In actual reality there would be wind and water currents diverting any ship sailing that route from the depicted “line” anyway so the whole argument is pointless
The only straight line paths in the universe are followed by electrostatically uncharged non-accelerating objects in free fall in a vacuum. Or massless particles.
Nuh uh. My fifth grade math teacher told me that if I drew a line with an arrow on graph paper and no other line intersected it, that it would continue on into infinity!
it’s a bit of a “spirit of the law vs letter of the law” kind of thing.
technically speaking, you can’t have a straight line on a sphere. but, a very important property of straight lines is that they serve as the shortest paths between two points. (i.e., the shortest path between A and B is given by the line from A to B.) while it doesn’t make sense to talk about “straight lines” on a sphere, it does make sense to talk about “shortest paths” on a sphere, and that’s the “spirit of the law” approach.
the “shortest paths” are called geodesics, and on the sphere, these correspond to the largest circles that can be drawn on the surface of the sphere. (e.g., the equator is a geodesic.)
i’m not really sure if the line in question is a geodesic, though
You are absolutely correct, but to add on to that even more:
When we talk about space, we usually think about 3D euclidean space. That means that straight lines are the shortest way between two points, parallel lines stay the same distance forever, and a whole bunch of other nice features.
Another way of thinking about objects like the earth is to think of them as 2D spherical manifolds. That means we concern ourself only to the surface of the earth, with no concept of going below the surface or flying up into the sky. In S2 (that’s what you call a 2D spherical manifold), and in spherical geometry in general, parallel straight lines will eventually cross, and further on loop back and form a closed loop. Sounds weird, right? Well, we do it all the time. Look at lines of Longitude, for example.
We call the shortest line connecting two points in curved manifolds geodesics, as you said, and for all intents and purposes, they are straight. Remember, there is no concept of leaving the sphere, these two coordinates is all there is.
What one can do, if one wants to, is embed any manifold into a higher-dimensional euclidean one. Geodesics in the embedded manifold are usually not straight in higher-dimensional euclidean space. Geodesics on a sphere, for example, look like great circles in 3D.
i think it depends on what you mean by “accurately”.
from the perspective of someone living on the sphere, a geodesic looks like a straight line, in the sense that if you walk along a geodesic you’ll always be facing the “same direction”. (e.g., if you walk across the equator you’ll end up where you started, facing the exact same direction.)
but you’re right that from the perspective of euclidean geometry, (i.e. if you’re looking at the earth from a satellite), then it’s not a straight line.
one other thing to note is that you can make the “perspective of someone living on the sphere” thing into a rigorous argument. it’s possible to use some advanced tricks to cook up a definition of something that’s basically like “what someone living on the sphere thinks the derivative is”. and from the perspective of someone on the sphere, the “derivative” of a geodesic is 0. so in this sense, the geodesics do have “constant slope”. but there is a ton of hand waving here since the details are super complicated and messy.
this definition of the “derivative” that i mentioned is something that turns out to be very important in things like the theory of general relativity, so it’s not entirely just an arbitrary construction. the relevant concepts are “affine connection” and “parallel transport”, and they’re discussed a little bit on the wikipedia page for geodesics.
Would clarifying words have helped? “If you only sailed with forward force…” or “Following along the surface of the earth…” or… what?
Obviously they mean that you don’t need to make any turns and that straight means an arc around the earth and not through the Earth, unless someone has a very different idea what sailing means…
Yes I think they mean it’s a continuous line, not a “straight” line. As in the line is uninterrupted (continuous). It’s also possible they mean the line qualifies as a nonlinear function since it also doesn’t double back over itself (A function is a relationship where each input value (X) will create only one output value (Y)).
Math is hard. Describing lines like this is math - calculus actually due to the curve, and actually not just basic calculus but vector calculus because it involves an x,y, and z axis. Most laypeople will struggle to describe a line with the correct jargon.
I assume you mean “both capes.” While this line does come within a few thousand miles of the Horn of Africa, that’s not known as an especially hard sailing area but maybe for pirates.
Sailing this line in the other direction would be considerably harder.
Not to mention, India’s coastline is very much not straight on a local scale. You’re bound to find a place where it turns perpendicular to the journey close to the theoretical starting point anyway.
Yeah but, I’m talking about this particular case, not making a mathematical rule. You have to move away from the coast, and then cannot turn, so you have to head towards Africa. You can’t set off toward Australia. Although I hadn’t considered that you can just move the starting point. So, there’s that.
No, that’s not Earth’s great circle, you’ll be turning slightly. It only seems straight on most map projections because they want latitudes to be horizontal.
It would, however, seem like a straight line to whoever was on the boat, because they’d be traveling due west the whole time, and the course corrections they’d have to make to keep going west would look the same as course corrections needed to account for wind, ocean currents, etc.
I know but you need to be the right amount of pedantic. Too little and any sufficiently large curve seems straight, too much and you point out that there is no straight line on the surface of a sphere.
Edit: I have just finished reading The Neverending Story and this reminds me of the last part where Bastian works in the picture mines until he finds the right picture.