Interpolation of Geodesic B-spline Curves on Manifold Triangulation

Abstract

In allusion to the deficiencies of the existing methods of designing free curve on the surface,a generation method for geodesic B-spline curve interpolated from manifold triangulation surface is proposed.The shortest geodesic curve between two points on the manifold triangulation surface is used to substitute for the straight line between the two points in Euclidean space;and de Boor algorithm in Euclidean space is expand to curved space,so that the representation for geodesic B-spline curve on manifold triangulation is obtained.The given range of points constrained on the mesh surface are operated base on the B-spline interpolation theory in Euclidean space;control points are gained by inverse computation,and then projected onto the mesh surface;points on the mesh surface are treated as the initial control points of expected curve,from which the initial geodesic B-spline curve is finally generated.A reverse error compensation strategy which obtains control points constrained onto the mesh by simple iteration is proposed in order to approximate the curve with the data points as much as possible.According to the convex hull property of the curve,the convex hull region is virtual partitioned from the whole mesh surface;and then the computation of interpolating geodesic B-spline curve is only related to the mesh vertices within the convex hull region rather than the overall mesh,which greatly reduces the computational complexity.The results show that,the method is robust,effective,and able to meet the requirements of curve interaction design on surface.