# 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**

Download zipped with *pkzip -p -r graphing*

Directions for use:
- Make a subdirectory & go in there
- Unzip (with
*pkunzip -d graphing*)
- Run
*hernoem.bat* (just do it)
- 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

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
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.