【作者】 张瑞坤；

【导师】 汪国昭；

【作者基本信息】 浙江大学 ， 计算机辅助几何设计与计算机图形学， 2006， 硕士

【摘要】 在CAGD/CAD中,阿基米德螺线是一个重要和基础的几何对象。曲线设计的有效方法是采用控制多边形来表示和控制曲线,现在多项式曲线都采用控制多边形的方法,它是CAGD/CAD系统中的主要方法。而阿基米德螺线不是多项式曲线,它是空间{1,sint,cost,tsint,tcost}中的曲线,至今还没有使得它像多项式一样应用于CAGD/CAD系统中。因此有必要创造新方法,使其能用控制多边形的方法来表示。 为了用控制多边形的方法表示阿基米德螺线,本文构造了拟Bernstein基。这组基有许多与Bernstein基类似的性质。例如:端点性质,线性无关性,对称性,归一性,正性等。 本文分别讨论了用4阶和5阶拟Bernstein基表示阿基米德螺线的充要条件,利用这个充要条件,可以得到用4阶和5阶拟Bernstein基表示的阿基米德螺线的控制顶点,从而可以方便地表示阿基米德螺线,实现了阿基米德螺线的控制多边形表示。并且我们还给出了控制顶点的几何作图法,有明显的几何意义。 最后,利用阿基米德螺线段和圆弧来构造凸轮,实现了凸轮的多边形控制,对构造和控制凸轮有一定的价值。更多还原

【Abstract】 In the CAGD/CAD system, Archimedes helix is an important and basic geometry object. The effective method of curve designing is to adopt control polygon. And now, almost all polynomial curves use this method, it’s the leading method in the CAGD/CAD system. At the same time, Archimedes helix is not polynomial curve, butthe curve which belongs to .spaw{l,sint,cost,t sint,tcost}, so it hasn’t been appliedin the CAGD/CAD system. Consequently, we need to create new method which can represent Archimedes helix with control polygon.In order to represent Archimedes helix with control polygon, we construct Bernstein-like basis. This Bernstein-like basis has many similar properties, for example: port property;linearly independent;symmetry;unitary property;plus property etc.This paper discusses the necessary and sufficient condition to represent the Archimedes helix with Bernstein-like basis of order four and five. Under this condition, we can obtain the control points of the Bezier-represented Archimedes helix. Then we can express the Archimedes helix expediently, and obtain the representation of Archimedes helix with control polygon. Otherwise, we study the geometrically constructing of control points, which has obvious geometric meaning.Finally, with the Archimedes helix and circular arc, we can construct a cam, so the control of the cam with a polygon comes true. This is useful to construct and control cam.更多还原