Mathematics 138
Identifying models of Cayley-Klein Geometries
Based on Yaglom, A Simple Non-Euclidean Geometry and its Physical Basis

	Elliptic length measure	Parabolic length measure	Hyperbolic length measure	
Elliptic
angle
measure

	Name: elliptic
Parallels – none
Can’t join – none
Betweenness - none	Name: Euclidean
Parallels – one
Can’t join – none
Betweenness - points	Name: hyperbolic, Lobachevsky
Parallels – two(many)
Can’t join – none
Betweenness - points	
Parabolic
angle
measure

	Name: co-Euclidean
Parallels – none
Can’t join – one
Betweenness - lines	Name: Galilean
Parallels- one
Can’t join – one
Betweenness - both	Name: co-Minkowskian
Parallels – two(many)
Can’t join – one
Betweenness - both	
Hyperbolic
angle
measure

	Name: cohyperbolic
Parallels – none
Can’t join – many
Betweenness - lines	Name: Minkowskian
Parallels – one
Can’t join – many
Betweenness - both	Name: doubly hyperbolic
Parallels – two(many)
Can’t join - many Betweenness - both