A Geometric Proof Demonstrating That the Path Described by Snell’s Law Follows Fermat’s Principle of Least Time

Refraction artifacts in ultrasound imaging can produce the appearance of side-by-side structures, and, in some cases, color Doppler jets. This phenomenon arises from the bending of the ultrasound wave at the interface between tissues with differing propagation speeds. The paths followed by both ultrasound waves and light rays can be described by Snell’s law and Fermat’s principle of least time, both of which are traditionally derived using differential calculus. However, such calculus-based proofs often lack intuitive clarity. In this paper, we present a simple geometric proof that clearly demonstrates the path taken is indeed the shortest—and therefore the fastest.