Wouldn't it be 0? Intuitively a straight line has no curvature, so I'd guess 0. Plus you'd need an infinitely large circle for a straight line to be tangent to it, so the radius r would be infinite, which gives us 0 curvature.

BryceSummers

@ShiaLaBeouf That sounds good to me! 1 / infinity = 0!

paluri

This begs the question, is there really such a thing as a straight line? Or are there just curves that have radius approaching infinity? :-|

BryceSummers

If a tree straightens itself in a forest, but no-one observes it, then did it ever have non positive curvature?

kmcrane

@paluri - I realize you may have been joking in your comment, but this idea that straight lines are not really distinguished from circles of finite radius is precisely the perspective adopted in a subject called Möbius geometry. In fact, there is a very nice animation illustrating how transformations of lines and circles in the plane can be thought of as just rotations and translations of a sphere sitting "above" the plane.

What would the curvature be for a straight line?

Wouldn't it be 0? Intuitively a straight line has no curvature, so I'd guess 0. Plus you'd need an infinitely large circle for a straight line to be tangent to it, so the radius r would be infinite, which gives us 0 curvature.

@ShiaLaBeouf That sounds good to me! 1 / infinity = 0!

This begs the question, is there really such a thing as a straight line? Or are there just curves that have radius approaching infinity? :-|

If a tree straightens itself in a forest, but no-one observes it, then did it ever have non positive curvature?

@paluri - I realize you may have been joking in your comment, but this idea that straight lines are not really distinguished from circles of finite radius is

preciselythe perspective adopted in a subject calledMöbius geometry. In fact, there is a very nice animation illustrating how transformations of lines and circles in the plane can be thought of as just rotations and translations of a sphere sitting "above" the plane.