The Digital Camera: The Principle of Least Time

Imagine you need to move from your current location to a site that is 3 miles to the south and 4 miles to the east. As all crows and most people know, the shortest path between those two locations is a straight line. To determine the length of that straight line, though, we need to use a famous rule that the ancient Greek mathematician Pythagoras of Samos developed in the 6th century BC. The Pythagorean theorem — as we call it today — is concerned with the mathematical relationship between the lengths of the three sides of a right triangle:

A right triangle. The angle between the sides marked 'a' and 'b' is 90 degrees.

The theorem tells us that the relationship between the lengths of the three sides is

a^2 + b^2 = c^2,

so, if we know a and b but want to know c, then we must evaluate the square-root of a sum of squares:

c = \sqrt{a^2 + b^2}.

To travel 3 miles to the south and 4 miles to the east then, the shortest distance is 5 miles because: 3 times 3 is 9; 4 times 4 is 16; 9 plus 16 is 25; and the square-root of 25 is 5.

The 17th century French lawyer and amateur mathematician Pierre de Fermat developed several mathematical theorems and physical rules during his lifetime. His famous last theorem, for instance, concerns equations like the one in the Pathagorean theorem, and the proof of this theorem puzzled scholars for many centuries until a British mathematician named Andrew Wiles discovered and published a proof in 1995. But it is Fermat’s principle of least time that provides our key to understanding the way in which light reflects from mirrors and bends when passing from air into surroundings like glass or water.

According to Fermat, light travels from one position to another through a path that is quicker than any nearby path. To see how this principle causes light to bend when traveling from air into glass or water, the Nobel Prize-winning physicist Richard Feynman provided an insightful example in his Feynman Lectures on Physics. To understand this phenomena, Feynman suggested, we should think about the fastest way to move from a particular location on land to another location in water.

To follow Feynman’s suggestion, suppose we can run at a speed of ten miles-per-hour on land and swim at a speed of four miles-per-hour in water. The ratio of these speeds is 2.5, which we can think of as our refractive index in water. Based on the following figure, then, we could determine where to enter the water so that our transition from location A to location B takes the least amount of time.

Three possible paths from location A on land to location B in water.

With Path 1 we minimize our time swimming by running from location A to a point on the shore that is directly above location B, then swimming the short path to our destination. We swim for 20 meters; we run for a little less than 61 meters. With our swimming speed of 4 mph we spend about 11.2 seconds in the water, and with our running speed of 10 mph we spend about 13.6 seconds on land. The total time for Path 1, then, is about 24.8 seconds.

With Path 2 we minimize our time running by entering the water directly below position A, then swimming a long path to our destination. We run for about 2.2 seconds, and swim for about 35.4 seconds. Our total time to reach the destination is about 37.6 seconds, which exceeds the time for Path 1 by more than 10 seconds. We should expect this to be true, though, because Path 1 minimizes the time we spend swimming and Path 2 minimizes the time we spend running.

Path 3 is the straight line between the two locations. For this path we spend about 5 seconds running on land and about 25 seconds swimming in water. The total time from location A to location B is about 30 seconds, which makes Path 1 the shortest of these three paths. This example teaches us that the shortest time between two locations is not a straight path when our speed changes along the path. But we have not learned whether or not Path 1 is the quickest way to move from location A to location B. To determine this, we need to check all possible paths between the two locations.

Just as we did for Paths 1, 2 and 3, we can compute the total time to move from location A to location B for every entry point between 0 meters – directly below location A – to 60 meters – directly above location B. The result is plotted in the figure below where we see that the path of least time occurs when we enter the water roughly 51.5 meters from location A. The time associated with this optimal path 23.9 seconds.

The total time required to move from location A to location B as a function of the point at which we enter the water. The path of least time requires us to enter the water a little over 51.5 meters down the shore.

When we take the route of least time, our path will bend substantially when we enter the water. This bending happens because we travel faster on land than we do in water. If we traveled with the same velocity on land and in water, then the path of least time would be a straight line between the two locations (Path 3 in the original figure). We take a bent path because we are 2.5 times faster on land than in water: we would enter the water slightly sooner if we were only 1.5 times faster on land than in water; and we would enter the water later if we were 5 times faster on land than in water. Because light travels faster in air than it does in water its rays bend in a similar way. Light also bends when it enters glass from air, and optical engineers design sophisticated lenses by understanding and controlling the precise angles at which light bends at the transition.

The path of least time requires us to enter the water approximately 51.5 meters down the shore from location A, then swim directly to location B.

© 2011 Timothy Schulz

The Digital Camera
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