Abstract To make a marked point on the boundary of a circular wheel of radius 1 trace the curve known as a cycloid, let this wheel roll along an arbitrary fixed curve called the base curve. There are two cycloids, each of which is drawn on one side of the base curve and share cusps with each other. Consider the two arcs joining the same pair of adjacent cusps, one arc from each cycloid. T. M. Apostol and M. A. Mnatsakanian showed that the sum of the lengths of these arcs is independent of the shape of the base curve. They also showed that the sum of the areas bounded by two cycloids is independent of the shape of the base curve. Their remarkable proof does not use integral calculus. We generalize these results to the general trochoid case. Our proof is based on simple undergraduate calculus.

中文翻译：

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摆线的长度和面积的不变性

摘要 要在半径为 1 的圆轮的边界上做一个标记点​​跟踪称为摆线的曲线，让这个轮沿着称为基曲线的任意固定曲线滚动。有两个摆线，每个摆线都画在基曲线的一侧，彼此共享尖点。考虑连接同一对相邻尖头的两条弧线，每个摆线各有一条弧线。TM Apostol 和 MA Mnatsakanian 表明，这些弧的长度总和与基曲线的形状无关。他们还表明，由两个摆线包围的面积之和与基曲线的形状无关。他们非凡的证明没有使用积分。我们将这些结果推广到一般的次摆线情况。我们的证明基于简单的本科微积分。