Self-Intersections of Rational Bézier Curves

As a curve fitting tool,rational Bézier curve is wildly used in geometric modeling.Determining and finding the self-intersections of rational Bézier curve is very important in computer aided design.In this paper,we present that a rational Bézier curve has no self-intersection for arbitrary positive weights if and only if its control polygon is compatible.The proof is based on the degree raising and toric degeneration of Bézier curve.