Is the shortest distance between two points always a straight line? I have been getting conflicting answers to this and would be grateful if someone could clear this up.
You would think, but no one on this board will let you get away with that saying.
In the essence of the saying, yes, it’s true.
But prepare thy self for discussions ranging from issues at the quantum level, to whether or not anything can ever be a “straight” line.
And also on a globe, the shortest distance is along a great circle…
It also depends on the geometry of the space that the two points occupy; the shortest distance between London and New York is a straight line, but to travel that line, you would need to dig a tunnel through part of the curved earth.
In the same way, Arguably, if three-dimensional space is somehow curved in a fourth spatial dimension(i.e. not time) that we conveniently cannot percieve, there might be a shorter route between two points than the straight line that is the only option when limited by three dimensions.
The shortest distance between 2 poits is through a wormhole.
As Alan Dean Foster has so sucintly put it in his Spellsinger series…the shortest distance between two points is NOTHING!
doing best impression of a drunken owl,“There is nothing to fear…there is nothing to fear.”
Within the limits of a given geometry, a straight line is defined as the shortest distance between any two points, so the basic answer is ‘yes’. In unusual geometries, it may not be simple or intuitive to see what a straight line actually is …
As stated above, a great circle is the shortest distance between any two points on the surface of a sphere (eg. the Earth, at least approximately a sphere). Now, a great circle won’t look like a straight line on a Mercator projection, especially if both points are far from the equator, but that’s due to the distortions of the Mercator projection. If you look at the two points on a globe, it’s not hard at all to see that the great circle is the shortest path, and that it’s perfectly straight.
Add in the third dimension (changing from a 2-dimensional space to a ‘flat’ Euclidean 3-dimensional space) and you get a different answer, a line burrowing through the planet. Add in a fourth dimension with variable curvature, per the theory of relativity, and you’ll get a path in spacetime, which minimizes the interval between the departure point/time and the arrival point/time. As you might imagine, this can get quite insanely complicated …
C’mon! Can you ask for better entertainment from a 20 cent question!?
Hey, how about for issues at the quantum level? Do particles take a ‘straight line’ anywhere? Don’t they appear here and there, but they aren’t exactly “travelling” to get from “here to there”. For particles at the quantum level, straight lines are terribly long.
If I’m confused, hey, sue me…it’s quantum mechanics…I’m supposed to be confused.
Sometimes there are several equal shortest distances between two points. Sometimes the shortest path has corners in it.
Here’s a look at Taxicab Space.
I thought it was "as the crow flies.
There can be multiple straight lines between two points, some of which are longer than others. This is what is behind “gravitational lensing”, the effect where a distant light source (typically a bright galactic center) is visible at several different spots in the sky because some intervening mass has curved the space around it. When such sources flicker, its different appearances in the sky flicker at slightly different times.
And, yes, all the lines you are looking along are straight - it’s the space they go through that curves, not the lines themselves.