Uniformly-solving Algorithm Of Interpolation And Reconstruction Of Complicated B-spline Surfaces

In this paper, we put forword a kind of uniformly-solving algorithm which is applied to work out control vertexes of complicated curves and surfaces interpolation on one hand, and a virtual data points method of complicated B-spline surface reconstruction on the other.When constructing free-form curves and surfaces based on parametric cubic B-spline curve P(t) = Nj,4(t)Pj+2 and surface P(u, ) = as fundamental mathmodel in egineering, we often meet with the problem of reversely-solving control vertexes through data points at first.For simple curves and surfaces ( including no multiple knots ),equation about control vertexes can be produced by knot reinsertion. Chasing method is valid for this system of equations, because its coefficients matrix characterized triadiagonal. While for complicated ones ( including multiple knots ), the coefficients matrix produced by the same method is odd, so we are not able to get the solution. In 1984, Pei-jun Xu and Xue-jun Situ proposed Piecewise Algorithm, in which multiple knots are all taken as end points after pieceing to avoid them successfully. Then a curve must be pieceing at each multiple knots. Additionally, the procedure is fussy. Thirdly, Pieceing Algorithm is only suitable for cases of one or two whole arrry of mutiple knots for complicated surfaces. Uniformly-solving Algorithm avoids these drawbacks.Pieceing is unecces-sary, multiple points can be dealt with only once and all the control vertexes points can be solved from an uniform system of equations. This algorithm is suitable for*all cases and all kinds of end conditions ,including free-form end, parabola end and clamp end . Corresponding formulas about coefficients are also given in the paper.Nowadays surface reconstruction is one of topic issues in the field of surface modeling. It plays an important role in many fields such as design of bodywork of saloon cars, cartoon facture about sculpture surface of human faces and 3D reconstruction of medical images ect. Representative methods mainly include piecewise linear method, subdivision surface method,implicit surface method and parametric surface method etc. While the method of surface reconstruction based on B-splineis still immature. Eck and Halstead developed some inaugurative work. Halstead proposed a method based on B-spline to approximate normal vector direction of submitted reconstruction surface. It's a pity that this method cannot reconstruct complicated surfaces with cusp and arrises. Virtual Data Points just breaks through the limitation.It succeeds to reconstruct complicated B-spline surfaces with crease, ruled surfaces, cusp, arrises or plain patches at random location. Its feasibility is illustrated by some examples given in this paper.