Section 3.3 - playing with hyperbolic tangents

I already defined tanh in section 2.4, but I can say it again :-)
tanh(z) = sinh(z)/cosh(z) = (ez - e-z)/(ez + e-z).

tanh(z) is near of -1 when z is small, and near of +1 when z is large.

And finally, like for sines and cosines, it is always between -1 and +1. This means that if you want a tanh to be between 0 and 255, you will do
tanh(...)*127+128, like for sines. You will have light in the areas where the argument is positive, and dark elsewhere.

Okay, maybe you wonder why I find that bunch of exponential interesting enough to devote a whole section to it?

Inequalities

First important use is for displaying the solutions of an inequation.
An inequation is an expression of the form a<0 or a>0. For instance, x^2+y^2-1<0 is the set of points inside the circle of radius one:
 r=g=b=tanh((x^2+y^2-1)*3)*127+128

So, to view the solutions of an inequation, you will write something like tanh(f(x))*127+128, where f(x) is the left-hand side of the inequation.
If its value changes quickly from negative to positive values, the border will be sharp, and will look blurred if it changes slowly. So if you want to make the borders sharp, multiply the whole inequation by some big constant.

 r(x,y)=tanh(k*(1-(x^2+y^2+.3*sin(7*angle(x,y)))))*127+128 k=30 k=3 k=0.3

Limiter

As tanh(z) looks a bit like z near zero and always remains inside the interval [-1;1], it can be used to constrain a function between two values.
Example:
 This is a view of the function sin(x)+x/2 (I showed the solutions of y=sin(x)+x/2 to obtain this picture), with blue lines at y=+1 and -1: Now, that one shows, in purple, the same function put inside an hyperbolic tangent (tanh(sin(x)+x/2)):
That way, you can for instance make sure that your function stays between zero and 255 without needing to write ugly min(255,max(0,..)) like stuff, using the same transformation I wrote above, tanh(...)*127+128.
this picture contains, on the left half, a function that got outside the [0;255], and that therefore got those holes due to modulo. On the right half is exactely the same function, but with a tanh applied to it. tanh((...-128)/127)*127+128 keeps the behaviour of "..." between 0 and 255 and makes sure it does not get outside this interval...

Rectangular signal

You get a funny effect when taking the tanh of a sine multiplied by a large number (something like tanh(90*sin(x))). As the sine will for most values be very far of zero, but alternatively with positive and negative values, taking its hyperbolic tangent will get you a function that switches between -1 and +1.
This picture was obtained doing tanh(30*cos(x*19)) (with x between -0.8 and +0.8), to which I substracted y and took the tanh (to highlight places where y>tanh(...) and keep dark other places. I took the opposite for the other colour component.
 tanh((y*17-tanh(30*cos(x*19)))*10)*127+128

(To Be Continued)

3.1: Monochrome Graphs
3.2: Implicit functions
3.3: Playing with hyperbolic tangents

Maxime Gamboni