Lemniscate Polygon

[Richmond]

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.
>