In continuation of the "Why Study Math" series of articles, here we look at another conic section: the ellipse. The four conic sections, in order from most popularly known to least, are the circle, the ellipse, the parabola, and the hyperbola. Remember that these shapes can all be obtained by slicing a right circular double-napped cone with a plane. As a visual exercise, picture an ice cream cone"without the ice cream"upside down standing on a table. This is a single-napped right circular cone. (To get a double-napped cone, add another cone on top, right-side up, balanced at the point.) To get the circle, take an imaginary plane (picture a piece of paper) and intersect the cone at a right angle to the base. The plane has just cut out a circle on the cone. Similarly, to get the ellipse tilt the plane slightly up or down and intersect it with the cone. What you have then is an elongated circle, or ellipse. Probably the most famous application of the mathematical curve called an ellipse is in the description of planetary orbits. Johannes Kepler, the famous German mathematician and astronomer, used the position of the planet Mars and the sun to work out the orbit of the earth. After twenty years of painstaking work and analysis, Kepler was finally able to put forth his three planetary laws of motion. The first of these laws states correctly that the planets move about the sun in elliptical orbits. This was a revolutionary breakthrough which finally put an end to the Copernican idea of circular orbits. Even though some might know that the planets revolve in elliptical orbits about the sun, a far less common application of this conic section is in machinery. The ellipse finds itself intricately involved in the manufacture of cams, which are rotating pieces of machinery that serve to transform rotating motion into up-and-down or back-and-forth motion. Examples of this are seen in the sewing machine, which uses the cam to generate up-and-down motion of the sewing needle. The punch press is another example that utilizes the cam. This machine is used in generating dies for the manufacture of metal objects. Because of the features of the ellipse, the punch press can function to produce all kinds of intricate metal objects running the gamut from rifle barrels to triggers to nuts and bolts to wire. Of course, let us not forget the use of the cam in the automobile engine. The car engine has to be one of the most important inventions of all time. Because of knowledge of the ellipse, man was able to design the engine, relying on the cam and camshaft to generate the up-and-down motion from the elliptical cams moving in a rotary manner. This up-and-down motion is intricately involved in the complicated tasks of the internal combustion engine. Not bad for a simple mathematical idea such as the ellipse! So the next time you hop in that new spit-shined Mercedes of yours and start to rev the engine, remember that had it not been for the study and application of the ellipse, you might be picking up your sexy date in a horse and buggy. Till next time... |
