Everything you always wanted to know about **Splines** but never dared to ask
...

The point I want to make here is that there is no need to use Cubic (or whatever higher order)
splines in two dimensions. Because they offer no significant advantage,
when compared with the simplest member of the family: the **Quadratic
Spline**.
Convince yourself by reading / running the stuff I have brewed on this subject:

**Mathematical Theory**

& Source Code**Theory in PDF**(with 'pdflatex')**Same in HTML/MathJax**. See Errata (*) below.**Executable Program****Treatise about Conic Sections**

and accompanying source**Some Obvious Arguments**:

Note: my son said to me that the prevalence of cubic splines is maybe due to the fact that one can easily build an interactive designing program with the latter, but not with the former. Well, I must admit that would be a "legal" consideration, at last.

(*) But .. I've made a mistake with terminology. The following is a comment by Rob Johnson.

*The second order counterpart of cubic splines are quadratic splines, which, as you have noted in your citations, are parabolic sections. However, [ your PDF ] seems to say that quadratic splines and conic splines are the same. While all quadratic splines are conic splines, not all conic splines are quadratic. In fact, quadratic splines are the intersection of cubic splines and conic splines, and proper subsets of both. Conic splines are rational splines, and as such are closed under not only affine transformations, but also perspective transformations.*

Kind of a Test Facility has been devised for Discrete / Random / Continuous
Densities and Senses. The purpose of it being able to test some theory, and
also getting familiar with the subject as such. A prototype of this has been
published earlier, in a chapter called *Continue Bovenbouw*
which is part of a whole book
(written in **Dutch**, my mother's tongue). The book is about what I thought
to be Foundational Issues in Mathematics, at that time. Meanwhile, I have come
to a quite different belief. I am convinced, nowadays, that all Mathematics is
complicated from the beginning and there is NO such thing as an evolution from
the simple to the more complicated. So far so good. Let's return to the point.
Our **Densities and Senses Test Bank** has been made available as:

**Executable Program****Source & Theoretical****Theory in PDF**(with 'pdflatex')

The **VaagZien** demonstration program was developed, because I have been
fascinated quite some time by the notion of **Fuzzy Optics**. The subject
already fills a chapter in my 1995 (Dutch) book:

*Wazige Optiek*

**Executable Program****Source & Theoretical****Theory in PDF**(with 'pdflatex')

What's highly frustrating for a computer minded musician is the fact that those
stupid machines still aren't capable of reading ordinary Music Score. No, they
can't ! Even though certain brave vendors
(as always) claim that they can. Have you ever fed a hand-written document into
your scanner and expect something sensible to come out ? Neat copies - having
been generated already with a Midi to Score comverter like the (excellent !)
NoteWorthy Composer - do not represent
a real-world problem, do they ? I'm So Sorry ...

What could have been the first step in Music Score Recognition, according to my
humble viewpoint, is the mere **Recognition of Straight Lines**. My *Pixel
Thinning* demonstration program doesn't actually do this, but may be thought
as a first step in the right direction:

**The art of Pixel Thinning**(**deprecated**2003-07-18)**And how it was done**

(demonstration BMP files included)

Have you ever seen the transformations of a Continuous (Lie-) Group in action ? I mean, 2-D transformations which do Continuous Scaling, Deformation, Rotation and Translation of an Image. Of course you have. Certainly such transformations are shipped with expensive Image Processing Packages (like Corel version x.y) . But I'm quite sure that nobody has ever told you how to develop such techniques entirely by yourself. Continuization instead of Discretization, again, makes it simple and straightforward to accomplish such things. Here comes:

**The****Continuous Transformation of B/W images****And how its was done**

(demonstration BMP files included

except this big one: use it with care)**Theory in PDF**(with 'pdflatex')

Nyquist Scaling has a potential for recognizing patterns in pictures. This may be demonstrated by setting the fuzzyness in X direction to a large value, while leaving the fuzzyness in Y direction normal. Now load the Temnapo Fortran listing and set the Contouring Level to a suitable value. You will see that the text lines in the listing are actually distinguished by the program, as a first step (and maybe also the last)-; to recognition.

Almost any search on the Web ends up in a Reference to an algorithm by Brendan McKay.

He was the first one to find all non-equivalent graphs up to order 11, for example.

Here comes a much more simple minded Program & Results

for Graphs of order (number of interconnected nodes) up to and including eight:

**All Graphs of certain order ≤ 8****And how its was done**

(output files included)

- 1 graphs of order 2
- 3 graphs of order 3
- 6 graphs of order 4
- 21 graphs of order 5
- 112 graphs of order 6
- 853 graphs of order 7
- 11,117 graphs of order 8

- 1 , 2 , 4 , 11 , 34 , 156 , 1044 , 12346

Here are some more, thanks to The Online Encyclopedia of Integer Sequences.

Sequence: 1 , 1 , 2 , 4 , 11 , 34 , 156 , 1044 , 12346 , 274668 , 12005168 , 1018997864 , 165091172592 , 50502031367952 , 29054155657235488 , 31426485969804308768 , 64001015704527557894928 , 245935864153532932683719776 , 1787577725145611700547878190848 , 24637809253125004524383007491432768 Name: Graphs with n nodes.

One of the methods used in
my FCP project has been the finding of **Cliques**.

But, of course
(and again), I have not been the first one who has done this:

BK73 C. Bron and J. Kerbosch. Finding All Cliques of an Undirected Graph. Communications of the ACM 9, 16(9):575--577, 1973.Anyway, here is my (recursive) program for Finding All Cliques in an Undirected Graph:

**All Cliques in undirected Graphs****And how its was done**

(with sample input files)

Here is a helper program for the Decomposition of Permutations into **Cycle
Factors** and the Composition of Permutations from Disjoint Cycle Factors:

How to do calculations in a
* Galois Field* ? And how to

- Modulo Rings, Polynomials & Galois Fields.

Source code for doing some elementary algebra in**Discrete Mathematics**.

The so-called **Term Structure of Interest Rates** can be explained a great
deal by
a theory which takes into account the psychology of (positive) interest-bearing
Money. This theory reveals that the primary interest (: mind the word !) of our
GOD, the Grand Omnipotent Dollar, is in the Depreciation of our goods.

This may be called the *Interest BY Depreciation*
theory.

There are two versions of the software: an English and a Dutch one. The Dutch
version is found at the
SP site.

The English version is found here:

- Term Structure of Interest Rates executable
- And how it was done
- Term Structure Data needed to run

Punch Cards have been a popular medium of data storage in the good old
days of computer technology. I still have a bunch of these items
at home. And once upon a day I wondered if it would still be
possible, not only to write these cards, but also to read them, back into our memory ;-)
This has lead to my software version of an **IBM PunchCard Reader**.

- IBM PunchCard Reader executable
- And how it was done.

Also contains data which are needed to run the program. - Accompanying documentation has been
*deprecated*.

Non ASCII: a Non Fortran: f Card Punch: O f a f f f a ffff ff f &ABCDEFGHI .<(+-JKLMNOPQR!$*);0/STUVWXYZ ,%_>^ 123456789:#@'=" ----------------------------------------------------------------- 2 | OOOOOOOOOOOOOOO | 2 1 | OOOOOOOOOOOOOOO | 1 0 | OOOOOOOOOOOOOOOO | 0 1 | O O O O | 1 2 | O O O O O O O O | 2 3 | O O O O O O O O | 3 4 | O O O O O O O O | 4 5 | O O O O O O O O | 5 6 | O O O O O O O O | 6 7 | O O O O O O | 7 8 | O OOOOO O OOOOO O OOOOOO O OOOOOO | 8 9 | O O O O | 9 ----------------------------------------------------------------- &ABCDEFGHI .<(+-JKLMNOPQR!$*);0/STUVWXYZ ,%_>^ 123456789:#@'=?Precautions and Limitations:

- Use a conventional flatbed (A4) scanner to scan the punchcards
- I think it's important to callibrate the scanner properly, before trying to read in any pictures (because the program employs the resolution)
- The background color of the (white) cards should be Black. This can be accomplished by covering them with black paper before scanning begins
- Cards must be scanned with their backside down as the surface which contains the data. The human-readable side should be for your eyes only ;-)
- More than one card can be processed simultaneously. However, punchcards should Not be mutually overlapping.
- All cards must be scanned in Black and White, resulting in monochrome BMP files to be processed
- The examples were all scanned are at 100 DPI. Therefore I am not quite certain what behaviour of the program is to be expected at other resolutions.
- Due to the kind of algorithm implemented, the program behaves
**best**if there are as "many" data on the punchcard as possible. Degenerate cases - such as almost empty cards - have not been tested to a sufficient extent.

**RSI**
is rapidly becoming a common disease amomg programmers. Not surprisingly -
but also somewhat contradictory - computer programs have been developed, in an
attempt to prevent (other)
people from being exposed to **R**epititive **S**train **I**njury.
Probably the best in its kind is WorkPace.
I have developed a much less advanced - say lightweight - version of this,
which does only the basic thing: force you to take a break from time to time.
Here comes:

**the WerkPauze program****And how its was done**

**Power Series in cos ^{k}(x) and Fourier Series in cos(k x) can
be converted into each other.**

I clearly re-invented the wheel again, since references on this subject could be found on the Internet at:

- SVIBOR - Collecting Data on Projects in Croatia.

Especially the numbers 28 and 43. (: broken link) - TrigTricks, a
*Mathematica*Notebook file.

> expand(cos(7*x));64 cos(x)

Nevertheless, here comes: **a ****mathematical
theory of Cosine Expansions**.