What is a line in hyperbolic geometry?

[mmmboh]

I'm reading a book on an introduction to non-Euclidean geometry, and it starts off with the usual Euclidean geometry. I didn't really need a line to be defined in that case, since it's obvious, but now that the parallel postulate has been replaced and we are working with non-Euclidean geometry, I'm not sure how to connect two points. The author hasn't mentioned a straight line being a geodesic, which is what I assume it would be, he is just proving stuff based on the first four of Euclid's postulates, and the new fifth one. Can you deduce what a line is based on that alone?

 

[phinds]

a line is the shortest distance between two points. Trite but true. Of course the SHAPE of the line depends on the geometry of the spacetime it is in

 

[AdrianZ]

I guess It depends on the Model that you're describing your Hyperbolic geometry in it. a saddle-shaped surface might be a good example to start with but there are many other models that can be used as well. like Poincare's models.

 

[mmmboh]

Ok, thanks. I was just confused because I didn't see him write anywhere that we consider a line to be the shortest distance between two points, which is what I assumed it would be. Can we replace the parallel postulate in Euclidean geometry with the postulate that the figures take place on flat planes, and get the same familiar Euclidean geometry? Or does that not make sense for some reason?...I don't see why you can't prove that parallel lines are at a constant distance from each other that way, or am I misunderstanding something?

 

[AdrianZ]

You can prove that Parallel lines are at a constant distance from each other but then You'll have to add a theorem in Euclidean geometry or a new postulate to your Axioms to prove it. Yes, You can think of Euclidean geometry as the geometry that works on uncurved surfaces. You can think of Euclidean geometry in different ways. one of them is to say that Euclidean geometry is a group of isometric transformations like translation,rotation and reflection. the other way is to say that Euclidean geometry is the geometry of uncurved spaces. I'm sure people on this forum can help you better than me on this topic.
the concept of a line will be replaced with the concept of a geodesic in a curved space. just to give you a sense of the terminology that you might see in the future.

 

[mmmboh]

But on a flat surface, can you prove that parallel lines are at a constant distance from each other, given that the surface is flat, and the first four postulates? If not, wouldn't that mean you can construct parallel lines on a flat surface that aren't at a constant distance from each other? Which doesn't seem possible... I've looked at the proofs in the book and understand why the parallel postulate (or equivalent) is necessary in the proof for this, but it seems to me like saying the geometry is on a flat surface would be equivalent to this.

 

[AdrianZ]

First you'll need to define what you mean by saying that the surface is flat. You're talking about curvature, but first you need to define what you mean by curvature. and then after you defined this, you could talk about a theorem that states in a flat space, the distance between parallel lines stays the same everywhere. That is also logically equivalent to saying that the sum of the angles of an arbitrary triangle is exactly equal to 180 degrees.

 

[mmmboh]

Ah. So to get hyperbolic geometry, could you just use the first four of Euclid's postulates, and for the fifth just define a different metric? (whatever a geodesic on an hyperbole is) and that would allow you to draw the shapes on a flat surface. Or for Euclidean geometry, replace the fifth postulate with the straight line metric? (basically the pythagorean theorem)

 

[AdrianZ]

Yes. You can do that. in fact you can obtain many properties of the space knowing only its metric function. you can define a flat space using its metric. a space is called to be flat if you can reverse its metric to be the Euclidean metric and vice versa. a Sphere is not flat because you can't do that. (Try this: the metric on a spherical surface of radius 1 is (sin^2(x'))(dx)^2 + (dx')^2. show that you'll fail at some point if you want to convert it into the Euclidean metric. also this metric leads you to a hyperbolic geometry: (ds)^2 = cosh^2(y/k) (dx)^2 + (dy)^2 where K is a constant that should be found experimentally).