Reverse Design Of Kinds Of Surfaces And Degree Reduced Approximation Of Curves With Constraints

The reverse design of kinds of surfaces with constraints and the reduced approximation of curves with constraints are two fundamental theoretical problems with considerable research values in computer aided geometric design (CAGD). The most operations in CAGD are based on curves and surfaces, and both reverse design of kinds of surfaces with the constraints of interpolating special curves which has potential applications in the field of industrial manufactures, and degree reduced approximation of curves with the constraints of interpolating higher order derivative vectors at endpoints which plays an important role on the process of data compression and transmission have become one of current research hotspots and difficulties in recent years.Here the so-called constraints of interpolating special curves are taking one or two given parametric curves as geodesics, lines of curvature, asymptotic lines or boundary line of target surfaces; the so-called reverse design of surfaces, which is the opposite to the traditional geometry calculation that is calculating the geodesics, lines of curvature, asymptotic lines or boundary line for a given surface, are reversely designing a surface pencil by taking the given curves as potential common special curves, which means that every surface in the pencil is taking the given curves as its geodesics, lines of curvature, asymptotic lines or boundary line so that users have infinite candidate surfaces to select; the so-called kinds of surfaces mean the surfaces are the kinds of developable, rational, rational developable and discrete minimal surfaces. Thus, the conception and idea in this dissertation is brand new which is different from the general and traditional ideas and our results have undoubtedly important theoretical significances and applicative values for CAGD. In this dissertation, we have made deeply researches on these two topics and presented abundant and innovative results as follows:1、Explicit Best multi-degree reduction of a WSGB curve interpolating higher derivative vectors at endpoints. WSGB curves are compatible with Bezier curves and better than Bezier curves in evaluation, so it has practical significance to study the degree reduced approximation of a WSGB curve. The main contributions of the dissertation on this topic are the two following points:one is deriving the transformation matrix between WSGB basis and power basis according to the theory of dual basis; the other is presenting the algorithm of the explicit multi-degree reduced approximation of a WSGB curve in the sense of L2, norm interpolating higher derivative vectors at endpoints, which has a lot of advantages such as explicit expression, constraints at endpoints, the best approximation, priori error estimation, multi-degree reduction at one time.2. Design of the discrete minimal surface with the constraints of taking a spatial closed polyline as the boundary line. The problem to find such a triangular mesh with a given spatial polyline as the boundary line whose mesh area is the minimum in the all meshes with the same boundary line, is called Plateau-Mesh problem. The former reference solved it by minimizing the mesh area, which as viewed from a function only obtained local minimum of the function formed by mesh area but not necessarily the global minimum. Based on this, we propose an algorithm of designing the minimal triangular mesh by optimizing the discrete mean curvature and present the derivative vectors of mean curvatures of the mesh surface. Then the correctness and effectiveness of the algorithm are verified and illustrated by programming examples and error analysis.3、Design of developable, rational Bezier and rational Bezier developable surfaces through one or two given curves as a geodesic or geodesics. By employing the local Frenet orthonormal frame, the explicit expression of the rational Bezier developable surface pencil through a given spatial curve as its geodesic is presented and the orders of the derived rational developable surfaces interpolating a given planar or non-planar curve as its geodesic are discussed. The formulae of the control net points for the derived surface are provided. Finally, effectiveness of our algorithms is confirmed by programming examples of interpolating a degree2or3Bezier curve as the geodesic; by employing cubic Hermite basis functions, the algorithm of designing the rational isoparametric Bezier surface through the two given spatial Bezier curves (jointed or disjointed with each other) as its own geodesics is proposed. For the case that the both geodesics are degree3Bezier curves and the derived surface is the degree3*6rational isoparametric Bezier surface, explicit expressions of the control points are given. Furthermore, a number of programming examples are presented with the special condition that the two Bezier geodesics are jointed or disjointed; the uniform expression of parametric surfaces which satisfy the two constrains that the two given isoparametric curves (jointed or disjointed with each other) are taken as geodesics and the derived surface can be developed. Furthermore, the necessary and sufficient conditions of the existence of designing such a developable surface are presented. For the three types of developable surfaces, a mass of programming examples are demonstrated.4、Design of the uniform expression of the developable surface and a rational Bezier developable surface pencil through a given curve as its curvature of line. By employing the local Frenet orthonormal frame, we obtain a uniform parametric representation of a developable surface pencil through an arbitrary parametric curve as its common line of curvature and explore the impact of two free variables in the expression to the type of the derived developable surface. Furthermore, the necessary and sufficient conditions that there exists the precise expression of the rational Bezier developable surface pencil interpolating a given Bezier curve as its common line of curvature are derived. Finally, the correctness and availability of the algorithm are illustrated and verified by the examples of designing the general or rational parametric developable surface pencil with a circle, circular helix, planner Bezier curve or spatial Bezier curve as its common line of curvature respectively.5、Design of the developable, rational Bezier developable and the general surface pencil through a given curve as its asymptotic line. The expression of the general developable surface pencil through an arbitrary parametric curve as its asymptotic curve is provided, and the type of designed developable surfaces is discussed. Furthermore, the rational Bezier form of the developable surface pencil through a given Bezier curve as its asymptotic curve is given. Finally, programming examples for the general and rational developable surfaces through a circular helix, conical helix or Bezier curve as its asymptotic curve respectively; for designing the surface pencil through two given orthogonal curves as asymptotic lines, by employing Frenet local orthogonal frames and cubic Hermite basis functions, algorithms of designing surface pencils for three cases which are both curvatures being vanishing, one being vanishing but the other being not vanishing, neither being vanishing for the two given orthogonal curves are discussed, and the algorithm of designing such a surface pencil when the two given orthogonal curves are rational Bezier curves is presented. Finally, the correctness and effectiveness of the algorithms are verified by illustrating a mass of programming examples.