waterfall start
### The Geometry of the Waterfall Start

In the diagram below the circle represents the measured line
in lane 1 of Manley track, the horizontal axis coincides with the finish
line, and the curve is the starting line. The curve has the
property that runners starting at any place along it and
running the shortest distance to the "pole" will run the same
distance as a runner who follows the measured line. Thus the
curve is the locus of points B such that the length of the tangent
AB is equal to the length of arc AC.

The technical name for this curve is the ** involute of a
circle. ** (Ignore the word "evolute" in the diagram above. It is incorrect, but the diagram is hard to edit.) There is no simple cartesian equation y = f(x)
for this curve, but it is simple to derive a pair of parametric
equations for x and y in terms of the parameter t which gives
the central angle COA in radians. Indeed, if R denotes the
radius of the circle then radius vector OA = (Rcos(t),Rsin(t)).
The key fact is that the tangent AB is orthogonal to OA, hence
has unit direction vector (sin(t),-cos(t)). Since AB has length
Rt ( = the length of arc

AC ), we obtain the coordinates of B
as the vector sum OA + Rt(sin(t),-cos(t)). The parametric
equations are thus

- x = Rcos(t) + Rtsin(t)
- y = Rsin(t) - Rtcos(t)

The equation for the polar coordinate r is even simpler. It is
r^2 = 1 + t^2. (Thanks to Vince Fatica for pointing this out.)
We only see a very small portion of this curve in Manley. The
entire curve looks almost exactly like an equilateral spiral
and, indeed, the distance between the two curves approaches zero
as t tends to infinity. ( Actually, since Manley is not a
circle but rather two semicircles joined by a straightaway, the
involute for Manley track will only coincide with the involute of
a circle for 0 < t < pi. )

There is more information about the involute and other
standard mathematical curves
available on the Web.
The * NCAA Track and Field and Cross Country Rulebook
*
describes a simple procedure for laying out a waterfall start
on tracks:

The curved starting line may be established by driving a row of
pins 3.05 meters (10 feet) apart, 0.3 meters (12 in.) from the
curb-- the first pin to be 0.3 meters (12 in.) from the curb
at the start. For a 9.75 meter (32 ft.) track, 10 pins are
sufficient.
Using a steel tape 30.48 meters (100 ft.) or longer, and with
the pin furthest from the start as center, scribe an arc from
pole to outer curb of track.

This will not be an arc of a circle as the radius will change
as the tape loses contact with each successive pin.

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