3.3 and one half - Spherical Geometry

3.3 and one half - Spherical Geometry - 3.3 Spherical...

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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. 1
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A10 (Ruler Postulate): Given line L and two points P and Q on L, the points of L can be placed into one – to – one correspondence with the real numbers x such that - α < x α (called coordinates) in such a manner that 1. points P and Q have coordinates 0 and k > 0, respectively 2. if A and B on the line have coordinates a and B, then AB is 2 a b if a b a b if a b α - - - - - A11 (Plane Separation Postulate): Let L be any line lying in any plane P. The set of all points in P not on L consists of the union of two subsets H1 and H2 of P such that 1. H1 and H2 are convex sets 2. H1 and H2 have no points in common 3. if A lies in H1 and B lies in H2 such that AB < α , line L intersects segment AB . A12 Each angle ABC is associated with a unique real number between 0 and 180 denoted m ABC. No angle can have measure 0 or 180. A13 (Angle Addition Postulate): If D lies in the interior of ABC, then m ABD + m DBC = m ABC. A14
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3.3 and one half - Spherical Geometry - 3.3 Spherical...

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