Lemniscate Polygon
Mark Waddingham
mark at livecode.com
Wed Nov 3 03:29:30 EDT 2021
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/
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