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 straightline segments. For a
flat track this would amount to
approximating a semicircular arc
by a half of a manysided
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
fourpin approximation as
3x2rsin(π/6) =
(41)x2rsin(π/(2(41))) (1)
with the angle given in radians.
If we generalize equation (1) to
consider an arbitrary number of
pins (1) becomes
2(n1)rsin(π/(2(n1))), (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,
(π2(n1)sin(π/(2(n1))))/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. 
