Essential Ideas for Section 6.7

This section discusses projective geometry and its relationship to three-dimensional representation in art. Next, non-Euclidean geometries are investigated with the idea in mind that Euclidean geometry is not the only possible geometry. The principle non-Euclidean geometries are hyperbolic and elliptic geometries.



A representative line in each geometry is shown in color for each models, and the shaded portion showing a Saccheri quadrilateral is shown directly below the respective models.
Geometry is on a plane:	Geomotery is on a pseudosphere:	Geometry is on a sphere:
The sum of the angles of a triangle is 180 degrees. Lines are infinitely long.	The sum of the angles of a triangle is less than 180 degrees. Lines are infinitely long.	The sum of the angles of a triangle is more than 180 degrees. Lines are finist in length.

The essential idea in classifying the correct geometry is the Lobachevskian postulate:
The summit angles of a Saccheri quadrilateral are acute.