Steinmetz Intersection of Cylinders—The Perpendicular Intersection of Two Identical Cylinders
Introduction
Charles Proteus Steinmetz was one of the great engineers at the turn of the twentieth century. His work caught the attention of Thomas Edison, who wanted very much to hire Steinmetz to work at his new company, General Electric. Although Steinmetz was interested in GE, he refused the job offer out of loyalty to his employer, Rudolph Eickenmeyer. Edison proceeded to buy Eickenmeyer's company specifically to employ Steinmetz, who then worked at GE for many years (Flynn, n.d.). He was the primary developer of the alternating current (AC) generator and AC power-distribution methods.
It is said that one day, several GE engineers came to Steinmetz about a problem they couldn't solve. The problem was how to determine the volume of the (orthogonal) intersection of two identical cylinders—that is, cylinders of the same radius. Steinmetz thought for less than a minute and gave them the answer. Being more mathematician than engineer, Steinmetz later became interested in the general intersection of several cylinders and wrote a paper of their solutions.
This problem has many applications. One such application is demonstrated at Metro Center in Washington, D.C., where two cylindrical subway tunnels intersect in the main area. The engineers who designed the Metro Center station needed to calculate the volume of that intersection to properly estimate the costs associated with earth removal, the amount of concrete needed, the volume of available space, air circulation, and related engineering tasks.
We will examine the Steinmetz intersection of cylinders in detail.
Problem Statement
We want to determine the volume of the orthogonal intersection of two identical cylinders. The views below show the intersection of the cylinders and the common volume being "lifted out" from the cross section. It is the volume of this "lifted out" intersection that we want to calculate.
Figure 1
The Perpendicular Intersection of Two Identical Cylinders:
Examining the Common Volume
Reference
Flynn, T. (n.d.). Charles Proteus Steinmetz, inventor. Yonkers Historical Society: Yonkers history. Retrieved from http://www.yonkershistory.org/stein.html