“The fastest path between two points is ________?”

  1. A straight line
  2. A curve

Most people instinctively pick 1. A straight line, because “the shortest distance between two points is a straight line.” But is that really true?

Shanghai Pudong to Los Angeles

For long-haul nonstop routes like China to the US, planes typically follow a great-circle route (the shortest path between any two points on the globe), as shown on the Google Maps image below:

PVG-LAX

The image below shows the actual flight path. Comparing the two, the real route is clearly longer. You might ask: isn’t that a detour? Your instinct is correct – it is a detour. But here’s the thing: the total flight time is actually shorter.

AL9706

This is because the route takes advantage of the prevailing westerly jet stream. The distance is longer, but the time and fuel consumption are both reduced.

Fermat’s Principle

You probably learned about the law of refraction in physics class:

When a light wave travels from one medium to another with a different refractive index, refraction occurs.

This law can be derived from Fermat’s Principle:

Fermat’s Principle was first proposed by French scientist Pierre de Fermat in 1662: light travels along the path for which the optical path length is an extremum. This extremum may be a maximum, a minimum, or even an inflection point. Originally known as the “principle of least time,” it states that light follows the path that takes the least time.

Fermat Snellius

Because of refraction, the path of light through different media is not a straight line. In other words, light takes a shortcut!

The Brachistochrone Curve

The animation below might shake your worldview: why does a curved path get there faster than a straight one?

Brachistochrone Curve

The answer lies in gravitational acceleration (G). This can also be derived from the law of refraction or Fermat’s Principle. For the detailed proof, see: https://en.wikipedia.org/wiki/Brachistochrone_curve . Shortcuts, it turns out, are hidden everywhere in nature – and there are many more waiting to be discovered.