Curvature (InstanceTopic, 3)

From Hi.gher. Space

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Revision as of 23:56, 16 June 2007

The curvature of a space or hypersurface defines several properties about it.

Flat objects

An object is said to be flat if the shortest route inside the object between any two given points A and B is equal to the shortest route in the object's bounding space between the same two points. Flat space observes Euclidean geometry.

Curved objects

An object is said to be curved if the shortest route inside the object between any two given points A and B is greater than the shortest route in the object's bounding space between the same two points.

Hyperbolic objects

An object is said to be hyperbolic if the shortest route inside the object between any two given points A and B is less than the shortest route in the object's bounding space between the same two points. Hyperbolic space observes strange effects.

Pages in this category (2)

Curved

Flat

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-A surface is said to be '''curved''' if the shortest distance between any two points ''A'' and ''B'' on that surface through the n-dimensional [[bounding space]] of that surface is greater than the straight line distance ''S'' between them:
+The '''curvature''' of a [[space]] or [[hypersurface]] defines several properties about it.
-''S'' = √((''A''<sub>1</sub>-''B''<sub>1</sub>)<sup>2</sup> + (''A''<sub>2</sub>-''B''<sub>2</sub>)<sup>2</sup> + (''A''<sub>3</sub>-''B''<sub>3</sub>)<sup>2</sup> + ... + (''A<sub>n</sub>''-''B<sub>n</sub>'')<sup>2</sup>)
+== Flat objects ==
+An object is said to be ''flat'' if the shortest route inside the object between any two given points ''A'' and ''B'' is equal to the shortest route in the object's [[bounding space]] between the same two points. Flat space observes Euclidean geometry.
-If the shortest distance between every pair of points is equal to the straight line distance between them, the surface is said to be ''[[flat]]''.
+== Curved objects ==
+An object is said to be ''curved'' if the shortest route inside the object between any two given points ''A'' and ''B'' is greater than the shortest route in the object's [[bounding space]] between the same two points.
+== Hyperbolic objects ==
+An object is said to be ''hyperbolic'' if the shortest route inside the object between any two given points ''A'' and ''B'' is less than the shortest route in the object's [[bounding space]] between the same two points. [[Hyperbolic space]] observes strange effects.
 [[Category:Geometric properties]]