Blending Of Two Cylinders With Axes In Different Planes

Blending of implicit algebraic surfaces is one of the basic theory problems of CAGDand is the theory basis on space geometry structure. Recently, with the proposing and application of Groebner basis and method of characteristic set given by Mr. Wu Wen - tsun, some conclusions of commutative algebra and algebraic geometry are constructive. Therefore, many problems of implicit algebraic surfaces are computable. In this paper, we will study the sufficient and necessary conditions of smoothly blending two cylinders whoes axes in different planes using cubic surfaces by the means of algebraic geometry.Primary hypotheses: Let gi ( i = 1,2) are two quadric irreducible polynomials. They determine two quadric cylinders s ( gi) (i = 1,2 ). hi ( i = 1,2 ) are linear polynomials. They determine two different planes s(hi)(i = 1,2), s(gi) , s(hi)(i = 1,2) intersect transversely with irreducible quadric curvesLet g1, g2 are the polynomials of the two cylinder, r1, r2 are radius respectively, d is the nearest distance between the two axes, is the angle that the two axes project on the. Choose the axes of s(gi) as x - axis. The origin is chosen as the cross point of public perpendicular line of the two axes z axe is the public perpendicular line. Aright- hand right angled coordinate system is created.In this coordinate system, g1 and g2 On be expressed as follows:Let s(h1) and s(h2) are two planes perpendiculer respectively to the axes of s(g1) and s(g2), then x,y,z can be uniquely expressed by h1,h2, h3Put x ,y, z in g1 and g2, choose properly, then the constant term in expression gi is non zero, and for any no zero, the surface determined by giand agi are the same, therefore we let the constant term of gi is 1. Let the expression of gi isWe have some conclusion:Theoreml Given two guadratic surfaces s(g1),s(g2) and corresponding clipping planes s(h1) and s(h2), there always exists cubic polynomail such that s(f) contains curves s(g1,h1) andProposition 2 There exists cubic polynomail f= u1g1 + b1h12 = u2g2 + b2h22,such that s(f) meet s(gi) along s(gi ,hi) with GC1 continuity respectively if and anly if one of the follawing conditions holds:WhereWhereWhereWhereWhereWhereWhereWherethe expression of blending surface is determined by(2.3.1).For the ptoblem of blendiny two aylinders with axesin different planes, the existing of cubic GC1 blending surface is determined by the redii r1, r2, the nearest distance d between two axes, the rotation angle and the clepping distance as also. We haveTheorems Let s (g1), s (g2 ) are two cylinders with axes in clifferen planes, there redii are r\ and r2 respectividy. d is the nearest distancs d and is the angle between two axes are clipping distances. Then there exit-sts a cubic surface s(f) meet s(gi) along s(gi,hi) with GC1 continuty respectively if and only if:Where 23 is non - zero constant, b1 and b2 satisfy(2.3.1).