By analogy, an infinite plane has zero curvature but infinite area, whereas an infinite cylinder is finite in one direction and a torus is finite in both.
12.
There is a margin of error in the results of any experiment, so all we can say is that zero curvature is within the range the experiments give.
13.
That is, a regular curve with nonzero torsion must have nonzero curvature . ( This is just the contrapositive of the fact that zero curvature implies zero torsion .)
14.
A transition curve can connect a track segment of constant non-zero curvature to another segment with constant curvature that is zero or non-zero of either sign.
15.
For the Universe to curve back on itself and maintain a zero curvature it would have to take on a toroidal shape ( I'm not sure off the top of my head if there are other shapes that would suffice ).
16.
In the cases of negative and zero curvature, the M�bius band can be constructed as a ( geodesically ) complete surface, which means that all geodesics ( " straight lines " on the surface ) may be extended indefinitely in either direction.
17.
Although, I suppose a circle of infinite radius will cancel out this graininess, but then is will also have zero curvature, not to mention, not fit inside the universe . talk ) 11 : 57, 3 May 2015 ( UTC)
18.
:It embeds into 4-dimensional space ( a two-dimensional subset of the three-dimensional sphere in "'R "'4 ) and has zero curvature everywhere . David Eppstein 00 : 46, 30 October 2006 ( UTC)
19.
If a straight line is considered a degenerate circle with zero curvature ( and thus infinite radius ), Descartes'theorem also applies to a line and two circles that are all three mutually tangent, giving the radius of a third circle tangent to the other two circles and the line.
20.
But that would imply that part of the torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction . ) On the other hand, according to the Nash-Kuiper theorem, proven in the 1950s, an isometric C 1 embedding exists.
