Lemniscate Polygon

[Roger Guay]

And thank you, Richmond for implementing what for me was overnight. Very nice clean and simple code too!!

Roger

> On Nov 3, 2021, at 1:39 AM, Richmond via use-livecode <use-livecode at lists.runrev.com> wrote:
> 
> https://forums.livecode.com/viewtopic.php?f=7&t=36429
> 
> Richmond.
> 
> On 3.11.21 9:29, Mark Waddingham via use-livecode wrote:
>> Hi Roger,
>> 
>> On 2021-11-02 22:27, Roger Guay via use-livecode wrote:
>>> Dear List,
>>> 
>>> Bernd has produced an absolutely beautiful animation using a
>>> Lemniskate polygon that was previously provided by Hermann Hoch. Can
>>> anyone provide some help on how to create this polygon mathematically?
>>> Since the equation for a Lemniskate involves the SqRt of negative
>>> numbers, which is not allowed in LC, I am stumped.
>>> 
>>> You can find Bernd’s animation here:
>>> https://forums.livecode.com/viewtopic.php?f=10&t=36412
>>> <https://forums.livecode.com/viewtopic.php?f=10&t=36412>
>> 
>> In general lemniscates are defined as the roots of a specific kind of quartic (power four) polynomials of the pattern:
>> 
>>     (x^2 + y^2)^2 - cx^2 - dy^2 = 0
>> 
>> So the algorithms for solving them you are probably finding are more general 'quartic polynomial' solvers - just like solving quadratic equations, the full set of solutions can only be computed if you flip into the complex plane (i.e. where sqrt(-1) exists) rather than the real plane.
>> 
>> However, there is at least one type of Lemniscate for which there is a nice parametric form - Bernoulli's lemniscate, which is a slightly simpler equation:
>> 
>>     (x^2 + y^2)^2 - 2a^2(x^2 - y^2) = 0
>> 
>> According to https://mathworld.wolfram.com/Lemniscate.html, this can be parameterized as:
>> 
>>     x = (a * cos(t)) / (1 + sin(t)^2)
>> 
>>     y = (a * sin(t) * cos(t)) / (1 + sin(t)^2)
>> 
>> Its not clear what the range of t is from the article, but I suspect it will be -pi <= t <= pi (or any 2*pi length range).
>> 
>> So a simple repeat loop where N is the number of steps you want to take, and A is the 'scale' of the lemniscate should give you the points you want:
>> 
>>     repeat with t = -pi to pi step (2*pi / N)
>>        put A * cos(t) / (1 + sin(t)^2) into X
>>        put A * sin(t) * cos(t) / (1 + sin(t)^2) into Y
>>        put X, Y & return after POINTS
>>     end repeat
>> 
>> Warmest Regards,
>> 
>> Mark.
>> 
> 
> 
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