Self-intersections Of Rational Bezier Curves

As a curve fitting tool, rational Bezier curve is wildly used in geometric modeling. Planer rational Bezier curve may have self-intersection, and its self-intersection has an important influence in the offset design of curves and surfaces, therefore judge and calculate the self-intersections of rational Bezier curve is important for the computer aided geometric design. Lasser used the de Casteljau algorithm to propose that a planer Bezier curve may have self-intersections if the sum Σ|αk|of the amounts of the rotation angles of the Bezier polygon edges is greater than π And using the convex hull property and the approximation property of the rational Bezier polygon, Lasser gave the algorithm for finding all the self-intersection points. Tiller et al. gave the algorithm for calculating the self-intersections of the B-spline curves. But this method may be fail, because a curve can have a loop even though its control polygon has no self-intersection. By defining the compatibility of the control polygon of the planer rational Bezier curve, the degree elevation, and the toric degradation of the rational Bezier curve, we propose and prove that a planer rational Bezier curve has no self-intersection for arbitrary positive weights if and only if its control polygon is compatible.The thesis is organized as follows.In the first chapter, we give a brief introduction of the background and the researches on self-intersections of the planer rational Bezier curves.In the second chapter, we give a brief introduction of the Bernstein basis, the rational Bezier curve and its basic properties. Moreover, the toric degeneration of rational Bezier curve is also introduced.The compatibility of the planer rational Bezier polygon is defined in the chapter3. By the degree elevation and the toric degradation of the rational Bezier curve, we propose and prove that a planer rational Bezier curve has no self-intersection for arbitrary positive weights if and only if its control polygon is compatible.