A nice feature of Bezier curves (and B-splines) is the convex hull property: since the curve must lie within the convex hull of the control points, any point ou

A nice feature of Bezier curves (and B-splines) is the convex hull property: since the curve must lie within the convex hull of the control points, any point outside the convex hull cannot be within the area bounded by the curve. So if you have a set of points and need to test which are inside a curve, you can immediately eliminate any that are outside the convex hull, potentially saving a lot of computation.