Blending Basic Curves And Surfaces Of Degree2by C-type Splines And Subdivision

After more than half a century of development, CAGD is constantly seeking newapplications, and more connection with Industrial products. At the same time, it alsoseeks a new spline curve modelling method. Some new forms of spline structure hasbeen proposed and people begin to discuss the application of the new form of spline inthe geometric modelling method. In the process of CAD/CAM geometric modellingdesign, control point curves and control mesh surfaces presented by B-splines,NURBS and B-like splines are more popular during the period of modelling. C-typespline in this paper is a kind of B-like spline, B-like spline is proposed to solve someproblems in geometric modeling with NURBS. As C-type curve can be used torepresent more kinds of curves, it avoids a rational form which reduces thecomputation complexity. Thus it has begun to be taken seriously in the geometricmodeling. spline function has been widely used in the geometrical modelling withregular grid, but in the structure of the relatively complex geometric modeling,subdivision has more advantages compared with spline function. First of all, thedrawing of curves and surfaces on a computer screen or on the machining in NC isoften a process of "discrete-continuous-discrete", so when we use the continuousspline function to represent the model, it needs the discretization during the processwhich increases more troubles and reduces the efficiency. If representing andadjusting curve and surface directly with discrete data, it will be beneficial to curveand surface’s representation, transformation, and intersection arithmetic. Secondly, thesubdivision method can process the geometric modelling of arbitrary rules, whendealing with irregular curve and surface modelling, it doesn’t need cutting on theoriginal model, cutting which reduces the error of calculation. In this paper, wepropose a new Geometric modelling method based on C-type splines and subdivisionin this paper.In the actual NC, cutting tool paths often composed of arc and straight line. So itoften needs to use circular arc spline to approach some discrete control vertices.This paper presents a G2circular spline interpolation algorithm with the data pointsand their tangents are given. For each data point, we construct an initial circle with acommon radius. And then we compute the tangent circles for each pair of adjacentinitial circles. At last we select suitable circular arcs from these initial circles and tangent circles, which can generate a G2circular spline curve. In addition, the wholecircular spline curve can be reproduced by subdivision. This is important fordesigning NC milling tool path.In Comparison with the existing blending method, this article presents a blendingmethod for basic curves including conics and circle arcs. The explicit formulas ofbasic curves represented by C-type splines of degree2are derived. The wholeblending curve is-continuous. The proposed method is simpler to represent andblend basic curves. Further, we design a subdivision method to generate the blendingcurve which is exact for those segments of basic curves and approximate for thosesegments of blending. By the theory of asymptotic equivalence, we also prove theC1-continuity of proposed subdivision curves.This article presents a blending method for basic surfaces including cylinder、circular cone、hyperboloid、paraboloid and so on. The basic surfaces and blendingsurfaced are presented by C-type splines surface of degree2, the whole blendingsurface is G1-continuous. Representation of Surface is very simple and the blendingmethod is easy to implement.