Gaussian Curvature Connection Of Bézier Surfaces On Rectangular Domains

The smooth connection of the surfaces is an important research topic in computer aided geometric design.It is widely used in the exterior modeling of industrial products.In practical application,there are three most common ways to splice the surfaces: the connection of the surfaces,the connection with tangent plane continuity(called G1 continuity)of the surfaces and the connection with curvature continuity(called G2 continuity)of the surfaces.This paper mainly discussed three aspects of the problem: the definition of Gaussian curvature connection of Bézier surfaces,the conditions of Gaussian curvature connection between two adjacent Bézier surfaces on the rectangular domains and the conditions of the Gaussian curvature connection of n Bézier surfaces on the rectangular domains with a public vertex.The first two problems are discussed in Chapter 3:1.The concept of Gaussian curvature connection between surfaces is proposed.The Gaussian curvature connection between surfaces means that the Gaussian curvature of the two adjacent surfaces at each point of their common boundary line is the same under the condition of tangent plane continuity.Gaussian curvature connection of the surfaces is a new type of splicing method,whose conditions are stronger than the that of tangent plane continuity,so its smoothness is also better.But the conditions are weaker than the connection of curvature continuity,so it is easier to realize.2.The conditions of Gaussian curvature connection between two adjacent Bézier surfaces on the rectangular domains are proposed.Based on the concept of Gaussian curvature connection of the surfaces,using the tangent plane continuity of two adjacent Bézier surfaces and the definition of Gaussian curvature,combining with differential geometry knowledge,the conditions of Gaussian curvature connection between two adjacent Bézier surfaces are found after the necessary calculation.This needs to satisfy three equations.The last problem is discussed in Chapter 4:3.A sufficient condition for the Gaussian curvature connection of n Bézier surfaces on the rectangular domains with a public vertex is proposed.For the smooth connection of Bézier surfaces on the rectangular domains with a public vertex,connection with tangent plane continuity has been realized.Because the conditions of curvature continuity are complicated,it hasn't been solved yet.So it is very meaningful to find a method which spliced smoothness is better than that of tangent plane continuity and the conditions are weaker than that of the connection with curvature continuity.The Gaussian curvature connection of the surfaces is such a splicing method.It can be seen from the introduction of the third chapter that the Gaussian curvature connection between two adjacent Bézier surfaces needs to satisfy three equations.According to the conditions of Gaussian curvature connection between two adjacent Bézier surfaces,the equations should be consistent.According to the consistency,the existence conditions of the solution of equations and the Gaussian curvature connection of Bézier surfaces on the rectangular domains with a common vertex are obtained.The smoothness of Gaussian curvature connection is better than that of tangent plane continuity,and this method is easy to be realized in practical application.