中文摘要：

旋轮线是数学中众多的迷人曲线之一，直线上的旋轮线、圆外与圆内旋轮线是经典的旋轮线。旋轮线方程依赖3个参数，不同的参数设置，曲线图案很不一样，其多变的图案可用于底纹防伪。经典的旋轮线的曲线图案比较单薄。对经典的旋轮线，利用曲线平移叠加、缩放叠加技术，使之更漂亮和丰满。同时，把旋轮线方程推广拓展为4参数或9参数情形，使得曲线变化更复杂和精妙，并利用程序生成图形丰富的花边、团花等防伪底纹。

英文摘要：

Cycloid is one of the most fascinating curves in mathematics. Cycloids, epicycloids and hypocycloids belong to the family of classic cycloids. The graph of cycloid depends on three parameters. Different parameter settings may lead to very different patterns. This kind of property ensures the diversity if they are used in security pattern designs. The patterns of classic cycloids are too simple. So we used translation, scaling and overlay techniques to make classic cycloids more beautiful and plentiful. Simultaneously, we extended classic cycloids to a general case with 4 or 9 parameters, in order to make the curve more sensible to the parameters. We also illustrate some security pattern exampies, such as borders and rosettes, by using our methods.