Section 2.4 - Other Functions

Summary

angle(h,v): the argument (angle), in radians, of vector (h,v).

sinh(a); cosh(a); tanh(a) are the hyperbolic functions. The inverses (a...) are available but do not work yet.

a^b: Raise a to power b. Use sqrt(a) instead of a^0.5 for square root.

ln(t); log(t,b)

r(c,x',y'); g(c,x',y'); b(c,x',y'): Inter-referencing. The possibilities of these functions are infinite and exposed in chapter three.
c is a project number, it must be a constant. (x',y') is the coordinate that will be read.

Operators priority is %; +; -; *; /; ^. When having the same operator several times, the left-most one has more priority. Example: a/b^2/c is read as a/((b^2)/c), which equals to "acb". This is not as we usually read them. Take care!

Angle(h,v)
This function takes an horizontal distance as first argument and a vertical one as second argument.
It returns a number between -pi and pi and some undefined value when both arguments are zero.

I have found two important uses to this function.
The actual reason why I created this function was for spirals.
You calculate the sine of the sum of the angle and the distance. You get the angle with angle(dx,dy) and the distance with sqrt(dx^2+dy^2)

Try the following equation and try to change the constants in it:

sin(angle(x-128,y-128)+sqrt((x-128)^2+(y-128)^2)/3)*127+128

If you replace the power of two by power of a larger even number (eg ten), you'll notice that your spiral gets square (you also have to adjust the other numerical constants for that).
If you multiply the angle() function in the above equation by a whole number, you will get more than one spiral on the same picture (actually as many as the number you multiplied by)
If you add a constant somewhere inside the sine function, it will make your spiral turn. If you multiply or divide the distance by some constant, you can set the distance between two rings of the spiral.

Note that instead of putting the angle() function on its own, you can put some function of it, you can do polar graphics.
For instance take its sine (having multiplied it by a sufficiently large number). You would get a picture like this:
If you want to have only one ring, then use an implicit function.

The other main use of the angle function is precisely for implicit functions.
If you want to get only the results of the equation on some area of the picture, then you can use an arctangent of the angle function, maybe having first taken its absolute value.
More on this in the implicit function chapter.

sinh(), cosh(), tanh() are hyperbolic functions. Their actual meaning will probably never be of any use for picFX, so I won't explain it here. (Well, shortly, it's the same as trigonometric function sin cos tan, except that we are not moving on a circle but on the right branch an hyperbola)
Their mathematical definition is sinh(z)=(e^z-e^(-z)), cosh(z)=(e^z+e^(-z)) and tanh=sinh/cosh.
What is important to see here is the exponentials e^... They cause the cosh and sinh functions to take really big values once the argument is a bit large.
sinh(x) has similar properties as x^3, ie it has the same sign as x and sinh(0) equals 0, and sinh(-x)=-sinh(x) (ie the graph is symmetric through the center).
cosh(x) is roughly equal to abs(sinh(x)), except that cosh(0) equals one, ie cosh is always bigger than one.

The physicians would tell you that the hyperbolic cosine is the equation of the chain, ie you hold a chain between your two hands, and the curve, looking like a parabolic but *not* a parabolic is a hyperbolic cosine :-)
Well, this interesting feature has no use in picFX, unless you want to draw realistic strings or chains in your picture [but remember then to take care of the perspective] ;-)

I use sinh and cosh very rarely. It is possible to replace a square somewhere by an hyperbolic cosine, just for fun...(But remember that cosh(x) is always bigger or equal to one, while x^2 goes down to zero!)

tanh, however, is always between -1 and 1, and can replace very handfully arctangent, as you do not have to divide by pi every time you use it :-)

(You can think of arctangent as being roughly equal to the sign of x (-1 or +1) but with a smooth change near zero...)

Mmh. Well.

You have a headache?

Shall I dear to speak about the ^ (power) operator before we go to easy stuff again? :-)

You know, the main reason why I created picFX was just to have fun, and I invite you to have fun as well... If you feel tired just write anything in the fields (I mean, anything but still mathematical :-) and look what comes, then try to transform it a little until it reaches to something nice...

...

Welcome again ;-)

a^b, as said above, is a to the power of b. Usually you will have b constant, eg for calculating the square of some value (eg for the famous Pythagoras theorem, when calculating the distance between two points).
When working in floating point mode, you can have b fractional. Note that a negative number raised to a fractional power is not defined as a real number and therefore returns some random value in picFX

Raising a variable to a large even number is gives near of (but superior to) zero when between -1 and +1, gives one on these two bounds and yields very large values out of this [-1,1] range.
If you do something like adding one and doing the inverse, you will get about one between -1 and +1, and about zero elsewhere.
(Then you can multiply an expression by this to put everything out of this strip to zero...)
This picture shows this function when using 20 as exponent:

And raising to an uneven number is the same except that the value is negative when the variable is negative.

sqrt(z) is equivalent to z^0.5 but is faster and more accurate (and works in 32bits mode)

ln(z), log(z,b) are the inverse functions of the exponential.
For instance, log(y,10) is the inverse of x^10, which means that log(y,10)=x is equivalent to x^10=y. For instance, log(1000,10) is three (as 10^3=1000)...
(b is the base. log(z,b) being for z the inverse of b^x; ln(z) is equal to log(z,e)).
b must be positive and different from one...

ln(z) is not defined (in real numbers) when z is smaller or equal to zero.
ln(z) is equal to zero when z equals one, and runs towards minus infinity when z gets near of zero, and slowly grows when z gets a little larger.
On this picture, you see the graph of the function, horizontal is the value you give to z and vertical is the value of the logarithm: .


Index

Chapter two: Mathematical function summary

2.1: Four operators
2.2: Trigonometric functions
2.3: Min, Max, Modulo, Abs and Neg
2.4: Other functions

Chapter three: Simple graphs for a start

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