Mathematical functions
What it determines the new value of rotation that our pendulum will have on every instant  i.e. frame  it is the associated mathematical function. It is well know that the function sine has an oscillating course, but constant in the time, and that it oscillates between values +1 and 1; therefore if we associate the simple function sine to the rotation of the pendulum, where the rotation value is given instant to instant  frame to frame  from the value that assumes sin(a) (a is our quarter variable, that it increases to every frame), the pendulum will oscillate between +1 degree and 1 degree, too much little for being appreciated visually:
this is the reason for which we use the variable maxang: it works as multiplier of sin(a); if sin(a) oscillates between +1 and 1, and we give maxang to a value of 60, then
maxang*sin(a)
it will oscillate between +60 (degrees) and 60 (degrees).
Now we want that the pendulum does not oscillate forever, but that it stops its movement, gradually. In order to realize this, we need a function that multiplied to "maxangle*sin(a)", at the beginning it does not have influence but with passing of the time reduces the oscillation until 0 (that is pendulum stopped). Moreover this function must have an effect that carry gradually to 0: the perfect function for this result is just the power function, that is
n^a
where n it is a real number >0. It is well know that
the function power has these property:
 if n>1 it has increasing graph;
 if 0<n<1 it has decreasing graph, and if a is "much large" the function catches up the 0 (and therefore multiplied for "maxangle*sin(a)", when a is "much large", it reduces the oscillation to 0);
 se n=1 it is constant of constant value 1;
 for a=0 it assumes value 1 any is the value of n (and therefore
multiplied for "maxangle*sin(a)" at the beginning,
when a=0, it does not have influence.
For how much said, it appears clearly that we must choose an n, than
in our case it will be the timesmorz variable, than 0
Now we can create the final function, that it encloses sin and power: according
to ActionScript syntax it will be
((Math.sin(a))*(Math.pow(timesmorz,a))*maxang
What about the n_osc variable? It is used in the function in this mode:
Math.sin(n_osc*a)
its duty is to increase the number of totals oscillations that happen in the same arc of time. Example: if n_osc=1 we will have 20 oscillations in 10 sec, if n_osc=2 we will have 40 oscillations in 10 sec and so on. So, the effect of this variable is, for the final motion, of having a "slowmotion" if n_osc<1 and an accelerated one to the growing of n_osc>1. The definitive function used in the movie is:
((Math.sin(n_osc*a))*(Math.pow(timesmorz,a))*maxang
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Added: 20010118 Rating: 9 Votes: 961 
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