# What is a straight line in curved space?

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 262947.  Mon Jan 21, 2008 12:28 pm Could anybody explain this? An illustration of it would be handy.

 262950.  Mon Jan 21, 2008 12:30 pm Certainly! If you'll just pass me that 4 dimensional piece of paper please... ;)

 262967.  Mon Jan 21, 2008 12:44 pm I think the 2D line in 3D space analogy comes into play here. pstotto: The equator is a straight line, yes? well, no it isn't, but in a way it is. do you get the idea? someone may need to come and explain it more eloquently.

 263012.  Mon Jan 21, 2008 1:56 pm The shortest path between two points in any space is called a geodesic. In flat space, which we call Euclidean, geodesics are segments of straight lines. On the surface of a sphere, they are arcs of great circle. (A great circle lies on the intersection of the surface and a plane passing through the centre of the sphere.) More exotic shapes have more exotic geodesics.

263105.  Mon Jan 21, 2008 4:19 pm

 pstotto wrote: Could anybody explain this? An illustration of it would be handy.

Find a really, really, big piece of chalk, a compass, and a very long straight edge (say, a 1000 metre ruler). Start on a large continent, say somewhere in Asia, not too near the sea (I'd say the USA, but you would undoubtedly get sued for something while performing your experiment). Make a note of how far North or South of the equator you are - you will need this information later.

Start chalking a long, straight line that runs East to West. Keep going until you get to the sea, using your straight edge to make sure the line is properly straight.

Now, find a space ship (anything capable of achieving orbit will do nicely). Go straight up about 200km, and find yourself a nice comfy orbit that has you looking at your nice East > West line from A Long Way North or South of your chalk line.

Does your line appear straight? Take careful notes on any curvature of the line, then extrapolate to 4 dimensions.

Job Done. Certain proof that travelling straight up into space can bend straight lines. Or not.

Regards,

Glenn.

 266500.  Sun Jan 27, 2008 8:51 pm to simplify, take a peice of string, find london & new york on a large map ( flat ) hold the string taut with one finger holding the string on london& one likewise on new york, that is the shortest route between the two, or would be if we lived on a flat suface. hold the string in place with sticky tack, mark the route with felt tip. next take a globe of approx same scale, mark on the globe the places where the felt tip passes through on the flat map, join them up...... see ? it's a curve. repeat this the other way round ( starting with the globe this time) see? similarly again , flat/straight on one is a curve on the other... the "true" straight line would pass through the earth's crust ( going through 3D space)

 266975.  Mon Jan 28, 2008 3:20 pm Unfortunately chalkie, you're quite wrong. Maps don't accurately represent the surface of the earth. They can't, after all, because the surface is curved and the map is not; that's sort of the point. I don't recall the theorem exactly, but I think that if two surfaces can be matched up so any point on one has a corresponding point on the other, and pairs of corresponding points are always the same distance apart, then the two surfaces have the same curvature. As I said before, the equivalent of a straight line on a sphere is a great circle. To find the path giving the shortest distance between New York and London, imagine a flat sheet of paper passing through your globe, touching New York, London, and the centre of the globe. As long as these three points are not on a line (if they are, you need a new globe), this defines the plane exactly. Now the points where the paper and the globe form a great circle; the shortest path between your two points is the shorter of the two segments of this arc between them. Fun things about spherical geometry: the angles of a triangle always add up to more than pi (or if you insist on using archaic measurements, 180 degrees). The amount by which the sum exceeds pi is proportional to the size of the triangle (relative to the size of the sphere). You can also have a two- and even a one-sided polygon. Hyperbolic space on the other hand has negative curvature, so the angles of a triangle sum to less than pi.

 269609.  Fri Feb 01, 2008 5:52 pm i can accept that neither a flat map nor a globe is totally accurate in representing the surface of the earth ( i think mr. fry said it was an " oblate spheriod" ) what i meant was point to point on a flat map gives a different answer to point to point on a globe, point to point through "actual " 3d space would mean leaving the surface of the earth & going underground ,undercrust & under a little bit of mantle given that planes don't go underground in that way, they follow a straight line over the surface, if memory serves,for the NY-from london trip, this goes through scotland,over the top edge of the atlantic,over greenland , finally following the coast of newfoundland & new england . following that path on a flat plan appears to be a long **** way round. ( i got a "u" for unclassified in o' level geography btw

 269760.  Sat Feb 02, 2008 7:33 am but the aformention long **** way round is probably more due to spending as little time over water as possible, rather than earth curvature, right?

 269780.  Sat Feb 02, 2008 8:33 am I think a straight line in curved space is curved. Sounds counter-intuitive. I come back to you on this one when I can.

 270054.  Sun Feb 03, 2008 6:45 am You can never draw a straight line with a ruler. The edge is not 100% straight anyway if looked at with a microscope. No straight lines exists in reality.

 270087.  Sun Feb 03, 2008 8:18 am I have a 'laser level' which projects a straight line onto any surface. Is this line straight? I soppose the answer might be that it is 'domestically straight'. In the same way, the lines of the National Grid on your Ordnance Survey map are straight - only they aren't, if you follow them from edge to edge. They can be taken as straight, within a journey (which is a day's walk, about 60 miles, as far as the eye can see).

270097.  Sun Feb 03, 2008 8:44 am

 chalkie wrote: if memory serves,for the NY-from london trip, this goes through scotland,over the top edge of the atlantic,over greenland , finally following the coast of newfoundland & new england.

Flights to the west coast of North America certainly overfly Greenland, but do flights to New York in fact do so - it's been a while, and to be honest I can't remember.

Out of idle curiosity, I just had a play on Autoroute and mapped the "straight line" routes from London to both New York and Vancouver - they look like this:

270396.  Mon Feb 04, 2008 12:07 am

 ficklefiend wrote: but the aformention long **** way round is probably more due to spending as little time over water as possible, rather than earth curvature, right?

i heard somewhere that , while concorde was operating, it was only allowed to go supersonic when out over the ocean. hmmm i've got something wrong here..... hang on was the flight which crashed in lockerbie going to new york ( normal jet, going to u.s.of a over scotland), or have i got that wrong?

 270464.  Mon Feb 04, 2008 5:10 am No, you're quite right about the Lockerbie flight: http://news.bbc.co.uk/onthisday/hi/dates/stories/december/21/newsid_2539000/2539447.stm

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