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@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. >> > > > _______________________________________________ > use-livecode mailing list > use-livecode@lists.runrev.com > Please visit this url to subscribe, unsubscribe and manage your subscription > preferences: > http://lists.runrev.com/mailman/listinfo/use-livecode _______________________________________________ use-livecode mailing list use-livecode@lists.runrev.com Please visit this url to subscribe, unsubscribe and manage your subscription preferences: http://lists.runrev.com/mailman/listinfo/use-livecode
