sphericon 2

$e-MATH$
The Differential Geometry of the Sphericon

2. An insider's guide to surface curvature and geodesics

Curvature: If you are a two-dimensional creature living on a smooth surface you feel its curvature in your guts. When you go from a flat region to one of positive curvature your insides get stretched with respect to your periphery; if you enter a region of negative curvature they get compressed.

For example, suppose you are a disc of radius 1 living on the flat part of the surface, so your circumference is 2 = 6.28.. and your area is = 3.14.

Positive:

Suppose part of the surface is curved like the graph of z= -(x²+y²) (this graph has positive curvature). If you slide over there so that the middle of your body is at (0,0,0), your perimeter will fit exactly on the circle at height -1, but your area will have been stretched to 5.33.. units.

Negative:

On the other hand suppose part of your surface is curved like the graph of z=x²-y² (this graph has negative curvature). If the middle of your body is at the point (0,0,0), your perimeter will fit the circle in the graph that lies over the circle r=.715.. in the plane; your insides will have to fit in the enclosed area which is only 2.26.. units.

Either way it is very uncomfortable.
Check these calculations?

Geodesics: The geographers in your two-dimensional universe need to be able to locate and measure areas of non-zero curvature without risking their insides. They may have discovered a theorem due to Gauss which permits these measurements. This theorem is stated in terms of geodesics. These are the paths on the surface which are as straight as possible: they turn neither to the left nor to the right, and their only bending is that which is forced on them by following the surface.

If the surface is an ordinary plane, the geodesics are ordinary straight lines.
If the plane is bent without stretching into a cylinder or a cone, the lines remain geodesics, even though they may bend with the plane.
On a sphere the geodesics are great circles. If you travel along the equator or along a meridian line, you always move straight ahead. If you travel along a latitude which is not the equator (think of one very near the North Pole) you have to keep turning to stay on the line. Such a latitude is not a geodesic.

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Positive:	Suppose part of the surface is curved like the graph of z= -(x²+y²) (this graph has positive curvature). If you slide over there so that the middle of your body is at (0,0,0), your perimeter will fit exactly on the circle at height -1, but your area will have been stretched to 5.33.. units.
Negative:	On the other hand suppose part of your surface is curved like the graph of z=x²-y² (this graph has negative curvature). If the middle of your body is at the point (0,0,0), your perimeter will fit the circle in the graph that lies over the circle r=.715.. in the plane; your insides will have to fit in the enclosed area which is only 2.26.. units.
Either way it is very uncomfortable. Check these calculations?