There are a great many such parametric formulae, such as
hypergeometric sequences and their sums. You can get very interesting
results by plotting the roots of a polynomial as the coefficients
vary, without ever letting two roots coincide. It is possible to
generate all knots and links in this way, by letting the graph wrap
around.
Given
* A parameter t in the range [0,1]
* A polynomial P = Σaᵢxⁱ for i in the range from 0 to n
* Continuous but not necessarily differentiable complex-valued
functions aᵢ(t), with aᵢ(0) = aᵢ(1)
for which P has no repeated roots at any value of t, we can take the
graphs of the n complex roots of P for each t, graph them in three
dimensions, and then topologically wrap the whole thing around into a
donut, joining t = 0 and t = 1, where the roots are the same, but may
be permuted.
The following diagram must be viewed in a monospace font in order to make sense.
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On Sat, May 21, 2011 at 17:11, Gregor Lingl <gregor.li...@aon.at> wrote:
> Hi all,
>
> recently I stumbled (once more) over a posting by our friend
> Daniel Ajoy about the superformula (I think it was in a Logo-Forum):
>
> http://en.wikipedia.org/wiki/Superformula
>
> On the other side Kirby wrote interesting suggestions
> about using generators.
>
> Now, weekend, bad weather, some sparetime, I
> assembled an implementation of a superformula-viewer
> using pygame. It runs with Python 2.6 or higher, also
> 3.x, of course
>
> You'll find it here for download:
>
> http://dl.dropbox.com/u/2016850/superformula.py
>
> or appended (which sometimes / for some of you)
> doesn't work well depending on the browser or whatever.
>
> The script uses generators for generating the pointlists
> of the graph of the superformula and also for generating
> colors for the segments of the graph-area.
>
> Morover I've prepared a q&d slider class, which possibly
> doesn't use the canonical way of processing events in
> pygame but works fine for this application.
>
> Critical comments and feedback and also questions, of course,
> are welcome.
>
> Perhaps someone is willing to amend the docstrings,
> which suffer from my clumsy English and could well
> be more clear.
>
> Amendments of the code, espercially ones, which make it
> more readable and easier to understand, also.
>
> Useful for classroom use? Perhaps to difficult?
>
> Best regards,
> Gregor
>
>
>
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>
>
--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
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