Research On Trigonometric Polynomial Curve Model And Surface Rendering Methods

Curve and surface modeling is a key technique in CAD/CAM system. NURBS as a uniform model, which can represent both free curves and traditional geometrical curves, has become a criterion in the field of industrial manufacturing. However, some constraints of NURBS, including its weight, parameterization and continuity remain a problem. Recently researchers have been trying to seek a solution in space of trigonometric function, hoping to overcome the constraints of NURBS model.After introducing the significance and reviewing previous research on curve and surface modeling in CAGD, this study focuses on the theory and method of trigonometric polynomial curve model and surface rendering. The main content and renovation include:(1) In order to adjust effectively the shape of a curve by using shape parameters and boost up its flexibility, the study investigates the representations and properties of five types of spline curves with shape parameters. By means of a large amount of mathematical derivation, we analyze the effect of shape parameters on curve shape. Besides, through comparing five modeling approaches, the applicable scale of shape parameters is also derived. Moreover, we also propose a new approach to representing free curves by means of different values of the shape parameters and give an example of it.(2) The study explores from the theoretical perspective the issue of completeness of T-Bézier and T-B spline curve model and puts forward their presentation and solution of (n+1)^th-order T-Bézier and T-B spline basic function. Besides, we propose G¹/C¹ geometrical continuity condition of T-Bézier curve, which offers a solution to the joining of several T-Bézier curves. At the same time, we also present G¹/C¹ continuity condition of C-B spline curve and T-B spline curve. Such G¹/C¹ continuity with C-B spline curves, which makes use of the property that T-B spline curves can represent semi-ellipses and semicircles, successfully makes up the constraint of C-B spline modeling, which cannot represent semi-ellipses and semi-circles accurately.(3) We construct a TC-Bézier curve with shape parameterλ, with [1,sint,cost, sin²t,cos²t] as its base, so as to avoid the complexity of the mathematical model. We also discuss the basic function and property of such curves. Within a certain scope, curves like semi-ellipses and semi-circles can be accurately presented by adjusting the value of λ. Meanwhile, we also give some examples of G¹/C¹ continuity condition between third-order and fouth-order TC-Bézier surface. The result bears explicit geometrical significance and can be conveniently applied to surface modeling.(4) In order to improve the accuracy of surface modeling and to increase its non-linear approaching capacity as well as the anti-noise capacity derived from radius basic function neural network (RBFNN), we establish an RBFNN model, which fits surface reconstruction. We also propose a method of using RBF neural network to eliminate noise and reform freedom surface. Besides, we conduct a rendering test of forth-order TC-Bézier surface. The test result shows that not only can the model eliminate the noise from surfaces but also learn fast and produce smooth surface.(5) In order to reduce the amount of storage and transmission and to achieve real time and dynamic display of 3-D objects with multi-resolution, we propose a new method of face cluster mesh simplification, i.e. to establish an Octree, which clusters the triangle mesh in the same space. The experiment demonstrates that the new face clustering method exceeds the old one because it accelerates mesh simplification.(6) Based on the exploration above, we design a series of algorithms of surface modeling. Under Microsoft Visual C++ 6.0, with OpenGL as graphics library, we also develop a Space Freedom Surface Modeling System, thus create a unified platform of different models in order to test the algorithms proposed in this study. The system can produce rotating surface and free surface in an easy way and make the graph look more real by adding illumination and texture.