defining a value along a uniform b-spline?

Im building a uniform b-spline using the cubic-bernstien-polynormials equation to define 3 cubic curves in 4 control points : p1,p2,p3,p4, as:

v = v1*(1-t)^3+vc1*3*t*(1-t)^2 + vc2 *3*t^2*(1-t)+v2*t^3

v1,vc1,vc2,v2 being tangent coordinates generated from the 4 control points. Now with bernstiens-polynormial t is defined as a value of 0 - 1 for a value alone its cubic curve. Given this fact is it possible to get a value along the entire b-spline constructed from 3 cbp's.

eg if it was 0.5 or 50% it would fall into the second cubic curve, will i have to in theory walk the t(i) across each curve and do some complex math algerbra? i could use the same math to build the v1,vc1,vc2,v2 on the cubic curves, but that will define a value thats a derivate of them so not really the answer.

This is the same with cardinal splines, there constructs of cubic curves, but how do you get a value along the entire spline?

eek