News from the American Sports Builders Association                                                       April 2013

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Measurement of Banked Indoor Tracks - Part 1
by Wayne T. Armbrust, Ph.D., ComputomarxTM

The Problem of Measuring Banked Tracks

Banked indoor tracks are designed to have significant banking in the curves to aid runners negotiating them, with a transition region at the beginning and ends of the curves to transition from the flat to banked surfaces. These tracks can either be permanently banked or raised hydraulically from a flat configuration. In the case of a flat track, since there is no banking, there is no variation in height above the straightaway as one traverses around a turn. Likewise, the radius of each lane on the track remains constant as one traverses the turn. Hence, locating markings on a flat track is a one dimensional problem; one of determining either the angular location for markings on a curve, or the distance on either straightaway from one of the Points of Curvature (PC). Locations of markings on a curve can also be determined using chord measurement, which also involves only one dimension.

By contrast, measurement on a banked track is much more complicated because the curves are much more complicated. Each curve consists of a transition from one straightaway at the PC to an ascending portion, the ascending portion itself, a transition from the ascending portion to a segment with constant banking, the constant banking portion, a transition from the constant banking segment to a descending portion, the descending portion, and finally a transition from the descending portion to the other straightaway. For each lane, the banking has a value of zero at the PC at the beginning of the curve, increases to a maximum value part way through the curve, maintains that value through the middle of the curve, then returns to zero at the PC at the end of the curve. Additionally, both the height above the straightaway z and radius r vary, with different values for each lane, as the ascending and descending potions of the curve, and the transitions from those portions to the straightaways and constant banking portions of the curve are traversed. Therefore, calculating the location of markings on a banked track is a three dimensional problem.

In contrast to the straightforward calculations required for a flat track, the calculations for a banked track will involve such parameters as the angle of banking in the constant banking portion of the curve, the details of the transition from the PC to the beginning of the ascending portion of the track and from the end of the descending portion to the PC, the angle traversed during the ascending and descending portions, the details of the transitions from the ascending portion to the constant banking portion and from that segment to the descending portion, and the angle traversed during the constant banking portion.

Because of the complexities mentioned above, track builders have to this point not attempted to locate markings on a banked track analytically. In the United States the usual process is to measure back from the finish line along the approximate measurement line by placing numerous pins on the curves 30cm out from the curb in lane one in the case of a track with a curb or 20cm out from the outer edge of the lane line for all other lanes as well as lane one for tracks with no curb. Alternatively, since the length of the straights are the same in each lane, and since the distances in the constant banking portion of the curve reduce to a one dimensional problem, measurement around the pins can be restricted to the transition, ascending, and descending portions of the curve. In effect the curve or portion thereof is being approximated by a large number of straight-line segments. For a flat track this would amount to approximating a semicircular arc by a half of a many-sided polygon.

Accuracy of Measurement using Current Methods

The question naturally arises: Using this method how many pins are required on a curve to achieve acceptable accuracy? The IAAF allows no short tolerance at all and only a one part in ten thousand long tolerance on outdoor tracks. This is relaxed to two parts in ten thousand for banked indoor tracks. This means that a banked track with a nominal lap length of 200 meters must have an actual measurement L of 200.000m ≤ L ≤ 200.040m.
Let us consider a simple case where we are approximating the arc length of a flat track of radius r using only four pins dividing the curve into three equal segments, as illustrated in Figure 1.

Referring to Figure 1, we see that θ is 1/3 of a semicircle, 60 or π/3 radians. s is seen to be rxsin(θ/2) = rxsin30 = r/2. We see in the case of a semicircle divided into three segments that the approximation to the arc length is 6x(r/2) = 3r. For a flat track we know the exact arc length to be πr, where π = 3.14159 With just four pins it looks like we may be getting close. How many pins are required for the required accuracy? We can rewrite our four-pin approximation as

3x2rsin(π/6) = (4-1)x2rsin(π/(2(4-1))) (1)

with the angle given in radians. If we generalize equation (1) to consider an arbitrary number of pins (1) becomes

2(n-1)rsin(π/(2(n-1))), (2)

with n the number of pins. It is apparent that all such approximations underestimate the actual arc length, so we want to use enough pins so that our approximation is within one part in 10,000, putting us in the middle of the allowable range. Thus, we want the difference between the exact value and our approximate value divided by the exact value to be less than or equal to 0.00010. Therefore,

(π-2(n-1)sin(π/(2(n-1))))/π ≤ 0.00010. (3)

Note that r has canceled in inequality (3), showing that our result is independent of the radius of the curve. By substituting values for n into (3), we find that n must be greater than or equal to 66 in order to approximate the arc length of a flat track to the required degree of accuracy. If we want the same one part in 10,000 long tolerance allowed for outdoor tracks the right side of inequality (3) would become 0.00005 and the corresponding number of pins would be 92.

If we now consider the case of a banked track, we can logically conclude that since the measure line is varying in three dimensions, at least as many pins would be required to produce the same level of accuracy as on a flat track. However, since the changes in z and r as one traverses the turn are a relatively small part of the total curvature of the measure line, we are probably safe to assume that 75 pins on a curve per lane would insure the track to be no more than two parts in ten thousand long, and that 100 would insure it be no more than one part in ten thousand long. If measurement around pins is restricted to the ascending, descending, and transition portions of the curve, the number of pins can be reduced in proportion the total that those portions of the curve bear to the total.

In Part II of this paper we will explore calculating arc lengths of banked tracks analytically.

2013 American Sports Builders Association 

8480 Baltimore National Pike #307 Ellicott City, MD 21043 410-730-9595

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