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:
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.
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:
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:
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:
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:
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:
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 define such a weird monster in the first place ?
The so-called Term Structure of Interest Rates can be explained a great
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:
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.
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:
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 Repititive Strain Injury. 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:
Power Series in cosk(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:
> expand(cos(7*x));64 cos(x)7 - 112 cos(x)5 + 56 cos(x)3 - 7 cos(x)
Nevertheless, here comes: a mathematical theory of Cosine Expansions.