Advance In The Research On Blending Of Implicit Surface

In the field of the Computer Aided Geometric Design (CAGD), surfaces can be classified into two categories: parametric surfaces and implicit surfaces. An implicit surface refers to the surface that is defined by the set of solutions of a real coefficient algebraic polynomial equation( f ( x , y , z ) = 0) , therefore we also call it the algebraic surface. The parameter surfaces have been at the center of research in geometric design for a long time due to their highly desirable properties such as the simple structure and easy computation. Compares with the parameter surface, the implicit surface has the following advantages: easily to judge a point in the surface, and the closure property under some geometric operations; for a parametric surface and an implicit surface, all results under geometric operations are denoted to implicit surfaces. Based on these advantages of implicit surfaces, it is very significant to study smoothly blending implicit algebraic surfaces.One of the elementary questions in CAGD is blending of implicit algebraic surfaces, which provides the theoretical bases for the design technology of space surfaces. Geometric modeling is an important topic of Computer Aided Geometric Design and Computer Graphics, it is widely used. In order to apply it in practice, Geometric modeling we need is usually realized by blending low degree algebraic surfaces. So the principle of blending of algebraic surfaces becomes the theoretical base of geometric modeling. It plays important role in the theory research and application. It is widely applied in blending cubic and designing body.There is no perfect results on blending of implicit algebraic surfaces until the late 1980's when the constructive algebraic geometric made great progress. In 1989, J.Warren described this kind of problem with ideal theory. The problem can be turn into finding the member in the intersection set of ideals. In 1993, Wu Wen-tsun studied the problem by using the characteristic set method and derived a sufficient and necessary condition for the existence of a GC¹ blending cubic surface of two cylinders whose axes meet at a right angle and of which the clipping planes are perpendicular to the axes. He proved that there exists a unique cubic blending surface if and only if r₁² + d₁² = r₂² + d₂², where r₁ ,r₂ are radii of the pipe, d₁ ,d₂ are the distances from the clipping plane to the intersection point of axes of the two cylinders. But the algorithm is complicated if used for the application of blending of general surfaces.It is possible for us to make the mentioned problem to be realized on computers with the development of symbolic computation and the system of computer algebraic. Many people are devoted in studying the blending algorithm and theory of surfaces, and have obtained a series of improved algorithms and the theoretical results. Under the influence of Mr. Wu, the people of the group of the researching computer algebraic in Ji Lin University have gained great achievement in the research of the blending algorithm and theory of surfaces, meanwhile solved some concrete problems.Recently, Yu Kai and his workmates discuss blending of algebraic surfaces and transform them into the solution of a set of linear equations by the method of computer algebraic. The existence condition of blending of surfaces is then given. In addition the blending algorithm of surfaces is presented under the assistance of the mathematical software. This article analyzes and surveys the mentioned research results on blending of implicit algebraic surfaces.Based on reading and understanding the literature on blending of implicit algebraic surfaces, we synthesize and think the research results of the question. It is classified into three chapters. In Chap 1, we introduce the history, the present condition and the development level of blending of algebraic surfaces; narrate some notions and definitions relating to the question; survey its important role in the theory research and application. In Chap 2, we have concluded some methods of blending of algebraic surfaces. Last by describing and comparing the research results on blending of two algebraic surfaces, blending of three algebraic surfaces as well as blending of several algebraic surfaces, the advance has been pointed out in the research on blending of algebraic surfaces. Thus, through the crosswise contrast, enables the people both to be allowed to understand advantages and shortages of each method and the achievement, and may determine further discussing question. Then this paper carries on the forecast to the research prospect of blending of algebraic surfaces.