Convexity Analysis And Convex-preserving Connection Of Parametric Curves And Surfaces

Parametric surface convexity and the continuity of surfacepatches are both important topics in the CAGD. This paper isa further study of the convexity in the modeling of two kindsof parametric curves and surfaces, i.e, Bezier, B-spline. Itmainly explores the convex relation between control vertex andthe parametric curve and surface convexity in geometry,fromwhich we have obtained some sufficient conditions. Based onthese conditions, we have got the Geometrical continuityconditions between two adjacent parametric surfaces patches.The first chapter introduces some main research related tothe convexity of surface-modeling in recent years.The second chapter introduces the definition of the globalconvexity for planar parametric curves, with which the globalconvexity theorem of Bezier curves has been proved. Whileapplying to local situation, we obtain the definition of localconvexity and prove that Bezier curve is convex whilecharacteristic polygon is convex. These properties are alsotenable for B-spline curves.The third chapter chiefly analyzes the convexity of Beziersurface with a given control mesh, deduces the condition whichcontrol vertex should satisfy when surface is convex and itsexpression. It also explains the convexity theorem of Beziersurface proven by Hua Xuanji is an essential condition of thisconclusion. We still have established the relation betweenmesh shape and convexity of surface in geometry. As for theuniform and quasi-uniform B-spline surface, the similarconclusion can be drawn.The fourth chapter will apply the convexity condition tosurface connection. We give out an convexity preservingalgorithm for the Geometrical connection between two adjacentparametric surfaces patches. Some examples are given in the endof paper.