How do programs like mathematica draw graphs and how can I make such a program?

Question

I've been wondering how programs like mathematica and mathlab, and so on, plot graphs of functions so gracefully and fast. Can anyone explain to me how they do this and, furthermore, how I can do this? Is it related to an aspect or course in Computer Programming or Math? Which then?

Answer 1

Well, with some encouragement from belisarius, here's a my comment as an answer: try looking at matplotlib. From the home page:

matplotlib is a python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms. matplotlib can be used in python scripts, the python and ipython shell (ala MATLAB®* or Mathematica®†), web application servers, and six graphical user interface toolkits.

It was originally inspired by MATLAB's plotting capabilities, though it's grown a lot since then. It's solid software - and it's open source, under a BSD license, so not only can you read the source, you can hack on it and use it in whatever you like.

Another place you could look is gnuplot. It's not one of the common open source licenses, but it's certainly open source, with some permissions to modify and such.

Gnuplot is a portable command-line driven graphing utility for linux, OS/2, MS Windows, OSX, VMS, and many other platforms. The source code is copyrighted but freely distributed (i.e., you don't have to pay for it). It was originally created to allow scientists and students to visualize mathematical functions and data interactively, but has grown to support many non-interactive uses such as web scripting. It is also used as a plotting engine by third-party applications like Octave. Gnuplot has been supported and under active development since 1986.

It does 3D plotting as well, which matplotlib doesn't do, and it's been around a lot longer. The reason I thought of matplotlib first is that it's intended as a library for a higher-level language, not a stand-alone application, so I'm guessing it might a bit easier for you to read.

One other suggestion, just to get an idea of the sorts of things Mathematica is doing under the hood, is to look at the documentation for Plot. In particular, if you look at the available options, you can deduce things.

MaxRecursion Automatic the maximum number of recursive subdivisions allowed Method Automatic the method to use for refining curves PerformanceGoal $PerformanceGoal aspects of performance to try to optimize PlotPoints Automatic initial number of sample points

From the MaxRecursion and PlotPoints, you can see that it's doing an initial sampling then somehow deciding which regions need to be subdivided (resampled) to get an accurate view of the plot. And from there on, it's magic: there is some Method for this, and a PerformanceGoal to guide it...