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Topic: Valid Definitions in Spherical Geometry
Replies: 5   Last Post: Jun 2, 2000 1:34 AM

 Messages: [ Previous | Next ]
 JIMMY Posts: 7 Registered: 12/6/04
Valid Definitions in Spherical Geometry
Posted: Jun 1, 2000 4:13 PM

How logical do you find the following definitions in spherical
geometry??

Similar polygons = all corresponding angles are congruent and lengths
of corresponding sides are proportional

Parallelogram = quadrilateral whose opposite sides are parallel

Rectangle = parallelogram whose angles are right angles

These definitions are really illogical in spherical geometry. Here are
more logical definitions in spherical geometry:

Similar polygons = all measures of corresponding angles are
proportional and all lengths of corresponding sides are proportional.
In other words, a spherical triangle whose angles measure 90 90 90 is
similar to one whose angles measure 120 120 120 but not to one whose
angles measure 90 90 120. This is more logical because if we defined
similar polygons the same way they are defined in Euclidean geometry,
"similar" and "congruent" would mean the same thing in spherical
geometry.

Parallelogram = quadrilateral whose opposite angles are congruent. In
other words, a spherical quadrilateral whose angles are 90 120 90 120
is a parallelogram but not one whose angles are 90 120 90 90. This is
more logical because if we defined a parallelogram in spherical
geometry as a quadrilateral whose opposite sides are parallel, then
there would be no parallelograms in spherical geometry.

Rectangle = parallelogram whose consecutive angles are congruent. In
other words, a spherical parallelogram whose angles are 120 120 120
120 is a rectangle but not one whose angles are 90 120 90 120. This is
more logical because if we defined a rectangle in spherical geometry
as a parallelogram whose angles are right angles but defined a
parallelogram as stated above, there would be no rectangles in
spherical geometry.

Any other terms that need to be defined differently in spherical
geometry??

Date Subject Author
6/1/00 JIMMY
6/1/00 Guy Brandenburg
6/1/00 John Conway
6/1/00 Guy Brandenburg
6/1/00 JIMMY
6/2/00 Walter Whiteley