Some Researches On Curve And Surface Modeling

With the improvement of product design and production automation, many products need geometrical modeling design of curve and surface before fabricating. For example, loft and design of hull, design of car shell, design of airplane wing, body and cabin, even the design of knife rest, blade, mold, cloth all attribute to the research of curve and surface modeling, this is also the core content of Computer Aided Geometric Design (CAGD). Curve and surface modeling experience parametric spline, Coons surface, Bezeir curve and surface, B-spline, etc. In addition, by using interpolating, fitting, approximation, blending to study curve and surface which satisfy some condition is very importance in practical application. In this paper, we study the blending of developable surface which contains a Bezier curve as a geodesic, designing approximation minimal parametric surfaces with geodesies and the construction of parametric and developable surface through line of curvature. We also discuss functional spline curves and surfaces with different degree of smoothness and apply it to piecewise approximate implicitization of parametric curves, give a multi-level univariate quasi-interpolation scheme and construct an easy and valid integration formula. The main work is as follows:1. Geodesic is an important curve on a surface. For example, in shoe-making industry the leather is considered as developable surface, and the girth is the geodesic on the shoe surface. In garment-manufacture industry, the waist line is also the geodesic of the cloth. However, only one developable surface is not enough in practical application. So the G1connection of some developable surfaces should be considered. We study the G1connection of the developable surfaces containing G1abutting geodesics, and derive the corresponding conditions. We primarily study the condition of G1connection of two developable surfaces possessing cubic Bezier geodesies, and give the equations that the control points satisfy.2. Minimal surface has many important properties. It has lots of applications in architectural design, biology, molecular chemistry and so on. Moreover, in product design, we not. only hope the surfaces interpolate some characterizing curves, such as geodesies, but also hope to consume more less material. Based on these, we consider the problem of designing approximation minimal parametric surfaces with geodesies.3. Line of curvature ia another important characteristic curves on a surface. A curve on a surface is a line of curvature if its tangents are always in the direction of the principal curvature. We express the surface pencil by utilizing the Frenet frame and derive the necessary and sufficient condition for the given curve to be the line of curvature on the surface and introduce two control functions θ(s) and λ(s) to control the shape of the surface. Moreover, we classify the condition according to the expression of θ(s), and derive the condition when the given curve satisfy the line of curvature and the geodesic.4. Developable surface and line of curvature play an important role in geodesic design and surface analysis. We propose a new method to construct a developable surface possessing a given curve as the line of curvature of it, and give the concrete expression of the surface. We analyze the necessary and sufficient conditions when the resulting developable surface is cylinder, cone or tangent surface. The control function can control the shape of the resulting surface.5. Implicit curves and surfaces are extensively used in curve and surface modeling. By adding auxiliary curves and surfaces, the functional spline curves and surfaces with different degrees of smoothness are presented. We analyze the interpolation, convexity, and regularity in detail. Based on functional spline curves (with different degrees of smoothness), we propose an easy and valid method for implicitization.6. Quasi-interpolation is very important in the study of the approximation theory and applications. Radial Basis Function (RBF) and cubic spline quasi-interpolation are two common schemes. We propose a multi-level univariate quasi-interpolation scheme with better approximation than LD, LR and cubic spline quasi-interpolation. Moreover, we apply it to numerical integration and construct an easy and valid integration formula.