The Smooth Connection Algorithm Between B-spline Patches

B-spline methods are widely used tools in CAD/CAM field such as solid modeling ,viaual reality and so on.It not only inherite all the advantages of Bezier method but also have more freedomfor shape modifying ,and be widely accepted in industry. However there are some important problems to resolve in CAGD.The geometric continuity between two adjacent surfaces and that of serval surfaces is one of the most inportant problems.The theory of geometric continity on Bezier surfaces have been almost perfect as yet.And there are a lot of results on the topic of geometric continuity of Bezier surfaces .But the theory about B-spline surfaces isn't so much as those about Bezier surfaces.An algorithm for G1, G2 blending of tensor product B-spline surfaces(for example bicubic tensor product B-spline surfaces) is presented. The capability conditions of conor with more then three patches are studied and the routines of the algorithms are given . The algorithm can be easily extended to coner with any number of patches around it. The algorithms presented avoid the drawback of decomposing the B-spline surfaces to serval Bezier patches ,so they recuce the complexity and the time for blending. Now we are going to introduce the main idea of the algorithms.Suppose there are two B-spline surfaces B(u,v) and C(s,v) with B(0,v)=C(0,v) for v [0,1](G0 continuity).Withour loss of generality,suppose B and C contain the same knot vector in v direction .Then the G1 continuity conditions are equvalent to that:with a(v),/3(v),7(t;) are piece wise polynomial functions.Generally we let a(v),/3(v] be constant and 7(1;) be a peicewise linear functions . Then from [3] we know that:C(0,i;) is a global cubic polynomial curve and 7(6) must be a global linear function .Let 7(7;) = 70(1 - v) + 7iu,then 7(u)^f |s=o is a global cubic polynomial curve. Because the cubic B-spline functions are the basis of cubic polynomial function space, =0can be represented in B-spline form.Let B(u, u) = E E bijNi,3(u)Nj:3(v),C(u,v) =E E CijN^MNtfM, 7(w)lg|s=o = E PjNj,s(v) .Then (*) is equvalent toSo if the {bilJ-}jn=0,{bolj}jn=0,t/ien{co,,-}7=0,{Pj}"=o are fixed ,we can get c1,j which satisfy the G1 continuity conditions.Algorithm 1:Step 1-.Firstly we convert (u)to a totally cubic B-spline curve,with Ferguson method[21]. We still use (v) to note the common boundry curve after being modified .Secondly ,we anticompute the B-spline control points of (u),that isStep2:Choose 1;Step3:Anticompute the B-spline control points of (v) /(v),that isStep4:Using (3.8) to compute the B-spline control points of CH and replace the corresponding points of C(s, v) For G2 continuity ,there are added conditions about the second derivative vectors of the two adjacent B-spline except of the first derivative vectors ,we can also imitate the idea of G1 continuity to resolve the c1,j.Algorithm 2 :StepltModify the control points of B(u,v)and C(s,v) to make them Continuous; Step2:Choose ;Step3:Anticompute the B-spline control points of (v)D6(v),2(v)D7(v), (v)D2(v) in (3.13);Step4:Replace the corresponding control points of C2i with the points we have compute from Step3.Now study the continuity problems of several patches around one corner. The key point is that if we blend every two adjacent B-spline surfaces with the algorithm abrove ,whether the constraints will be antinomy? In general case they are. So we must find a method to avoid the contradicts.Firstly wo consider the intance of a corner with three (odd number) patches around it. The equations that would make condtradicts are the following:(**)mainly imply that E1 - O,E2 - O,E3 - O are coplanar.So we can express with i0 according to the relationship of E1 - O, E2 - O, E3 - O.Suppose, then we can express ai, i with i0 and a,b,c. And a,b,c can easily be resolved from the equation above. Then in (***),only Gi are unknown and the matrix of coefficient is nonsingular.So after fix i0,we can resolve Gi from (***).Algorithm 3:Step1:Anticompute the B-spline control points of B,C,D.Let Hi,Ii, Ji be the...