How to recreate the math behind photoshop curves

Question

Basically, what I want to do is understand how to calculate the values along a \'curve\' as represented here in the photoshop curves box:

Answer 1

I'm not familiar with C-R, but if it's like beziers, than the T parameter varies from 0 at one endpoint of a curve section to one at the other end point. It's how you "step along" the curve. So you can't just plug an x value you in. You can either sample a long at some arbitrary interval or you can use some sort of goal seeking algorithm to approach a given x value to whatever amount of precision you deem necessary.