math - Transform 2d spline function f(t) into f(x)

Question

So I've got a special case set of cubic splines, whose 2d control points will always result in a curve that will never cross itself in the x axis. That is, the curves look like they could be a simple polynomial function such that y=f(x). I want to efficiently create an array of y coordinates along the spline that correspond to evenly-spaced x coordinates running the length of the spline segment.

I want to efficiently find the y coordinates along the spline where, for instance, x=0.0, x=0.1, x=0.2, etc., or approached another way, effectively transform the f_x,y(t) style function into an f(x) function.

I'm currently using a 4x4 constant matrix and four 2d control points to describe the spline, using matrix constants for either Hermite or Catmull-Rom splines, and plugging them into a cubic function of t going from 0 to 1.

Given the matrix and the control points, what's the best way to obtain these y values over the x axis?

EDIT: I should add that an approximation good enough to draw is sufficient.

Answer 1

Often people will use a root finding technique (like Newton's Method) if an numerical approximation is good enough.