杨之先生在其名著[1 ] 中倡导研究的“双圆多边形”是指既有外接圆又有内切圆的多边形 .仿此 ,我们给出下面的定义　若一条封闭折线的顶点都在一个圆上 ,每条边都与另一个圆相切 ,则称该折线为双圆封闭折线 .相应地 ,若它的边数为 n,环数为 k,则称为 n边 k环双圆封闭折线 .图 1 The “dual-circular polygons” advocated by Mr. Yang in his famous book [1] refer to polygons with both circumcircle and inscribed circles. For example, we give the following definition if the vertices of a closed polyline are all in one On the circle, each side is tangent to another circle, then the fold line is called a double-circle closed foldline. Correspondingly, if its number of sides is n and the number of rings is k, then it is called n-edge k-loop double-circle closure. Polyline. Fig. 1