Research On Interpolation And Reconstruction Of Curves And Surfaces In CAD

Interpolation and reconstruction of curves and surfaces are important topics in the CAD fields. In this paper we profoundly research on interpolation of PH curves and surface reconstruction. PH curves are widely used such as designation of mechanism part, highway and railway, robot, etc. Surface reconstruction is increasing important in geometric modeling for generating surfaces from cloud points captured from real objects, often by laser range scanners but also by hand-held digitizers, computer vision techniques, edge detection from medical images, or other technologies. The main contributions are listed as follows.Firstly, the main attention is paid on geometric characterization and interpolation of planar PH curves. We present the geometric characterization of control polygons for a planar quartic Bezier curve to be a PH curve. Based on the definition of PH curve and complex representation of planar curve, we deduce geometric conditions in terms of the legs of the control polygon which guarantee the PH property. We also discuss the problem of G~1 Hermite interpolation by planar PH quarrics. Given two end points with associated tangent directions, we first choose the middle control point on some curve segments, and then find another control points which satisfy the deduced conditions. For PH quintics, as there are two main classes of curves: primitive and non-primitive. The first class is studied with detail by lots of researchers, but few work have been focus on the latter case. In this paper, we deduce the geometric characterization of planar non-primitive PH quintics in the same way as quartic case. Also, we discuss the problem of C~1 Hermite interpolation by non-primitive PH quintics. Given Hermite data, by numerically solving a quartic equation with respect to some auxiliary edges we have the control polygon of the PH quintic.Secondly, the main attention is paid on reconstructing curves and surfaces from unorganized point clouds. We introduce two algorithms. For point clouds with oriented normals and few noise, we present a radial basis function based algorithm. As data interpolation us- ing radial basis function always produce large dense matrix which seems to be beyond the capabilities of most present PCs. After the bounding box of the point cloud was partitioned into pieces, radial basis functions were interpolated for points for every piece, and an implicit function can be got by summing all the functions restricted by window functions. Marching cubes is then used to get simplicial mesh. The method is very suitable to be applied to fast reconstructing system because it can be implemented in parallel. For point clouds with much noise which can hardly get oriented normals, we introduce a new Delaunay-based shape definition and an efficient surface reconstruction algorithm. Delau-nay triangulation of point clouds is constructed and refined such that it is uniformly sampled in the neighborhood of every point. Inner points, outer points and boundary points are defined from the basic concept of set theory, and different level shapes of point clouds are also well-defined depending on the choices of parameter setting. The final reconstructed curves or surfaces is obtained by thinning the shape with appropriate parameter.