Subtangents - An Aid to Visual Calculus

1. MAMIKON’S THEOREM. Areas and volumes of many classical regions can be determined by an intuitive geometric method that does not require integral calculus or even equations for the boundaries. The method and its many applications was first announced by Mamikon A. Mnatsakanian in [1], but this paper seems to have escaped notice, probably because it appeared in Russian in an Armenian journal with limited circulation. A recent exposition is given in [2], expanding on special cases discussed in [3] and [4]. The work is based on Mamikon’s theorem illustrated in Figure 1. The lower curve on the left of Figure 1 is a more or less arbitrary smooth plane curve. A set of tangent segments to the lower curve defines a region that is bounded by the lower curve and an upper curve traced out by the other extremity of the tangent segments. This set is called a tangent sweep. It can be visualized dynamically as the region swept out by tangent segments moving along the lower curve. When each point of tangency is brought to a common point as shown on the right of Figure 1, the set of translated segments is called the tangent cluster. Mamikon’s theorem states that the area of the tangent cluster is equal to that of its tangent sweep.