Cone Spline Approximation And Refinement Scheme For Triangular Mesh

(1) Cone spline approximation via fat conic spline fittingFat conic section and fat conic spline are defined. With well-established properties of fat conic splines, the problem of approximating a ruled surface by a tangent smooth cone spline can then be changed as the problem of fitting a plane fat curve by a fat conic spline. Moreover, the fitting error between the ruled surface and the cone spline can be estimated explicitly via fat conic spline fitting. An efficient fitting algorithm is also proposed for fat conic spline fitting with controllable tolerances. Several examples about approximation of general developable surfaces or other types of ruled surfaces by cone spline surfaces are presented.(2) A refinement scheme based on point-normal triangle for triangular meshA refinement scheme based on PNT(point-normal triangle) for triangular mesh with vertex normal is proposed, In each step of the refinement, a planar triangle with three vertex normals is subdivided into four planar triangles of the same form. The new vertexes, that associate with an edge of the planar triangle, is on a cubic Bezier curve determined by the two end points of the edge and their normals. The normal of the new vertex is interpolated by the normals of the two end points of the edge. We subdivide very single planar triangle independently, and finally get a smooth surface interpolating the vertices of the original triangle mesh and their normals.