Fit B-splines Curves Using Algebraic Splines Curves

In recent yeas, the algebraic spline has gradually been a tool with increasing concern, research and application in the computure graphics,because the implicit curve has particular advantages,for example, the implicit algebraic curve is closed under several algebraic operations (intersections, union, offset, etc.) and the implicit algebraic curve of degree n has more degrees of freedom ( n+3)) compared with rational function (2n+1) and rational parametric curves (3n-1) ,etc.Bajaj and GuoLiang Xu have defined a Algebraic splines in reference [1],it is a G k continuous piecewise real algebraic curves that can achieve local interpolation and approximation. Every real algebraic curve segment is the zero contour of the bivariate polynomial over a triangle in the BB form. In this paper, such degree n A-splines can achieve in general G 2 n?3 continuity by local fitting and still have degrees of freedom to achieve redundant data approximation.Let f ( x,y) be a bivariate polynomial of degree n with real coefficients,And p1 , p2,v1 be three affine independent points in the xy-plane. Then the Transform where 0≤αi≤1 andα1 +α2+α3=1. (α1 ,α2,α3) is a barycentric coordinate about ( x ,y) piont on the triangle [ ]p1 , p2,v1. In the barycentric coordinate system , F (α1 ,α2,α3) can be expressed in BB form where and bi jk be the coefficients of BB Form.In this paper ,we show how to construct locally cubic A-splines to fit the given cubic B-splines and give the calculable process and design algorithm and provide Matlab program of construct.In this paper,we consider two join configurations : convex join and nonconvex join.First ,we fit a segment B-spline curve without inflection point. As a example, we consider a two segment A-spline curve on two neighboring triangles .so this is a convex join. First, Form two G 1 control triangles [ p1 (1)V1(1)p2(1)] and [ p1 (2)V1(2)p2(2)],where p1 (1)= p2(2). To guarantee continuity of A-spline at every join point, we reqire that every join point of B-spline segment is vertice p1 or p 2 of triangles. Now we get vertices V1 (1) and V1 (2), to guarantee our A-spline passing through p1 and tangent with multiplicity three withα2 =0 at p1 in a triangle, we give a tangent lineα2 =0 at p1 = p and a tangent lineα1 =0 at p 2(1), then we have a intersect pointα2 =0 andα1 =0 as V1 (1) ,similarly we have vertice V1 (2) of other triangle. Hence, we have formed two triangles [ p1 (1)V1(1)p2(1)] and [ p1 (2)V1(2)p2(2)]. Second, we provide coefficients of two segment A-spline curve by fitting to a parametric curve with convex join .last, we obtain a sequence of point (α1 ,α2,α3) in terms of barycentric coordinates, by using A-spline curves, then we obtain asequence of points ( x ,y) in terms of Descartes. From which we can obtion a desirable shape of A-spline curve.Second, we fit a segment B-spline curve with inflection point. As a example, we consider a two segment A-spline curve on two neighboring triangles .but this is a nonconvex join. First, similarly form two G 1 control triangles [ p1 (1)V1(1)p2(1)] and [ p1 (2)V1(2)p2(2)],but this is a nonconvex join of two triangles. It requires inflection point of curve as vertice p1 (1)= p2(2). Second and last are the same as convex join configuration.Last by an actual example, we show the algorithm in this paper. By compare with two curve picture, the degree of fitting is good and the algorithm is actually effective and we obtain a desirable aim.