How to draw Bezier curves
This sketch demonstrates how Bezier curves are constructed. Move the blue control points to change the curve's shape.
Given a fixed number (n) of control points, you create a new set of n-1 control points by moving a proportion of the way between successive points. Repeat until you only have one control point, and the locus of that point is the Bezier curve. What would the equation of a curve generated in this way look like?