Precondition
Be sure to know the basic math: Math HomeIntroduction: Identity Functions
If h(x) with inverse hInv(x) is a bijective function (one-to-one), define the identity function I(x) = hInv( h(x) ).Because hInv is the inverse of h: I(x) = hInv(h(x)) = x; I(x) = x.
Examples for Identity Functions I(x)
- I(x) = a(x) = sqrt(x*x)
- I(x) = a5(x) = pow(pow(x,5),1/5)
- I(x) = log(exp(x)) = iExp(x)
- I(x) = asin(sin(x)) = iSin(x)
- I(x) = acosh(cosh(x)) = iCosh(x)
Identity Functions and Fractals
E.g. the generation function for the nova set is nova(z,c) = z + 1/z + c.Generated is a sequence dependent on z, with a "disturbing parameter" c.
Transforming nova(z,c) with an identity function I(x) there are four possibilites:
- I(nova(z,c)), e.g. iSin(nova(z,c))
- nova(I(z),c)), e.g. nova(iExp(z),c)
- I(nova(z))+c, e.g. iSin(nova(z))+c
- nova(I(z))+c, e.g. nova(iSin(z))+c