Minimize the travel time for light from point-to-point

Problem statement:

In optics Fermatís principle states that light traveling from point-to-point follows the path for which the travel time is a minimum.

If we assume that the light is in one medium, like air or water, then the path that minimizes time is also the one that minimizes the distance traveled. Here we also assume that the light travels in a straight line, and is reflected off a mirror towards an observer as in the following diagram.

The light source is a distance a > 0 from the mirror, which has length c, and the observer is a distance b > 0 from the opposite end of the mirror. If the light ray strikes the mirror at V, then the total time to travel to the observer is the same as the total distance the light travels, namely L + R. If we can move the point V where the light ray strikes the mirror then we can determine the point V that minimizes L + R, hence minimizes the time to travel from the source to the observer.

In the diagram above angle is called the angle of incidence and angle is called the angle of reflection.

One of the animations supplied here was derived from the accompanying interactive routine for approximating the minimum time which is a Geometer's Sketch Pad file. This file was developed by

Catherine A.Gorini
Maharishi University of Management
1000 North Fourth Street
Fairfield, IA 52557

and is used with her permission. For more details on classroom use of this file see "Dynamic Visualization in Calculus", by Catherine A. Gorini in, (MAA Notes 41) Edited by James R. King and Doris Schattschneider, The Mathematical Association of America, Washington, DC, 1997, pp.89-94.

Special thanks to Nicholas Jackiw for his technical assistance in updating the Geometer's Sketch Pad file and to Mike Simpson of Key Curriculum Press for his support for the Demos with Positive Impact project.