# Programming in Delphi

While everybody seems to know nowadays how to solve (even Partial) Differential Equations - the Art of Numerical Integration - know-how of the inverse, the Art of Numerical Differentiation is still lacking a great deal. This problem isn't truly hard, though. Essentially, it has been known to me for a long time, since the method of Gaussian sensing is already desribed in my (Dutch) book. What the world needs is no more Discretization, but Continuization instead! Convince yourself by running the Gaussian sensed Numerical Differentiation Demo: Numerical Differentiation may find numerous applications in pattern recognition with chemical spectra and optimization theory, to mention a few areas.
Due to a debate with Dik T.Winter, there has been some software development. Also the need for a better documentation was felt.

Kind of an update of my contouring techniques for gray-valued bitmaps:

## Graph Drawing

Take a look at the Graph Drawing Tutorial in the first place. This project was started in 2002 and has never been finished.
I'll be glad if someone picks it up and put an end to it in a more decent way. Anyway, here comes:
• Introduction
• Some documentation
• Sources & Executables
Directions for use:
1. Make a subdirectory & go in there
2. Unzip (with pkunzip -d graphing)
3. Run hernoem.bat (just do it)
4. Run Project2 (executable)
When running the program, you might not expect that the vertices of a displayed graph can be dragged with the mouse to other positions.
Meant as kind of a compensation for the lack of a fully automated planar graph layout mechanism.

Yes ! Some of you might not expect this, but it's a problem to decide whether a point is inside or outside a boundary. And I am only talking about the planar (2-D) case here: think of a contour which encloses a certain area of interest. And how about the following: to calculate how much of a line is inside a region, if only the boundaries of that region are known. Here is accompanying

### WebPage documentation software and input files

Usage: Make a directory. Unzip in that directory and Run the 'exe' files. Open a file, do the Preprocessing and select an Experiment. Eventually set some Options. Then click on the image with your mouse. And see what happens. All source code has been included as well, for the serious programmer.

Related document: Ellipses made Useful .

## Audiolization of Graphs

We all know how to make more or less advanced visualizations of mathematical objects, like the graph of a function. But how about making an audiolization of graphs ? That is, instead of seeing them: let's try to hear them ! OK, I've accomplished just that. Just for fun. Here comes accompanying

### source code and executable

Usage: Make a directory. Unzip in that directory and Run the 'exe' files. All source code has been included as well, for the serious programmer. You can also start the executable in Explorer. Don't be affraid to to so. The only danger is that it will create a scratch-midifile 'afspelen.mid' on your desktop.