R E P R I N T
SIAM Review
Volume 39, Number 4, December 1997, pp. 745-754
(Society for Industrial and Applied Mathematics)
FINDING THE
CENTER OF A
CIRCULAR
STARTING
LINE IN AN
ANCIENT
GREEK
STADIUM
by
CHRIS RORRES
Department of Mathematics and Computer Science
Drexel University
Philadelphia, PA 19104
and
DAVID GILMAN ROMANO
Mediterranean Section
The University of Pennsylvania Museum of Archaeology and Anthropology
Thirty-third and Spruce Streets
Philadelphia, PA 19104
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.
Fig. 1. The curved starting line of a racecourse in Corinth, ca. 500 B.C.. (Courtesy American School of Classical Studies, Corinth Excavations.) 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.) 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.
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 two successive toe grooves with respect to the center of the least-squares circle (-20.940, 33.618). |
1. 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 poros 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 -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 two-and-a-half millennia; errors in locating points to measure due 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 -coordinates and -coordinates of these "triplet" circles were then averaged to determine the - and -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 (-22.943, 32.506) and their average radius was 56.242 meters. The range of their 680 radii was [27.643, 160.668] and their sample standard deviation was 12.058 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 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]. In general, let be the coordinates of points imperfectly measured along a circle whose center and radius are to be determined. Let be the center of an arbitrary circle in the plane and let be its radius. The least-squares error associated with the maximum-likelihood circle is then
We need to find values of , and that minimize . By setting the derivative of with respect to equal to zero, we quickly find that
That is, for any given center the corresponding optimal radius is the average of the distances of the points to that center. By substituting (2) into (1), the problem reduces to minimizing the following function of two variables
This problem does not appear to have a closed-form solution. Using numerical techniques (the "fmin" function of MatLabĒ [4], which implements a Nelder-Mead simplex search [5]) the center of the circle determined by (3) was found to be (-20.940, 33.618) and the corresponding radius determined by (2) was 53.960 meters (Fig. 4). The 21 data points encompass an arc of 12.134 ° with respect to its center. The range of their distances from the center is [53.938, 53.991] and their sample standard deviation is 0.0133 meters. In Fig. 5 we show the surface determined by Eq. (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 is 0.0036 and at the center of the three-points circle, slightly down the valley away from the data points, the value of is 0.0084. 4. Comparison of the Two Methods. From Figures 4 and 5 we notice that the centers of the 680 triplet circles lie along the valley of the least-squares surface. Thus 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 2.282 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 of a circle on which two points and lie whose positions we know (Fig. 6). Let the distance between these two points be 2. We imperfectly locate the midpoint of the arc between and by measuring the distance along the perpendicular bisector of the chord . Specifically, we take measurements of . Suppose that is a normal random variable with standard deviation whose mean is the exact distance to . A little geometry then shows that the exact radius of the circle is
Taking the average of the measurements of as an estimator for , we then obtain the following estimate for the radius of the circle:
This procedure is equivalent to finding the least-squares circle that passes through and determined by the data points. The analog of the three-point method would be to first compute the radii of the circles through all triplets consisting of the points and and one of the measured points. Then these radii would be averaged to estimate the radius of the final circle. Now, if is the normal distribution function of the random variable , then the distribution function of the radius of the triplet circle is
Actually, this distribution function is correct only if the support of lies in the interval [0, ]. We assume that the mean and standard deviation of are such that this is approximately the case. Figure 7 is a graph of when = 0.0740 meters, = 0.0133, and = 2.8240. These values where chosen so that: (1) the radius of the exact circle is 53.960 meters, the same as the least-squares radius of the starting-line circle, (2) the points and subtend an arc of 6 °, about half of the 12.134 ° arc subtended by the original 21 data points, and (3) the standard deviation 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 using these values is 55.910 meters (computed numerically). This compares favorably with the radius, 56.242 meters, of the three-points circle. The distribution function in Fig. 7 likewise compares favorably with a histogram of the 680 triplet circles shown in Fig. 8. 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 1.019 °. 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. [6]. He adopted the Babylonian practice of dividing the twelve signs of the zodiac into 30 equal parts [7]. 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. 6. Acknowledgements. 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 under the auspices of the Corinth Excavations of the American School of Classical Studies at Athens, Charles K. Williams, Director. During the years 1989-91, Mr. Benjamin Schoenbrun was Research Intern for the Corinth Computer Project and assisted with the analysis of the staring line data. I (D. G. Romano) am grateful to all of the University of Pennsylvania students who assisted me 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. |
[1] | C. K. WILLIAMS, II AND P. RUSSELL, Corinth excavations of 1980, Hesperia, 50(1981), pp. 1-44. |
[2] | D. G. ROMANO, Athletics and Mathematics in Archaic Corinth: The Origins of the Greek Stadion, Memoirs of the American Philosophical Society, Volume 206, American Philosophical Society, Philadelphia, 1993. |
[3] | J. E. DENNIS AND R. B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, NJ, 1983. |
[4] | D. HANSELMAN AND B. LITTLEFIELD, The Student Edition of MATLAB: Version 4: User's Guide, Prentice Hall, Englewood Cliffs, 1995, pp. 419-421. |
[5] | J. A. NELDER AND R. MEAD, A simplex method for function minimization, Computer Journal, 7(1965), pp. 308-313. |
[6] | I. THOMAS, Selections Illustrating the History of Greek Mathematics (Volume II), Harvard University Press, Cambridge, MA, 1941, pp. 395-397. |
[7] | O. NEUGEBAUER, The Exact Sciences in Antiquity, Second Edition, Brown University Press, 1957, pp. 97-144. Reprinted by Dover Publications, Inc., New York, 1969. |