平面图形的等积变换

The Equivalent Transformation of Plane Figure

把紧密相连的圆环构成的圆盘以渐开线展开成等面积的直角三角形,得出了较为直观的由周长和半径表示的圆面积公式.更一般地可以得出:圆扇形由弧长和半径表示的面积公式,面积与其弧长的原函数关系;圆扇形与三角形之间、圆扇环形与梯形之间进行的等积变换,以及它们中弧长半径和边角间的对应关系,而对于曲边是变曲率的曲边扇形不具有这些关系和结论.

The disk, which is composed of continuous annuluses, can expand equivalent trangle by asymptotice line. We can obtain the circular area which is visually expressed by circumference and radius. Generally, the area formula of circular section can be expressed by arc length and radius. At the same time, we can obtain the relationship between area and primitive function of arc length, the equivalent transformation between circular sector and trangle and between ring of circular section and trapezoid, and the respondence between radius of arc length and its edge or angle. But these conclusions are not true for curvel sector with curvature of change.