jlp-GeneralDiskAutomorfism: (GDA)
This is the first one of a series I would call "complex automorfisms".
For those who are familiar to complex analisys, it's just an
implementation of the general disk automorfism (it was clear, wasn't it?)
For those less familiar, complex automorfisms, or bi-holomorfic complex
functions are functions that take a piece of the complex plane into
itself in a VERY smooth way (the so-called holomorfic way) which is
smoothly invertible. In practical sense that means that no point will be
loosed with the aplication of the transformation, all points will be
moved, thus the transformation will result in a shape deformation (which
can be rather complicated). In this case, the GDA will take a disk of
arbitrary center and radius into itself, the whole transformation is
completely definied just giving one point inside the circle that will be
fixed by the transformation, if this point coincides with the conter of
the circle, the GDA will be just a simple rotation.
Since UF deals quite well with singularities, I've allowed the GDA to
exted outside the selected circle, beign holomorfic (smooth) almost
everywhere (in fact, everywhere but in a single point) If you choose as
fixed point a point outside the circle, the transformation will be an
automorfism in the region outer the circle; if you take the fixed point
exactly in the circunference... well, I'll assume no responsability if
you do that ;-)
jlp-CayleysTransform
The Cayley's transformation just take the selected circle (with center
and radius) into half a plane. Inverting the transformation you take
half a plane into the selected circle.
Cayley's transformation is one particular application of the Riemann
mapping theorem, for the particular shapes "Cricle" and "Half a plane";
I intend to develope some other applications of this theorem in the
future, adding some more shapes.
jlp-MöbiusTransform
The Möbius transformations, also called "Extended Plane Automorfisms"
are the transformations given by the formula
f(z)=(az+b)/(cz+d)
(where a,b,c,d are complex constants) which are all and the only
bi-holomorfic
transformations of the Riemann sphere onto itself (the Riemann sphere is
the complex plane with an additional point, the "Point at th infinity")
So can obtain all the smooth shape deformations of the whole plane just
by choosing a,b,c and d (called in the formula a1,b1,a2 and b2).
jlp-GeneralAstroid-Rose {
I arrived to this transform while trying to develope the Circle-->Sqare
transformation. My code was worng, and Samuel Monnier gave me the
correct formula, but the effect I got beofre Sam's help was already
interesting, so I decided (as I understood where my error was after
studying Sam's code) to implement and extend it in a nicer way. This
transforms a circled shape into an (Order)-peeks thing similar to an
astroid. Decimal values of Order can be used to get Tear-like effects
(try as an example Order=0.75) .
Inverting the transform takes a circle into a Rose-Like shape. Again,
decimal values of Order are interesting, too.
The default formula seems to be nicer with the "Mited" parameter set to
"TRUE", while the inverse transform looks better turning it off.
jlp-SimplePorwering
This is such a simple transformation that I couldn't believe it wasn't
implemented. It simply applies the formula
f(z)=z^n
that have very nice symmetrical effects for integer values of n (both
possitive and negative), and a not so nice effect for decimal values of
n.