lap, lead, and travel, at what point in the piston's stroke does the valve open and close, to steam and to exhaust? Calculating an exact answer to that question before computers was too much work. The easy approximation (used in Zeuner's and Realeaux's diagrams) is to pretend that both the valve and the piston have a sine-wave motion (as they would, if the main rod were infinitely long). Then, for instance, to calculate the percent of the piston's stroke at which steam admission is cut off: Calculate the angle whose cosine is twice the lap divided by the valve travel Calculate the angle whose cosine is twice the (lap plus lead), divided by the valve travel Add the two angles and take the cosine of their sum; subtract 1 from that cosine and multiply the result by -50. As built the Pennsylvania's I1s 2-10-0 had lap 2 inches, lead 1/4 inch and valve travel 6 inches in full gear. In full gear the two angles are 48.19 deg and 41.41 deg and the maximum cutoff comes out 49.65% of the piston stroke. See also Slide valve Steam locomotive components Valve gear References Locomotive valve gear Steam locomotive technologies Engine valves