Here are the first four axioms of Euclidean Geometry Any two points can be joined by a straight line. Any straight line segment can be extended indefinitely in a straight line. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre. All right angles are congruent.
In Non-Euclidean Geometry we use the first four axioms of Euclidean geometry and then replace the fifth postulate (axiom) with an alternative version.

One version of the fifth postulate is that given a line and a point outside it, there is exactly one line through the point parallel with the given line. We therefore get two alternative versions:

• No possible parallel lines
• Spherical Geometry
• Angles in triangle add up to more than 180 degrees
• Many possible parallel lines
• Hyperbolic geometry
• Angles in triangle add up to less than 180 degrees
Related: Hyperbolic Function
Axiom (none) (none)
PoincaresDisc (none)

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NonEuclideanGeometry
Axiom
EuclideanGeometry
HyperbolicFunction
(none) Euclid Function
KurtGoedel
TrigonometricFunction
UnitCircle