On Hermite-type Vector Interpolating Subdivision Schemes And Applications

Along with the great progress in 3D data measurement device and the popularization of complex solid modeling, recursive subdivision has been a research focus in the Computer Aided-Geometric Design and Computer Graphics. However, most interpolating subdivision schemes are Lagrange-type schemes whose smoothness is not able to be achieved easily. Because the traditional subdivision may not be represented in the analytical forms, the geometric features of points on the subdivided curves and surfaces are given difficultly, and the offset curves and surfaces can not be obtained simply, which limits applying subdivision methods to computer animation, reverse engineering, precision machine design and medical image reconstruction.Just in order to change the passive situation mentioned above, we are devoted to the applied research of Hermite-type vector interpolating subdivision schemes. The main contents include one-order and two-order Hermite-type vector interpolating subdivision curves and their geometric features generation; Hermite-type vector interpolating subdivision surfaces on quadrangular meshes and their applications.After briefly reviewing the origin and research history of recursive subdivision, a survey of subdivision schemes is presented including the classification, the characters and advantages, the computation of geometric attributions, several classical subdivision curves and surfaces and their convergence and smoothness.Firstly, using five-point tangent estimation to construct initial vector-type Hermite element series and parameterization by length of chord, a non-stationary Hermite-type vector interpolation subdivision scheme with two factors is proposed, and its convergence and continuity are analyzed. The scheme's iterative layers areestimated for given error. The sufficient conditions of C1 continuity are proved by constructing the stationary scheme to the original non-stationary one. Geometric features of subdivision curve, such as line segments and cusp etc., are obtained by appending some conditions to initial Hermite elements. An algorithm is presented for generating geometric characters and offset curve. For the initial Hermite element series from the circle, the numerical error between the C1 subdivision and original curve is 0(10'3).Secondly, a non-stationary two-order Hermite-type vector interpolation subdivision scheme with four factors is given. The sufficient conditions of C2 continuity are proved, and the iterative layers are estimated for given error. An algorithm is obtained to generate geometric features and offset curve by appending some conditions to initial Hermite elements. For the initial two-order Hermite element series from the circle, the numerical error is (^(lO"4).On Hermite-type vector interpolating subdivision surfaces on quadrangular meshes, a non-stationary Hermite-type vector interpolation subdivision scheme with five factors is proposed and its convergence and continuity are analyzed according to the theory of matrix infinite norm. The sufficient conditions of C1 continuity and some properties are given to approximately represent some special surfaces by the scheme. Sharp features of subdivision surface, such as creases, crease vertices, cone and corner, are generated by appending some conditions to initial Hermite elements and using the same scheme. The method of generating subdivision-skinning surface by Hermite-type vector interpolating subdivision scheme is built.