Why do two lines have to be 'great circles' on a sphere?

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In summary, elliptical geometry is based on rejecting Euclid's fifth postulate, which states that there is only one parallel line. Instead, in this type of geometry, there are no parallel lines and straight lines are represented by great circles on a sphere. This is because great circles are the shortest distance between two points on a sphere. The concept of straight lines in this geometry is different from those on a flat plane, and can be understood as spatial geodesics. This can be seen by considering the shortest paths between two points on a globe, which will always form a portion of a great circle. Therefore, in elliptical geometry, two lines must be great circles in order to be considered parallel.
  • #1
bjgawp
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I was just doing some reading about elliptical geometry and came across a problem. I've read that this type of geometry is pretty much based off the 'rejection' of Euclid's fifth postulate where instead of having one parallel line, you don't have any and this can be pictured by two 'great circles' intersecting on a sphere.

My question is why do these two lines have to be 'great circles'. I mean, it's possible for two lines to be drawn on the sphere without touching each other at all - i.e. making it parallel. So, I don't really understand the explanation in which my book / other internet sources are saying ...

Thanks in advance.
 
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  • #2
"Straight lines" are the shortest paths between two points. On a plane, they are literally straight lines. On a sphere, you can show with a bit of calculus which escapes my memory that the shortest path between two points on a sphere is a section of a great circle. Hence, if you're considering the 5th postulate, you're still considering 'straight lines', but just not in flat space (or at least not with a flat metric).

A slightly more physical way of thinking about it is to consider yourself on the Earth's surface. If you walk in what you think is a straight line, what path do you trace out? A great circle. If you walked on what would be one of those smaller circles you mentioned, you could know immediately you weren't walking in a straight line, because you're always be turning in a particular direction.

Since parallel lines must be 'straight', you want to work only with straight lines. Technically, you're actually working with a notion of spatial geodesics since they extremise distances.
 
  • #3
Hmm .. I still don't think I understand. If I had two points and connected them in one of those 'smaller circles', wouldn't that be a straight line - even though it doesn't form a great circle ...

Thanks for replying :)
 
  • #4
No, it wouldn't be. A "great circle" is the shortest distance between two points on a sphere. That's the crucial point that makes a great circle the choice for a "straight line".
 
  • #5
http://img294.imageshack.us/img294/6821/circlexs7.png [Broken]

What about line a? We could have two points along that line that will yield the shortest distance without producing a great circle per se, right?
 
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  • #6
bjgawp said:
http://img294.imageshack.us/img294/6821/circlexs7.png [Broken]

What about line a? We could have two points along that line that will yield the shortest distance without producing a great circle per se, right?

NO.

if you choose any two points on the latitude circle a, walking to one from the other along a will be a longer path than if you got there via a great circle.

A great way to see this is: get yourself a globe and some string. pick two points on the same line of latitude (not the equator) and stretch the string tightly against the globe so that it connects the two points. you will find that the string -- which automatically will stretch to the shortest path between them -- will not follow the line of latitude but will form a portion of a great circle.

this is the reason why cross-country airflights look so strange on a map. they are following great circles across the Earth (in order to minimize fuel consumption and time).
 
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1. What is spherical/Riemann geometry?

Spherical/Riemann geometry is a branch of mathematics that deals with the properties and measurements of curved surfaces. It is named after mathematician Bernhard Riemann and is based on the concept of a curved space or manifold, as opposed to the flat space of Euclidean geometry.

2. What is the difference between Euclidean and spherical/Riemann geometry?

The main difference between Euclidean and spherical/Riemann geometry is that Euclidean geometry deals with flat surfaces, while spherical/Riemann geometry deals with curved surfaces. In Euclidean geometry, the sum of the angles in a triangle is always 180 degrees, while in spherical/Riemann geometry, the sum can be greater than 180 degrees due to the curvature of the surface.

3. How is spherical/Riemann geometry used in real life?

Spherical/Riemann geometry has many practical applications in fields such as physics, astronomy, and cartography. It is used to model the curvature of the Earth and to calculate distances and angles on the Earth's surface. It is also used in Einstein's theory of general relativity to describe the curvature of spacetime.

4. What are some key theorems in spherical/Riemann geometry?

Some of the key theorems in spherical/Riemann geometry include the Gauss-Bonnet theorem, which relates the curvature of a surface to its topology, and the Poincaré-Hopf theorem, which states that the sum of the indices of singular points on a surface is equal to its Euler characteristic.

5. How does spherical/Riemann geometry relate to non-Euclidean geometries?

Spherical/Riemann geometry is one of the three main non-Euclidean geometries, along with hyperbolic geometry and elliptic geometry. These geometries differ from Euclidean geometry in terms of the parallel postulate, which states that given a line and a point not on the line, there is exactly one line through the point that does not intersect the given line. In spherical/Riemann geometry, there are no parallel lines, while in hyperbolic geometry, there are infinitely many parallel lines.

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