Lemniscate Polygon

[Mark Waddingham]

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.

-- 
Mark Waddingham ~ mark at livecode.com ~ http://www.livecode.com/
LiveCode: Everyone can create apps