I have a control that's used as an exponent in a pow function. This changes the values in an ease in fashion. Is there a way to turn this into an ease out mathematically?
I know for example, ease in cubic is:
pow(t, 3)
but ease out cubic is:
(pow((t - 1), 3)) + 1
and ease out quart:
float t = t2 - 1
-(pow(t, 4) - 1)
So the formula changes quite a bit, and I need a generic way so I can use values like 4.2, 9.7, as the exponent etc.
For integer exponents this is called Hermite interpolation.
One can also interpret this task as an application of the chinese remainder theorem in polynomial rings
f(x) == 0 mod x^p
f(x) == 1 mod (x-1)^p
The solution has the form
f(x) = a(x)*x^p+b(x)*(x-1)^p with deg a(x) < p, deg b(x) < p.
Inserting into the second equations gives
a(x)*x^p == 1 mod (x-1)^p <=> a(y+1)*(y+1)^p == 1 mod y^p
which can be solved via binomial series, i.e., direct power series arithmetics without solving linear systems of equations,
a(y+1) = 1 - p*y + p*(p+1)/2*y^2 - p*(p+1)*(p+2)/2*y^3 +-... ...*y^(p-1)
For non-integer exponents, why would you want to do that?
Also look up "sigmoid functions".
Update
If continuity of derivatives is not so much a concern then for any positive a you can use c*x^a for 0<=x<=0.5 and 1-c*(1-x)^a for 0.5<=x<=1. To close the gap at x=0.5, the constant has to be chosen such that
1 = 2*c*0.5^a <=> c = 2^(a-1)
which can be implemented for 0<=x<=1 as
y = (0.5>x) ? 0.5*pow(2*x,a) : 1-0.5*pow(2*(1-x), a);