3.3 ½ Spherical Geometry
Neutral Geometry (axioms below) is a structure that works for all of the “Big Three”
geometries:
Euclidean, Spherical, and Hyperbolic.
You just make certain choices
about the distance boundary and the situation about parallel lines and you choose one of
the 3.
Absolute Geometry is what we’re building in our textbook – it’s is only about
Euclidean and Hyperbolic Geometries.
They’re really not the same thing at all though
some website authors seem to use the terms interchangeably.
So, get a 6” diameter ball, a handful of rubber bands that fit snugly on the ball, and some
sticky dots. Also have on hand a little protractor – one of those hard plastic semi-circle
ones – clear plastic if you can get it.
Let’s get started.
The Axiomatic Structure for Neutral Geometry
Undefined terms:
point, line, plane, space
Axioms:
A1
To each pair of points (A, B) is associated a unique real number, denoted AB,
with least upper bound,
α.
A2
For all points A and B, AB
≥
0, with equality only when A = B.
A3
For all points AB = BA.
A4
Given any four distinct collinear points A, B, C, and D such that
A – B – C, then either D – A – B, A – D – B, B – D – C, or B – C – D.
A5
Each two points A and B lie on a line, and if AB <
α
, that line is unique.
A6
Each three noncollinear points determine a plane.
A7
If points A and B lie in a plane and AB <
α
, then the line determined by A and B
lies in that plane.
A8
If two planes meet, their intersection is a line.
A9
Space consists of at least four noncoplanar points, and contains three noncollinear
points.
Each plane is a set of points of which at least three are noncollinear, and
each line is a set of at least two distinct points.
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