# Intuition - The Shortest Curve Between Two Points is a Line

The references below aver that the following is not crudely trivial:

The shortest curve between two points is a (straight) line.

An elementary school teacher construed it as follows (which I now register as the contrapositive):
If you don't walk to the other point in a straight line, then you must be walking more distance to get there.

Is there an improved intuition of this result? I am not asking for any proof or formal argument. Please forgive me should this be a duplicate.

I referenced ♦ The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?,
http://www.thestudentroom.co.uk/showthread.php?t=52233,
♦ P117-119 of A Guided Tour of Mathematical Methods for the Physical Sciences by Roel Snieder.

• I have in mind the triangular inequality. – Tony Piccolo Sep 2 '13 at 10:51
• If you are allowed a piecewise linear approximation to a path (which is one of the ways of defining the length of a path which has a meaningful length) you can look at the first two line segments from the start. and create a shorter path with fewer segments by connecting the endpoints together and using Ton's idea of the triangle inequality. Eventually you end up with one straight path. – Mark Bennet Sep 2 '13 at 10:55
• Your reference assumes only certain kinds of curves (differentiable) and also uses the distance metric of a straight line (are we being circular in logic)? So the Q is even more difficult! – Macavity Sep 2 '13 at 10:57
• The reason why this is not trivial is that the shortest path between two points depends on which kind of surface you are sitting on or how you measure the distance. Any intuition that is going to give answer to this question must use the fact that we are measuring the daily euclidean distance using "standart metric", that is the one you get by using a "straight ruler". With this in mind, a start could be to give an intuitive explanation for triangle inequality and subadditivity property of metrics. – Sina Sep 2 '13 at 10:57
• @MarkBennet that still is a straight line (but you are in a non euclidean plane) – Willemien Sep 2 '13 at 12:37

## 3 Answers

You asked for something that wasn't a proof or formal argument, so I hope this helps.

In any geometry, including non-Euclidean geometry (e.g. hyperbolic, or spherical geometry), "straight lines" are really called geodesics, which are defined to be the shortest line between two points. This means you stand somewhere holding one end of some rope, your friend stands somewhere else holding the other end, and together you pull the rope taut and this gives you your shortest path.

For example, say you're standing on the surface of a ball (i.e. a 2-sphere, such as the surface of the Earth), and your friend is some way away also on the surface, both of you holding the rope tight. This is spherical geometry. The taut rope or our "straight line" or geodesic is really the shortest path between us that lies on the surface, i.e. where the rope goes. This geodesic will look curved to someone in Euclidean space because there, the geodesic/"straight line" would pass through the ball.

Therefore it turns out that our definition of "straight" depends on the geometry we're using and how we pull the rope taut (the metric we use). It just so happens that in Euclidean geometry, this gives us lines that we call straight.

Interesting note: in other spaces, there are some super cool and peculiar metrics that make the taut rope (shortest path) go into weird shapes in Euclidean geometry! One example: https://en.wikipedia.org/wiki/Taxicab_geometry

I think the explanation that's simplest, oldest and easiest to understand and believe is Euclid's Proposition 20:

In any triangle the sum of any two sides is greater than the remaining one.

http://aleph0.clarku.edu/~djoyce/elements/bookI/propI20.html

If you want to think about more complicated paths, imagine them as built from triangles, as @Abi suggests.

The comments and answers about other geometries (spherical, taxicab) are interesting and correct, but go beyond what I think you are asking.

Let's call the two points A and C. Suppose I go from A to B, then B to C. When I apply the cosine rule, because cos180 is the least value of cosine, the distance I have is minimized when the angle is 180.

This deals only with the case where the curve bends once. If it bends multiple times, you can repeatedly apply the cosine rule to the particular bends to show that they are not optimal (because they are not at 180.) If you have a curved line, I just like to think of it as an infinite number of bends, but that is where the intuition starts to get fuzzy.