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

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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