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Areas and Shapes of Planar Irregular Polygons Forum Geometricorum b Volume 18 (2018) 17–36. b b FORUM GEOM ISSN 1534-1178 Areas and Shapes of Planar Irregular Polygons C. E. Garza-Hume, Maricarmen C. Jorge, and Arturo Olvera Abstract. We study analytically and numerically the possible shapes and areas of planar irregular polygons with prescribed side-lengths. We give an algorithm and a computer program to construct the cyclic configuration with its circum- circle and hence the maximum possible area. We study quadrilaterals with a self-intersection and prove that not all area minimizers are cyclic. We classify quadrilaterals into four classes related to the possibility of reversing orientation by deforming continuously. We study the possible shapes of polygons with pre- scribed side-lengths and prescribed area. In this paper we carry out an analytical and numerical study of the possible shapes and areas of general planar irregular polygons with prescribed side-lengths. We explain a way to construct the shape with maximum area, which is known to be the cyclic configuration. We write a transcendental equation whose root is the radius of the circumcircle and give an algorithm to compute the root. We provide an algorithm and a corresponding computer program that actually computes the circumcircle and draws the shape with maximum area, which can then be deformed as needed. We study quadrilaterals with a self-intersection, which are the ones that achieve minimum area and we prove that area minimizers are not necessarily cyclic, as mentioned in the literature ([4]). We also study the possible shapes of polygons with prescribed side-lengths and prescribed area. For areas between the minimum and the maximum, there are two possible configurations for quadrilaterals and an infinite number for polygons with more than four sides. The work was motivated by the study of two Codices, ancient documents from central Mexico written around 1540 by a group called the Acolhua ([9],[10] or online [3]). The codices contain drawings of polygonal fields, with their side- lengths in one section and their areas in another. We had to find out if the proposed areas were correct and to confirm the location of some of the fields. The fields are not drawn to scale and it is not known how the Acolhua com- puted (or measured) areas. There are no angles or diagonals therefore we could not compute the actual areas but one thing we could do was compute maximum and minimum possible areas for the given side-lengths and say the areas in the docu- ments were feasible if they were between those two values. Maxima are easy to Publication Date: January 17, 2018. Communicating Editor: Paul Yiu. We thank Ana Perez´ Arteaga and Ramiro Chavez´ Tovar for computational support, figures and animations. This work was supported by Conacyt Project 133036-F.. 18 C. E. Garza-Hume, M. C. Jorge, and A. Olvera find for quadrilaterals by applying Brahmagupta’s formula ([5], [6]) but we could not find in the literature a formula for the maximum possible area of polygons with more than four sides. There are some results in [8] but not a formula. We could also not find a complete treatment of minima in the literature. The study of the documents also required an analysis of the possible shapes of polygons with prescribed side-lengths and area to try to localize the fields. For polygons with more than four sides there is no easily available, explicit formula for this. We explain the mathematics of the problem, provide an algorithm to find the possible shapes and a computer implementation in javascript of the algorithm. Observe that finding areas of polygons is just an application of calculus if the coordinates of the vertices are known but the problem we are addressing here is, what can be done when only side-lengths are known, not angles, diagonals or co- ordinates. The question is relevant in surveying, architecture, home decorating, gardening and carpentry to name but a few. For some quadrilaterals it is possible to change orientation by deforming con- tinuously and in others it is necessary to perform a reflection. This observation led to a classification of quadrilaterals into four classes. The case of triangles is special because they are the only rigid polygons, that is, the shape and area are determined by the side-lengths. Also, all triangles can be inscribed in a circle whose center is located at the intersection of the perpendicular bisectors. If the side-lengths are called a, b, c the area is given by Heron’s Formula, AH (a, b, c)= s(s − a)(s − b)(s − c) (1) 1 where s = 2 (a + b + c) is the semi-perimeter.p In what follows we will discuss polygons with more than three sides. 1. Computing the shape with maximum area for any polygon In this section we will describe how to compute the maximum possible area of a planar polygon with n sides and how to construct the shape with maximum area given its ordered side-lengths. It is well-known that no side-length can be larger than the sum of the others if they are to form a closed polygon. We will call this the compatibility condition. It was proved in [4] that the configuration with maximum possible area is given by the cyclic polygon, that is, the configuration that can be inscribed in a circle. The problem is that we do not know which circle; we do not know its center or its radius. We will solve this problem in the present section. We will first give an analytical solution for quadrilaterals and then describe an iterative method that works for any polygon and its numerical implementation. Analytic solution for n=4: Circle-Line construction. For n = 4, Bretschneider’s formula gives the (unoriented) area A of a quadrilateral with side-lengths a, b, c, d and opposite angles α and γ as α + γ A = (s − a)(s − b)(s − c)(s − d) − abcd cos2( ), (2) r 2 where s =(a + b + c + d)/2, see Figure 1. Observe that A is always non-negative. Areas and shapes of planar irregular polygons 19 Figure 1. Two examples of a quadrilateral ABCD. The maximum value of this expression is Amax = (s − a)(s − b)(s − c)(s − d), (3) and it is attained when the sump of either pair of opposite angles is π radians, that is, when the quadrilateral es cyclic. But this does not give directly the coordinates of the shape with maximum area. There is a formula for the vertices of the shape with maximum area which is described in [6] SI Appendix p.414, for a quadrilateral with given side-lengths a, b, c, d and prescribed area Ac. We give here the geometric idea. As explained in [6], vertex A is placed at the origin and vertex D at (a, 0); it remains to find the coordinates of vertices B and C. The coordinates of vertex B are (u,v) and of vertex C, (w,z) (see Figure 2). Figure 2. Shape with maximum area of a quadrilateral with side-lengths 4,6,4,7 arbitrary units. After some lengthy computations with analytic geometry which we include in Appendix A, it is shown that vertex B is given by the intersection of a circle C of radius b and center at the origin, and a line L which depends on the side-lengths and on the prescribed area Ac. The maximum area is attained when the line is tangent 20 C. E. Garza-Hume, M. C. Jorge, and A. Olvera to the circle. The coordinates of vertex B are given explicitly and the coordinates of vertex C and found from the lengths of the remaining sides. Figure 2 shows an example of maximum area produced using a program we wrote using routines from JSXGraph which can be found under “Circle-line con- struction” in www.fenomec.unam.mx/ipolygons. The value of the maximum area can be computed from equation (3). In this example, in arbitrary units a = 4, 1 b = 6, c = 4, d = 7 therefore s = 2 (4+6+4+7) = 21/2 and the maximum pos- sible area is approximately 25.8 square units. The program allows the computation of the shape with maximum area. The construction also shows that if Ac > Amax then the line and circle do not intersect and hence no quadrilateral exists with those side-lengths and area. If Amin < Ac < Amax then L intersects C twice giving two possible shapes for the given quadrilateral data. Here Amin represents the minimum possible area. We will say more about it later. We recommend playing with the computer program to understand the construc- tion and its implications. Iterative method for any n ≥ 3 and its numerical implementation. For n > 4 there is no explicit formula for the center or radius r of the circumcircle of the configuration with maximum area. We describe here an iterative method that works for any n ≥ 3 for finding the circumcircle and thus the maximum area of the polygon and the coordinates of its vertices. Observe first that the center of the circumcircle of a cyclic polygon can be in- side, outside or on the boundary of the polygon, as illustrated in Figure 3, and this determines how the circumcircle is computed. Figure 3. Position of the center of the circumcircle. Now consider n side-lengths a1,a2,...,an that satisfy the compatibility condi- tion, placed in clockwise order in the shape of maximum area. Referring to Figure 4, by the cosine rule 2 2 2 aj = 2r − 2r cos θj therefore 2r2 − a2 cos θ = j .
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