Blending Of Quadric And Cubic Implicit Algebraic Surfaces Along Planar Sections

Blending of implicit algebraic surfaces is one of the central problem in CAGD. Although low degree algebraic is applied abroad in CAGD, the problem didn't get the perfect results until the late 1980's when the constructive algebraic geometry 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 c¹ blending cubic surface of two cylinders whose axes meet at a right angle and of which the clipping planes are perpendicular to the axes, and proved that thereexists a unique cubic blending surface if and only if r₁² + d₁²=r₂² +d₂², where r₁,r₂ is radius of the pipe.d₁,d₂ is the distance from the clipping plane to the intersection point of the axes of the two cylinders. But the algorithm is complicated if used for the application of blending of general surfaces.In this article we will study the constructive theory of a kind of implicit GC¹ blending algebraic surfaces by using algebraic geometry .The present paper deepens about the work of quadratic surfaces problem, pilot study to smooth blend different quadric surfaces, and obtain the condition and algorithm of cubic and biquadratic blending surface, and make maple program to calculate cubic and quadratic blending surface.。 Finding the c blending surfaces to smooth blend different quadric surfaces is equivalent to find the member in the intersection set of ideals, and is equivalent to be existence of nonzero solution. of some homogeneous linear equations 。 The essential results are as follows:(1) A quadratic algebraic surfaces and a cubic algebraic surface Theorem 1.1 A quadratic algebraic surfaces s(g₁) and a cubic algebraic surfacess(g2)> cubic blending surfaces exist if and only if the coefficient matrix A, of some linear equations satisfy Rank (/I,) -< 11 .Theorem 1.2 A quadratic algebraic surfaces s(gi) and a cubic algebraic surfaces s(gj), quartic blending surfaces exist if and only if the coefficient matrix A 2 of some linear equations satisfy Rank (A2) -< 34 .A quadratic algebraic surfaces and a cubic algebraic surfaces exists algorithm of blending surfaces.(2) Two quadratic algebraic surfaces and a cubic algebraic surface . Theorem 2.1 Two quadratic algebraic surfaces s(g ,?)(i = 1,3) and a cubic algebraic surfaces s(g2), cubic blending surfaces exist if and only if the coefficient matrix A 3 of some linear equations satisfy Rank ( j43 ) -< 21 . Theorem 2.2 Two quadratic algebraic surfaces s(g j)(/ = 1,3) and a cubic algebraic surfaces s(g2)> cubic blending surfaces exist if and only if the coefficient matrix A4 of some linear equations satisfy Rank(4,)^60 .Last, two quadratic algebraic surfaces and a cubic algebraic surfaces existsalgorithm of blending surfaces.