[2005] Finding a Arbitrary Pointa on a Cardinal Spline

I'm moving bitmaps along lines drawn by the user in a football play creation application. That works fine, but I am now attempting to have my bitmap players move along cardinal spline curves which is much more math intense.

Is there a way to get the points the GDI pixelates when drawing a curve like this? My plan would be to get several points and then have my player move along a straight line between them; I'd have enough points so the user can't tell it's not a true curve.

I've searched high and low without much success so far, so any help would be greatly appreciated. :)