Types Of Parametric Curves, Geometric Properties Of The Analysis Method

It is undesirable that there occur singularities or unwanted inflection points on curves where one wants to design 'fair' curves in computer aided geometic design. Hence the study of geometric properties, which include the distribution of singular and inflection points and the convexity of parametric curves, is the key for designer to control the curves' shape. This problem of planar parametric polynomial curve of degree 3 had been discussed perfectly many years ago, making use of invariant of affine transformation. But the method of affine invariant may be no use to non-algebraic planar curves.In this paper, the method, which is based on the theory of envelope and continuous map, is investigated. According to this method, conditional curve of cusps is envelope curve of the distribution region of inflection points, and the distribution region of loops can be easily obtained with aid of continuous map. This method is effective not only for algebraic parametric curves of degree 3, but also for non-algebraic planar curves of order 4. Applying the method to some curves, we obtain the necessary and sufficient conditions for those curves containing or not containing cusp, loop, and inflection point in terms of the relative position of their control vertices. Major works are as follows:Firstly, the properties of singular and inflection points on the general curve, which are the linear combinations of the basis functionsl,t,φ(t) and ψ(t), are discussed.Curves whose forms are similar to Bezier curve or B-spline segment are constructed based on the same functions, their properties are also discussed.Secondly, the geometric properties of planar C-curve, C-Bezier curve, C-B-spline segment, rational C-Bezier curve and rational C-B-spline segment are obtained, including the distribution of cusps, loops, infection points on those curves, and necessary and sufficient conditions for those curves containing one or two inflection points, or a loop, or a cusp, or none of the above points in terms of the relative position of their control polygons. The method we used is more geometrically visual than that used in [15], our results are simplier and easier to judge. We alsoinvestigated the influnces of shape parameter or weight coefficients on distribution of singular and inflection points.Thirdly, we obtain the representations of integral circle, cycloid and sinusoid on a period by uniform C-B-splines of degree 3, along with adding control points the control polygons approach the generated curves, the approximation accuracies are better than that in [16] and [18].Lastly, we discuss distributions of singularities and general inflection points on spatial polynomial curve of degree 4 and spatial Bezier curve of degree 4 in details, and obtain the necessary and sufficient conditions for those curves containing or not containing singularities and general inflection points. So the geometric problems of spatial curves of degree 4 are solved.