FINDING THE CENTER OF A CIRCULAR STARTING LINE IN AN ANCIENT GREEK STADIUM

Transcription

1 SIAM REV. c 1997 Society for Industrial and Applied Mathematics Vol. 39, No. 4, pp , December FINDING THE CENTER OF A CIRCULAR STARTING LINE IN AN ANCIENT GREEK STADIUM CHRIS RORRES AND DAVID GILMAN ROMANO Abstract. Two methods for finding the center and radius of a circular starting line of a racetrack in an ancient Greek stadium are presented and compared. The first is a method employed by the archaeologists who surveyed the starting line and the second is a least-squares method leading to a maximum-likelihood circle. We show that the first method yields a circle whose radius is somewhat longer than the radius determined by the least-squares method and propose reasons for this difference. A knowledge of the center and radius of the starting line is useful for determining units of length and angle used by the ancient Greeks, in addition to providing information on how ancient racetracks were laid out. Key words. circles, archaeology, stadium, racetrack, starting line, least squares, degree, history of mathematics AMS subject classifications. 01A20, 62P99, 62G07 PII. S Introduction. The Greek city of Corinth contains a stadium enclosing several racecourses for foot races constructed at different periods in antiquity. In 1980 a curved starting line for one of the racecourses dating from about 500 B.C. was excavated by the American School of Classical Studies at Athens, Corinth Excavations, Charles K. Williams, II, Director [1]. The starting line is constructed of rectangular porous blocks of limestone with a thin plaster coat covering the stone blocks (Fig. 1). The width of the starting line is between 1.25 and 1.30 meters, and its excavated length is about 12 meters along a circular arc. We desire to determine the center and radius of the circular arc in order to ascertain how the starting line and racetrack were laid out and to determine the units of measurement used by the Corinthians at that time. Also visible in Fig. 1 are the front and rear toe grooves of twelve starting positions for the runners cut into the starting line. We wish to determine the angles between the center of the arc and the twelve starting positions to see what angular units might have been used at that time. To determine the center and radius of the starting line, measurements were taken along its front edge at 21 points where the edge was fairly well defined [2]. These measurements were taken using an electronic total station (Fig. 2), which includes an electronic theodolite and an electronic distance meter. Figure 3 shows the locations of the 21 points on the starting line, and Table 1 lists their xy-coordinates with respect to an arbitrary coordinate system. Of course, it is not possible to fit a circle exactly through these 21 points due to many cumulative errors. These include errors made by the original surveyors in laying out the starting line, errors made by the stonecutters and builders of the starting line, shifting of the starting line over 2.5 millennia, errors in locating points to measure due Received by the editors April 30, 1996; accepted for publication June 6, Department of Mathematics and Computer Science, Drexel University, Philadelphia, PA (crorres@mcs.drexel.edu). Mediterranean Section, The University of Pennsylvania Museum of Archaeology and Anthropology, Thirty-third and Spruce Streets, Philadelphia, PA (dromano@sas.upenn.edu). 745

2 746 CLASSROOM NOTES FIG. 1.The curved starting line of a racecourse in Corinth, ca. 500 B.C. (Courtesy American School of Classical Studies, Corinth Excavations.)

3 CLASSROOM NOTES 747 FIG. 2. The archaeological team from the University of Pennsylvania with the instruments used to measure the data points. One of the authors (Romano) is in the center behind four graduate assistants (left to right: Chris Campbell, Mimi Woods, Elizabeth Johnston, and Ben Schoenbrun). The cliffs of Akrocorinth loom in the background and a portion of the starting line is barely visible in the foreground. (Courtesy Irene Bald Romano.)

4 748 CLASSROOM NOTES FIG. 3.The curved starting line and the 21 data points along its front edge. TABLE 1. Coordinates of 21 data points along the front edge of the starting line. Data x-coordinate y-coordinate point (meters) (meters)

5 CLASSROOM NOTES 749 FIG. 4. The 2 circles determined by the 2 methods. Also plotted are the centers of the 680 triplet circles used to determine the three-points circle. to erosion and other damage along the edge, and errors in the measuring instruments. Below we discuss two methods for determining the best circle that fits the data. 2. Three-points circle. The approach taken by the archaeologists who measured the data points is based on the fact that if only three points were measured along the starting line, then by any (reasonable) criterion the best circle would be the exact circle passing through those three points. Consequently, they first determined the exact circles passing through various triplets of the 21 data points. The x-coordinates and y-coordinates of these triplet circles were then averaged to determine the x- and y-coordinates of the center of a final circle, and the radii of the triplet circles were averaged to compute the radius of the final circle. The triplets were chosen to be all triplets that have at least two other data points between any two of them using the ordering in Table 1 a criterion that resulted in 680 triplets. The average center of the resulting 680 triplet circles was found to be at ( , ), and their average radius was meters. The range of their 680 radii was [27.643, ] and their sample standard deviation was meters. The resulting three-points circle is plotted in Fig. 4, together with the centers of the 680 triplet circles. 3. Least-squares circle. We next discuss a method of finding the best circle that fits the 21 data points using a least-squares criterion. This method determines the maximum-likelihood circle assuming that the coordinate errors of the measured data points have identical normal distributions with mean zero. The resulting circle is the one for which the sum of the squared orthogonal distances of the data points to the circle is a minimum [3], [4]. In general, let (x i,y i ),i=1,2,3,...,n, be the coordinates of n points imperfectly measured along a circle whose center and radius are to be determined. Let (α, β) be the center of an arbitrary circle in the plane and let ρ be its radius. The least-squares

6 750 CLASSROOM NOTES FIG. 5.The surface determined by the least-squares error in (3). error associated with the maximum-likelihood circle is then n ( ) 2. (1) F (α, β, ρ) = (xi α) 2 +(y i β) 2 ρ i=1 We need to find values of α, β, and ρ that minimize F. By setting the derivative of F with respect to ρ equal to zero, we quickly find that (2) ρ = 1 n n (xi α) 2 +(y i β) 2. i=1 That is, for any given center (α, β) the corresponding optimal radius is the average of the distances of the n data points to that center. By substituting (2) into (1), the problem reduces to minimizing the following function of two variables: n (3) E(α, β) = 2 (x i α) 2 +(y i β) 2 1 n (x j α) n 2 +(y j β) 2. i=1 This problem does not appear to have a closed-form solution. Using numerical techniques (the fmin function of MatLab [5], which implements a Nelder Mead simplex search [6]), the center of the circle determined by (3) was found to be ( , ), and the corresponding radius determined by (2) was meters (Fig. 4). The 21 data points encompass an arc of with respect to its center. The range of their distances from the center is [53.938, ], and their sample standard deviation is meters. In Fig. 5 we show the surface determined by (3). This least-squares error surface has a long narrow valley that points toward the center of the arc determined by the 21 data points. The valley reveals the small sensitivity of the location of the center point along a line perpendicular to the arc subtended by the 21 data points. At the center of the least-squares circle (the global minimum of the surface) the value of E is , and at the center of the three-points circle, slightly down the valley away from the data points, the value of E is Comparison of the two methods. From Figs. 4 and 5 we notice that the centers of the 680 triplet circles lie along the valley of the least-squares surface. Thus j=1

7 CLASSROOM NOTES 751 FIG. 6.Points A and B are known to lie on the circle. The location of midpoint C is uncertain. both methods agree as to the location of a line along which the center of a best circle should lie. But the radius of the three-points circle is meters longer than that of the least-squares circle a significant difference if one were searching for physical evidence of the center in the stadium. To suggest the reason for this difference, let us look at a considerably simpler problem. Suppose we wish to determine the radius R of a circle on which two points A and B lie whose positions we know (Fig. 6). Let the distance between these two points be 2b. We imperfectly locate the midpoint C of the arc between A and B by measuring the distance c along the perpendicular bisector of the chord AB. Specifically, we take n measurements c 1,c 2,...,c n of c. Suppose that c is a normal random variable with standard deviation σ whose mean µ is the exact distance to C. A little geometry then shows that the exact radius R of the circle is (4) R = µ 2 + b2 2µ. Taking the average c of the n measurements of c as an estimator for µ, we then obtain the following estimate for the radius of the circle: c (5) R = 2 + b2 2 c. This procedure is equivalent to finding the least-squares circle that passes through A and B determined by the n data points. The analogue of the three-point method would be to first compute the radii of the n circles through all triplets consisting of the points A and B and one of the n measured points. Then these radii would be averaged to estimate the radius of the final circle. Now, if f(c) is the normal distribution function of the random variable c, then the distribution function g(r) of the radius R of the triplet circle is (6) g(r) =f(r ( ) R 2 b 2 R ) R2 b 1. 2

8 752 CLASSROOM NOTES FIG. 7.Probability distribution of the radius of the triplet circle in the simplified example. FIG. 8.Histogram of the radii of the 680 triplet circles using 5-meter cells. Actually, this distribution function is correct only if the support of f(c) lies in the interval [0,b]. We assume that the mean and standard deviation of f(c) are such that this is approximately the case. Figure 7 is a graph of g(r) when µ = meters, σ =0.0133, and b = These values where chosen so that (1) the radius of the exact circle is meters, the same as the least-squares radius of the starting-line circle; (2) the points A and B subtend an arc of 6, about half of the arc subtended by the original 21 data points; and (3) the standard deviation σ is equal to the sample standard deviation of the distances of the 21 data points to the least-squares center of the starting-line circle. The mean of g(r) using these values is meters (computed numerically). This compares favorably with the radius, meters, of the three-points circle. The distribution function g(r) in Fig. 7 likewise compares favorably with a histogram of the 680 triplet circles shown in Fig. 8.

9 CLASSROOM NOTES 753 TABLE 2. Coordinates of the center points of the front toe grooves of 11 of the 12 starting positions on the starting line. The last column is the angle subtended by 2 successive toe grooves with respect to the center of the least-squares circle ( , ). Starting x-coordinate y-coordinate Angle to position (meters) (meters) next position In summary, the nature of the sources of the errors in this problem suggests that the measured coordinates of the data points should be regarded as normal random variables. This, in turn, implies that the least-squares circle would best locate the center employed by the ancient surveyors. The three-points method overestimates the true radius because when the errors in the three points used for a triplet cause them to line up they yield a circle that can have a significantly larger radius than the true radius. This is evident in the way that the distribution in Fig. 7 and the histogram in Fig. 8 are skewed to the right. 5. Angles of the starting positions. As mentioned in the introduction, a primary interest for determining the center of the starting-line circle was to determine the angles subtended by the twelve starting positions of the runners. The measurements of the coordinates of eleven of the front toe grooves are given in Table 2. The toe groove for position 05 was in too poor a condition to be measured accurately. The last column gives the angles between the starting positions with respect to the least-squares center of the starting-line circle. The average angle between the twelve starting positions is Although the range of the angles is rather large, this average angle is sufficiently close to one degree to warrant attention. Among the Greeks, the earliest known use of the degree as a unit of angular measurement is found in the writings of Hypsicles in the second half of the second century B.C. [7]. He adopted the Babylonian practice of dividing the twelve signs of the zodiac into 30 equal parts [8]. The choice of 30 is probably because the sun takes about 30 days to pass through each sign of the zodiac. That is to say, the division of a circle into 360 parts can probably be traced to the fact that there are roughly 360 days in a year. It may be that in laying out the starting positions, the Corinthians used the angle traveled by the sun along the zodiac in one day as the angle allotted each runner. If so, the Corinthian starting line is evidence of an early use of the degree as a unit of angular measurement among the Greeks. Acknowledgments. The archaeological field survey on which this study was based is the Corinth Computer Project of the University of Pennsylvania Museum of Archaeology and Anthropology, David Gilman Romano, Director. The project is run

10 754 CLASSROOM NOTES under the auspices of the Corinth Excavations of the American School of Classical Studies at Athens, Charles K. Williams, Director. During the years , Mr. Benjamin Schoenbrun was Research Intern for the Corinth Computer Project and assisted with the analysis of the starting line data. D. G. Romano is grateful to all of the University of Pennsylvania students who assisted him both in the field and in the laboratory. The Corinth Computer Project is supported by the Corinth Computer Project Fund of the University of Pennsylvania Museum of Archaeology and Anthropology. REFERENCES [1] C. K. WILLIAMS, II AND P. RUSSELL, Corinth excavations of 1980, Hesperia, 50 (1981), pp [2] D. G. ROMANO, Athletics and Mathematics in Archaic Corinth: The Origins of the Greek Stadion, Memoirs of the American Philosophical Society 206, American Philosophical Society, Philadelphia, PA, [3] J. E. DENNIS AND R. B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, NJ, [4] B. A. JONES AND R. B. SCHNABEL, A comparison of two sphere fitting methods, Proc. Instrumentation and Measurement Technology Subgroup of the IEEE, Boulder, CO, March [5] D. HANSELMAN AND B. LITTLEFIELD, The Student Edition of MATLAB: Version 4: User s Guide, Prentice Hall, Englewood Cliffs, NJ, 1995, pp [6] J. A. NELDER AND R. MEAD, A simplex method for function minimization, Computer J., 7 (1965), pp [7] I. THOMAS, Selections Illustrating the History of Greek Mathematics, Vol. II, Harvard University Press, Cambridge, MA, 1941, pp [8] O. NEUGEBAUER, The Exact Sciences in Antiquity, Second Edition, Brown University Press, 1957, pp