What is the quickest way for light to get from point A to point B? The speed of light varies with the medium through which it travels. The quickest route through any one medium is a straight line segment. The only question is where on the boundary should the refraction point be? In this sketch, let x be the horizontal distance from A to the refraction point. Time is distance divided by speed.
The time will be minimized where the derivative of time with respect to x is zero.
This last equation is a version of Snells Law. If light (or anything else) conforms to Snells Law, then it must be taking the quickest path.
Back to the Brachistochrone
Back to Whistler Alley Mathematics
Last update: August 3, 2011 ... Paul Kunkel firstname.lastname@example.org
For email to reach me, the word geometry must appear in the body of the message.