Scientific Approach to the Yin-Yang Geometry by Sergey Yu.
Shishkov (RUSSIA
, Shishkovser@rambler.ru)
Here is given (below) the most generalized definition of the
astroid-like hypocycloid as the trajecory of a point P of a
rotating with angular velocity "omega1"=1 circle of radius
"radius1"=a, with centre of which also being rotating around the
origin by the circle of radius "radius2"=1-a , and angular
velocity "omega2"=-3, so that"radius1" +"radius2"=1, and
"omega2"/"omega1" =-3. Then for coordinates X[t], Y[t] of this
point P we have: X[t]=(a)*cos(t)+(1-a)*cos(3*t);
Y[t]=(a)*sin(t)-(1-a)*sin(3*t);
1-X[t]^2-Y[t]^2=factor(simplify(expand(1-((a)*cos(t)+(1-a)*cos(3
*t))^2-((a)*sin(t)-(1-a)*sin(3*t))^2)))=16*a*cos(t)^2*(cos(t)-1)
*(cos(t)+1)*(-1+a)=16*a*cos(t)^2*(cos(t)^2-1)*(-1+a)=16*a*cos(t)
^2*(sin(t)^2)*(1-a)=FULL SQUARE!=>
If Z[t]=4*cos(t)*sin(t)*(a*(1-a))^(1/2), then
X[t]^2+Y[t]^2+z[t]^2=1 ;(i.e., For every time t {X[t],Y[t],Z[t]} is on the unit SPHERE!!!).
With different values of the parameter a we obtain the
whole class of astroid-like hypocycloids with FOUR PARTS. Below
is given the Maple 5.4 Text programm for plotting of these
trajectories.;
> a=0.6339;plot([(a)*cos(t)+(1-a)*cos(3*t),(a)*sin(t)-(1-a)*sin(3*t)
,t=0..2*Pi]);
plot([(a)*cos(t)+(1-a)*cos(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2),
t=0..2*Pi]);
plot([(a)*sin(t)-(1-a)*sin(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2),
t=0..2*Pi]);
Look also in Maple 5:
> factor(simplify(expand(1-((b)*cos(t)+(1-b)*cos(3*t))^2-((b)*sin(
t)-(1-b)*sin(3*t))^2)));
> The Optimal Value for the parametr a is a=(1/2)*(3-3^(1/2))=0.6339, as will be
shown elsewher;-). It corresponds to the most "BEAUTIFUL" 3D-hypo-astroid. Such a configuration may serve also as the Yin-Yang
MAGNETIC TRAP for adiabatic freezing of Bose condensate in modern Atomic Beam Lasers and for hot plazma in thermonuclear fusion systems. However, this is beyond the scope of this site.
Let us call this unique value of the parameter a as "THE YIN-YANG PLATINUM SECTION", analogous to the famous "GOLDEN SECTION GS" (i.e.,GS=1/2*5^(1/2)-1/2=0.6180339887), suggested by Leonardo da Vinci!
>
[>a:=0.6339;plot([(a)*cos(t)+(1-a)*cos(3*t),(a)*sin(t)-(1-a)*sin
(3*t),t=0..2*Pi]);
plot([(a)*cos(t)+(1-a)*cos(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2),
t=0..2*Pi]);
plot([(a)*sin(t)-(1-a)*sin(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2),
t=0..2*Pi]);
Научный Подход к Yin-Yang Геометрии (Sergey Yu. Shishkov ,РОССИЯ, Shishkovser@rambler.ru) Здесь дается (ниже) наиболее обобщенное определение астроидно-подобной гипоциклоиды как траектории точки P вращающейся с угловой скоростью "omega1" =1 по окружности с радиусом "radius1" =a, центр которой также вращается вокруг начала координат по окружности с радиусом "radius2" =1-a, и угловой скоростью ""omega2" = -3, так, чтобы "radius1" + "radius2" =1, и "omega2" / ""omega1" = -3. Тогда для координат X [t], Y [t] этой точки P мы имеем: X [t] = (a) *cos (t) + (1-a) *cos (3*t); Y [t] = (a) *sin (t) - (1-a) *sin (3*t); 1-X[t]^2-Y[t]^2=factor(simplify(expand(1-((a)*cos(t)+(1-a)*cos(3 *t))^2-((a)*sin(t)-(1-a)*sin(3*t))^2)))=16*a*cos(t)^2*(cos(t)-1) *(cos(t)+1)*(-1+a)=16*a*cos(t)^2*(cos(t)^2-1)*(-1+a)=16*a*cos(t) ^2 * (sin(t) ^2) a* (1-a) =ПОЛНЫЙ КВАДРАТ! = >, если Z [t] =4*cos (t) *sin (t) * (a* (1-a)) ^ (1/2), то X [t] ^2+Y [t] ^2+Z [t] ^2=1; (то есть, Для каждого момента времени t {X [t], Y [t], Z [t]} С различными значениями параметра a мы получаем целый класс astroid-подобных гипоциклоид с ЧЕТЫРЬМЯ ЧАСТЯМИ. Ниже дается Maple5.4 Текст программы для составления графика этих траекторий.; > a=0.6339;plot([(a)*cos(t)+(1-a)*cos(3*t),(a)*sin(t)-(1-a)*sin(3*t), t=0 .. 2*Pi]); plot([(a)*cos(t)+(1-a)*cos(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0 .. 2*Pi]); plot([(a)*sin(t)-(1-a)*sin(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0 .. 2*Pi]); Просмотр также в МАПЛ 5: > factor(simplify(expand(1-((b)*cos(t)+(1-b)*cos(3*t))^2-((b)*sin (T) - (1-b) *sin (3*t)) ^2))); > Оптимальное Значение для параметра a = (1/2) * (3-3 ^ (1/2)) =0.6339, как будет показано в другом месте ; -). Этому значению соответствует наиболее "КРАСИВЫЙ" 3D-hypo-astroid. Такая конфигурация может служить также как Инь-Яньская МАГНИТНАЯ ЛОВУШКА для адиабатического охлаждения Бозе-конденсата в современных Лазерах на пучке атомов и для горячей плазмы в системах термоядерной реакции. Однако, это выходит за рамки данного сайта. Позвольте нам называть это единственное значение параметра как " ИНЬ-ЯНЬСКОЕ ПЛАТИНОВОЕ СЕЧЕНИЕ ", аналогично известному " ЗОЛОТОМУ СЕЧЕНИЮ З.С. " (то есть., З.С.=1/2*5 ^ (1/2) -1/2=0.6180339887), предложенному Леонардо да Винчи! > [> a:=0.6339;plot([(a)*cos(t)+(1-a)*cos(3*t),(a)*sin(t)-(1-a)*sin (3*t), t=0 .. 2*Pi]); plot([(a)*cos(t)+(1-a)*cos(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0 .. 2*Pi]); plot([(a)*sin(t)-(1-a)*sin(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0 .. 2*Pi]);
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