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May 30

Are there any other non-Euclidean geometries?

If you take Euclidean geometry and delete the parallel postulate, are the hyperbolic geometry and the elliptic geometry the only two possible geometries? Is it not possible to have, say, a finite number but more than one straight lines intersecting a given point and straight line? Or, is it possible to have a point through which there exists infinite straight lines intersecting another given straight line, and another point in the same space where there are none? Widener (talk) 16:59, 30 May 2012 (UTC)[reply]

In the latter case you lose a defining property of the big three, that one neighborhood is like another (I don't know the name of this property). There are useful geometries without that property, such as the geometry of the physical universe, with varying spacetime curvature. —Tamfang (talk) 18:49, 30 May 2012 (UTC)[reply]

Oh, Tamfang, the word you're probably looking for is homogeneity. --Trovatore (talk) 19:36, 30 May 2012 (UTC) [reply]

Sure, let's go with that. —Tamfang (talk) 20:41, 30 May 2012 (UTC)[reply]

I think the issue you're running into here is the contrast between classic axiomatic geometry (such as Euclidean, Lobachevskyan, or the original version of Riemannian geometry), where all geometric notions are axiomatized in and of themselves, and the more modern sort of geometry, where geometric notions are defined in terms of notions from a larger theory such as real analysis or set theory. These larger theories have the flexibility to accommodate varying curvature and so on and still prove interesting stuff, whereas if you took one of the classic axiomatic geometries and simply dropped an axiom so as to allow for varying curvature, you probably couldn't prove very much at all.

Anyway, Widener, for the most general framework that's still a reasonably straightforward generalization of the stuff you're talking about, see semi-Riemannian geometry (where the "semi" part allows for stuff like the time–space distinction in relativistic spacetime). There are certainly even more general geometric frameworks but they get less obviously connected to the classic theories. --Trovatore (talk) 19:04, 30 May 2012 (UTC)[reply]

I'd argue that a more natural "most general framework" is the notion of a manifold with projective connection. Roughly, this is any space in which the notion of "straight line" makes sense, and looks like the familiar projective geometry on small scales. Sławomir Biały (talk) 13:01, 31 May 2012 (UTC)[reply]

The question is discussed rather thoroughly at Non-Euclidean geometry#Axiomatic basis of non-Euclidean geometry. Looie496 (talk) 19:22, 30 May 2012 (UTC)[reply]

Honestly, no, that doesn't strike me as a thorough discussion at all. Unless I missed it, it doesn't discuss the case Widener asked about, where the number of parallels may vary from place to place. Still, a useful link. --Trovatore (talk) 19:35, 30 May 2012 (UTC)[reply]

There are plenty of other geometries in mathematics, which differ more and more from Euclidean geometry. There are projective geometries, which includes a set of points at "infinity" - so in the projective plane, for example, any pair of lines intersects in just one point. There are finite geometries, with a finite number of points and lines. There is the taxicab geometry, in which a circle has four corners etc. etc. Gandalf61 (talk) 09:54, 31 May 2012 (UTC)[reply]

I think Sławomir has the best answer in the spirit of Widener's last case. I was going to drop in to mention arbitrary manifolds, but stopped because I didn't know the right words /definitions necessary for projective connection :) SemanticMantis (talk) 18:16, 31 May 2012 (UTC)[reply]

Q. What is the technical term for the innermost edge of a curved plane?

Q. What is the technical term for the innermost edge of a curved plane?

For a real-life example, imagine a race car taking the shortest path possible through each curve in a gran prix course.

I have searched via Wikipedia and Google and failed to find an answer to what I thought would be a rather simple question.

Thank you in advance for satisfying my curiosity!

19:37, 30 May 2012 (UTC) — Preceding unsigned comment added by Pstreeter (talk • contribs)

How about "inside edge", or "inside lane", in the car case ? StuRat (talk) 19:48, 30 May 2012 (UTC)[reply]

For motorsports the term is Racing line. A mathematician would think of this a curve which minimises distance.--Salix (talk): 20:20, 30 May 2012 (UTC)[reply]

The racing line minimizes only the time needed to complete the course however and not necessarily the distance traveled. Whether or not these two things coincide will depend on the shape of the race course. For an oval course, the distance traveled will be further than the most minimum distance which is the path of the innermost lane or the track's concave edge. --Modocc (talk) 04:08, 31 May 2012 (UTC)[reply]

Euclidean geometric planes are always flat with zero curvature everywhere. Hyperbolic planes are not flat, but these are, of course, not typical planes! However, the edges and surfaces of figures can be either convex or concave. With an oval race track, its innermost edge is concave and the outer edge is convex. --Modocc (talk) 04:08, 31 May 2012 (UTC)[reply]

The technical term for the shortest path between two points, given a set of constraints, is a geodesic. Looie496 (talk) 23:18, 31 May 2012 (UTC)[reply]

Not sure if this is pedantic, but a geodesic is only locally a shortest path. To illustrate, if we take the space of

\mathbb {SO} (3)

, and take the start and finish points to be the identity transform, a motion of constant angular velocity that consists of a rotation by

2\pi

(about some axis through the origin) from start to finish still maps out a geodesic path, though it is not the shortest path.--Leon (talk) 21:35, 1 June 2012 (UTC)[reply]