitrary functions often masks behavior and can lead to erroneous values in interpolation and (especially) extrapolation. There are times, such as with controls, that you’ll specifically pick a polynomial of a particular order to fit to and then use a least squares approach to find the “best” polynomial in the range that you intend to operate in (e.g. one that can be most easily made piecewise linear). If you are just fitting data mapped to a surface, piecewise interpolation to a spline is probably the best approach rather than trying to fit to a specific global function.
If you do want some kind of arbitrary data fitting there are a wide variety of different packages and tools to do so for both commercial and open source applications and frameworks. Mathematica/Wolfram, the Matlab Curve Fitting and Optimization toolboxes, and scipy.optimize. The GNU project has a number of different regression and fitting tools. All of these will be of much more utility in trying to find a “best” fit than looking through a book of curves and truing to discern a suitable function.
Hey, this looks useful! Especially the gallery of curves linked near the top! I had what looked like one branch of the swastica curve to model today and now I have a form to apply.
Math programs can generate curves on request, but the need is to know what to ask for. When I’m looking at a scatter plot and wondering what form of function could imitate it well, the math programs aren’t helpful.
I haven’t been at university (college) for awhile, but as i recall the mathematical programs and/or scripts Stranger mentioned (and others generally) will both plot the data, then identify the most appropriate curve or curves based on a variety of regression analysis tools.
So they should actually perform the step you hope to do by a visual comparison.
Abramowitz and Stegun’s Handbook of Mathematical Functions and the CRC Handbook of Chemistry and Physics have already been mentioned. There’s also J. Dennis Lawrence’s * A Handbook of Special Plane Curves*, published by Dover, and now in print for over 40 years:
There’s a lot in this book, but there’s also a lot to confuse the non-mathematical reader, and it’s not all adequately explained.