By
joining smoothly with the segments at either end, we mean that not only
do the segments join, but that the first derivatives of α with respect
to θ are continuous at the junctions. The limits on acceleration in the
z direction will put limits on the lengths of the transition zones
depending on the maximum angle of banking α_{0}, and to a lesser
extent, the radius of the curve.

The
simplest mathematical form to produce the smooth transition zones is a
quadratic equation of the form

α = a(θ_{
}– θ_{s})^{2}+d (15)

with α the angle of banking at an angle traversed of θ, θ_{s}
the angle traversed at the beginning of the transition zone, and a and d
constants to be determined.

For
the transition zone at the beginning of the curve, α equals zero when θ_{
} equals zero and since at point 0 θ_{s} equals zero, d
equals zero for the transition segment from point 0 to point 1. The
constant a must be chosen such that the transition joins the ascending
portion of the curve at point 1 smoothly.

We can rewrite Equation
11 as

α = b(θ_{
}– θ^{’}) (16)

with

b = (α_{2}
- α_{1})/(θ_{2 }– θ_{1}) (17)

and
θ^{’} the value of θ at which α equals zero in Equation 11. The
smoothness requirement determines a; taking the derivative of Equation
17 and setting it equal to the derivative of Equation 15 at θ_{ }
= θ_{1} we obtain

a = b/2θ_{1}
(18).

Thus, for the transition
from point 0 to point 1, the equation for α as a function θ of becomes

α = (b/2θ_{1})θ_{
}^{2} = [((α_{2} - α_{1})/(θ_{2 }–
θ_{1}))/2θ_{1}]θ_{ }^{2}. (19)

Using similar arguments, we can determine the equation for α in the
transition region from point 2 to point 3. In this transition zone we
obtain

α = α_{0}
- (b/2(θ_{3} – θ_{2}))(θ - θ_{3})^{2} =
α_{0} - [((α_{2} - α_{1})/(θ_{2 }– θ_{1}))/2(θ_{3}
– θ_{2})](θ - θ_{3})^{2}. (20)

Since we assume that the angle of banking is constant across the cross
section of the track, Equations 19 and 20 apply for all lanes.
Substituting into (8a) and (8b), z_{i} and r_{i} become

z_{i}
= d_{i}sin[(b/2θ_{1})θ^{2}] (19a)

r_{i}
= r_{0} + d_{i}cos[(b/2θ_{1})θ^{2}]
(19b)

for the transition
segment from point 0 to point 1, and

z_{i}
= d_{i}sin[α_{0} - (b/2(θ_{3} – θ_{2}))(θ
- θ_{3})^{2}] (20a)

r_{i}
= r_{0} + d_{i}cos[α_{0} - (b/2(θ_{3} –
θ_{2}))(θ - θ_{3})^{2}] (20b)

for the
segment from point 2 to point 3.

∂z_{i}/∂θ and ∂r_{i}/∂θ
are given by

∂z_{i}/∂θ
= d_{i}[cos((b/2θ_{1})θ^{2})](bθ/θ_{1})
(21a)

∂r_{i}/∂θ
= - d_{i}[sin((b/2θ_{1})θ^{2})](bθ/θ_{1})
(21b)

for the first transition
segment and

∂z_{i}/∂θ
= -d_{i}[cos(α_{0} - (b/2(θ_{3} – θ_{2}))(θ
- θ_{3})^{2})](bθ/(θ_{3} – θ_{2}))
(22a)

∂r_{i}/∂θ
= d_{i}[sin(α_{0} - (b/2(θ_{3} – θ_{2}))(θ
- θ_{3})^{2})](bθ/(θ_{3} – θ_{2}))
(22b)

for the second.
Substituting into (7) and integrating (after simplification), we obtain

S_{t1i = ∫}√(r_{0}^{2}
+ 2r_{0}d_{i}cos[(b/2θ_{1})θ^{2}] + d_{i}^{2}cos^{2}[(b/2θ_{1})θ^{2}]
+ d_{i}^{2}[bθ/θ_{1}]^{2})dθ (23)

where S_{t1i} is
the arc length of the transition segment from point 0 to point 1 in lane
i and

S_{t2i} = _{∫}√(r_{0}^{2}
+ 2r_{0}d_{i}cos[(α_{0} - b/2(θ_{3} – θ_{2}))(θ
- θ_{3})^{2}] + d_{i}^{2}cos^{2}[(α_{0}
- b/2(θ_{3} – θ_{2})) (θ - θ_{3})^{2}]
+ d_{i}^{2}[bθ/(θ_{3} – θ_{2})]^{2})dθ
(24)

where S_{t2i} is the arc length of the transition segment from
point 2 to point 3 in lane i. As in the case of the ascending and
descending segments, Equations 23 and 24 are not integrable in closed
form, but again the integrals can be accurately evaluated using standard
numerical methods.

From the symmetry assumed in Part II of this paper, we obtain the arc
length of lane i for the entire curve as

S_{i}
= S_{ci} + 2S_{ai} + 2S_{t1i} + 2S_{t2i}.
(25)

**Vertical Acceleration Considerations**

We now consider accelerations in the z direction.
If we differentiate Equation 19a with respect to θ twice and apply the
chain rule, and evaluate the resultant expression at θ equals 0, we
obtain an expression for the vertical acceleration ∂^{2}z_{i}/∂t^{2}
at that point.

∂^{2}z_{i}/∂t^{2}
= (d_{i}b/ θ_{1})(dθ/dt)^{2} (26)

where dθ/dt is the angular velocity of the runner equal to v/r, where v
is the linear velocity and r is the effective radius of the curve at
that point (reduced by the banking when applicable). Performing similar
operations on Equation 20a and evaluating the resultant expression at θ
equals θ_{3}, the beginning of the constant banking segment,
yields

∂^{2}z_{i}/∂t^{2}
= -[d_{i}b/(θ_{3} – θ_{2})](dθ/dt)^{2}cosα_{0}.
(27)

Table 1 shows vertical accelerations at the beginning and end of the
curve for each lane for a track similar to the IAAF Standard Indoor
Track having an angle of banking of 10° in the constant banking
segment. The constant banking segment is 60° of arc with 5° arc
transitions joining the 50° of arc ascending and descending segments to
the straightaways and constant banking segment. A running speed of 10
m/s is used for the calculations, which is about the maximum ever
expected. Positive vertical accelerations (making the runner feel
heavier) increase going out the lanes, but never exceed 0.21 the
acceleration due to gravity. This is acceptable for the short distance
over which this acceleration occurs.

Table 2 shows vertical accelerations at the beginning and end of the
constant banking segment for a slightly different configuration. Here
the constant banking segment is 70° with 6° transitions and 43°
ascending and descending segments. Here the negative vertical
accelerations (making the runner fell lighter) are never greater than
0.19 g.

In
general maximum vertical accelerations increase with the running speed
and angle of banking. They decrease as the transition segments are
lengthened and the constant banking segment is shortened. However, as
transition segments are lengthened, the slope encountered by the runners
as they ascend and descend the banking increases. The parameter b is a
measure of this slope. Reducing the length of the constant banking
segment negates some of the purpose of having a banked track.
Parameters may be adjusted to suit individual preferences and to match
anticipated competitions. Since vertical accelerations vary with the
square of speed and are significant only in the outer lanes, they really
are only a consideration for shorter races run all or partially in
lanes.

**
Alternate Approaches**