The arithmetico-geometric mean of gauss

Abstract

Before going further, it is worthwhile to recall how length of (part of) a curve is defined. Suppose first tha t the curve is given on the plane and is described by the pair of equations x = x ( t ) , y = y ( t ) in terms of a pa ramete r t. Intuitively, one would think of the length of any par t of the curve to be tha t of parts of a polygon which increasingly approximates tha t part of the curve. Thus, one divides the interval [a, b] where t varies into finitely many parts a = to < tl < " " < tn = b. The points P~ = ( x ( t , ) , y ( t i ) ) can be joined by line segments and the sum PoP1 + PIP2 + "'" + P n l P n of the lengths of these segments is an approximation to the length of the part of the curve which corresponds to the pa ramete r t varying from a to b. This approximat ion might be a crude one but it is at once clear (see Figure 1) tha t one can take more points to obtain a be t te r approximation. At this point, calculus comes to the rescue. It is heuristically clear tha t the best possible notion of length is obtained as the l imi t ing case as one increases the number n of points Pi indefinitely while s imultaneously let t ing the lengths of all the line segments PiPi+l diminish indefinitely. In other words, if the limit exists, it can be defined as the length of the curve. A curve for which this limit exists is called rectifiable for evident reasons. Clearly, circles, ellipses and hyperbolae are rectifiable