12
$\begingroup$

Snell's law of refraction can be derived from Fermat's principle that light travels paths that minimize the time using simple calculus. Since Snell's law only involves sines I wonder whether this minimum problem has a simple geometric solution.

  • 1
    This is treated in "What Is Mathematics" by Courant and Robbins. See the bottom half of [this page](http://books.google.com/books?id=_kYBqLc5QoQC&lpg=PA330&vq=heron&pg=PA330#v=onepage&q&f=false).2012-06-04
  • 1
    @Michael really? I don't have a copy at hand right now, but from the title of the section I would be more inclined to think that the geometric construction relates to _Heron's principle_, which is the principle of reflection that incidence angle is the same as the reflection angle, and which indeed has a simple geometric proof. I would be very pleasantly surprised if there is in Courant and Robbins a geometric proof of Snell's law of refraction.2012-06-04
  • 4
    Huygen's gave a somewhat geometric proof of Snell's law, however, he did not start with Fermat's principle, but rather the assumption that light is a wave, that wave speed equals the product of wave length and frequency, that frequency is invariant across a boundary, and a continuity criterion. I suppose, however, that is not what you are looking for.2012-06-04
  • 0
    @WillieWong Ah you're right - I read the OP's question too quickly. They mention Snell's law at the end of that section, but don't derive it geometrically.2012-06-04
  • 4
    There is a very detailed proof, using Ptolemy's Theorem, in Ivan Niven's bwautiful book *Maxima and Minima Without Calculus.*2012-06-04

2 Answers 2