New Insight into Cycloidal Areas

نویسندگان

  • Tom M. Apostol
  • Mamikon A. Mnatsakanian
چکیده

1. INTRODUCTION. A cycloid is the path traced by a point on the boundary of a circular disk that rolls without slipping along a straight line. It forms a sequence of arches resting on the line, as shown in Figure 1. It is known that the area of the shaded region below one arch is three times the area of the rolling disk. Our analysis shows that the factor 3 reflects a much deeper property of cycloidal sectors. Figure 2 shows various stages of rotation of the disk, together with cycloidal sector OPC bounded by arc OP and two line segments OC and PC. At each stage we have:

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عنوان ژورنال:
  • The American Mathematical Monthly

دوره 116  شماره 

صفحات  -

تاریخ انتشار 2009