The Construction Of G～1 Smooth NURBS Surfaces Over Arbitrary Topology Type

In the real world, the shape of objects are very complex. One single NURBS patch can only represents a surface with the simple topology type, so it is necessary to adjoin several different NURBS patches together for meeting the demand.of free-form surfaces modelling, that is, the surfaces of objects are usually divided into some small patches, then each patch is fitted by a NURBS patch, at last, the smooth surfaces model is obtained by joining all the NURBS patches smoothly. This is just the problem of geometric continuity for NURBS surfaces that we consider. In order to construct G¹ smooth NURBS surfaces over arbitrary topology type, we consider the NURBS patches with arbitrary degrees and single interior knots. The main results of this paper include the following three parts.1. The G¹ smooth conditions and algorithm of constructing adjacent G¹ NURBS surfacesSuppose two NURBS surfaces(p×qth,(p|₎× qth) are denned aswhere {P_i,j} and {(P|-)_i,j} are the 3D control points, {ω_i,j} and {(ω|-)_i,j} are the weights, and are B-spline basis functions defined on the knot vectorsboundary R(0, v) = (R|-)(o, v), and relabel the knot vectorsUtilizing the necessary andsufficient conditions of G1 continuity between two adjacent NURBS surfaces in [44], and supposing the connection function a(v) = a, 7(1*) = 7 and 5(v) = S are constant , /3(v) = av + b(l — v) is a linear polynomial function, we obtain a sufficient conditions of G1 continuity between two adjacent NURBS surfaces. Theorem 1 Surfaces R(u, v and R(u, v) are of G1 continuity if the following Eq.(l) holds,I = 1,2, ? ? ? , m - q(1) j = 0,1, ? ? ? , mwhere X, = ^,*o = 0, tm^1 = 1, ABJ = Blj+1 - B), AkB\ =B\ - Blj_v VfcBJ = VCV"-1^.),B'j(j = 0,1, ? ? ? ,q - 1,a = 1, ? ■ ? ,m - q + 1) are therational Bezier control points of i^(O,t;)(t;€ [t,-i,t,]).ibi = ^(^7 +2. G1 smooth NURBS surfaces with a n-Patch corner and their constraint conditionsSuppose $ = {/^(uj.Ui+i)}"^1 be a collection of n NURBS patches around a corner P and i := (n + t)mod n, gjth-degree B-spline basis functions {Nji,z are the weights that correspond to P,Ei,Fi,G{ respectively. Denote P — u>oP,Ei = uitiEi,Fi = oJi^Fi,Gi = u>i^Gi. To each common boundary curve Fj, denote the connection functions by Qi,/3j = ajti + h(l - u),7j(qoi > 0) and 5i, and let Aij = Aj,Cij = Cj (i - 0, ■ ■ ■ ,n - \,j = 0, ? ? ? ,77ii). Based on Theorem 1, we obtain two groups of constraint conditions firstly when we stich the corner.The first constraint conditions are as follows- P)- W0) = 0- P) = 0(2)where kn = "l^"1 , fo = -r3*-, A*3 = Tffi1"' ? We can determine the tangent points and the corresponding weights by the first constraint conditions. The second constraint conditions can be expressed asfwhere u)j = (kn + ki3-qtki5-5i)uiiti -,dn-l)T,2i i,ck should satisfy the following supplementary constraintn-2 n-2 T,Then we haveTheorem 2 If Eq.(2) and Eq.(3) hold, and every two adjacent NURBS patches satisfy Eq.(l) with j'? = 2, ■ ■ ? ,mj, then NURBS surfaces with n-patch corner are of G1 continuity.3. The construction of Gl smooth NURBS surfaces over arbitrary topology typeBased on Theorem 1 and Theorem 2, we present a local scheme forconstructing G1 smooth biquintic NURBS surfaces over arbitrary topology type. The biquintic is the lowest degree which exists the local scheme. The local scheme is described as following steps,Step 1;Determine the tangent points, curvature points, twist points and the corresponding weights by the two groups of constraint conditions, and the connection functions are determined at the same time.Step 2;Calculate other m - 5 control points and weights of the common boundary curve by the first equation of Eq.(l).Step 3;Calculate the two rows closest control points and weights to the common boundary by the second and third equation of Eq.(l).The result of this paper can be applied to the irrational surfaces, such as Bezier surfaces and B-spline surfaces. In reverse engineering, it can be used to construct the smooth open surfaces and closed surfaces. It also can be used to adjust the local shape of the continuity surfaces.