A Sector Isoperimetric Problem

Abstract

We find the sector of largest area that can be formed by taking curves of fixed length that begin at some point on the positive x axis and connecting their endpoints to the origin with radial lines. We also discuss a more general question. 1. Statement of Problem The standard isoperimetric result in two dimensions is that of all possible closed curves of fixed length, the circle is the one that encloses the greatest area. See for instance the article [2] and its references. In this paper we are interested in looking not at a closed curve but at open curves of fixed length s which we will assume start at some point (r, 0) on the x axis and we want to maximize the area of the sector created by connecting its endpoints to the origin. So the idea is to find which curves maximize the area shown in figure (1) and how do they depend on r and s. We will begin by assuming that the curve has length s = 1 and starts at the point (1, 0) and show later the result for arbitrary s and r values which turns out to be no more difficult. 1.1. Examples with area = 0.5. There are two particularly interesting curves we would like to look at first. One curve is part of the unit circle and the other is a vertical line segment. Both curves are shown in the figure (2). 2000 Mathematics Subject Classification. Primary 53A04. Figure 1. Maximize the shaded region 1 2 WILLIAM CALBECK LSU-ALEXANDRIA BILLC@LSUA.EDU Figure 2. Equal Ratio Curves What is interesting about both of these curves is that they each produce the same area of 1/2. In fact these two curves have a property which has been called the equal ratio property [4] meaning that the ratio of the area of the sector to the length of the curve remains constant regardless of how long the curve is. In this case it will always be 1/2. These are the only types of curves that have this property. Besides providing examples in which the area of the sector is easy to compute, these examples also show us that a) Circular arcs are not necessarily the shapes that produce the maximum area. b) There is reason to believe there may be a curve that produces a greater area whose path lies ”between” these two curves. As it turns out circular arcs are the shapes that produce the maximum area if we are not required to stay ”between” the two curves, it is just that the circles are not centered on the x axis. But before we find the formulas for these circular arcs we want to give some of the basics of working with arclength and area for curves given both parametrically and in polar coordinates. 1.2. Polar Coordinates. In polar coordinates we can state the problem this way. Among all piecewise differentiable functions f(θ) defined on some interval [0, β] such that f(0) = 1 and