Spherical geometry

This case led to the discovery of spherical (or elliptical) geometry. Spherical geometry takes place on a sphere, rather than a plane. The best example is navigation around the Earth, and the concept of a great circle. We will learn about spherical geometry and what stays the same or turns out to be different than Euclidean geometry.

Hyperbolic geometry

The second case led to the discovery of hyperbolic geometry. Hyperbolic geometry is sort of the opposite of spherical geometry. If we think of turning a sphere inside out, or the opposite shape of a sphere, it would look something like a saddle. However, this surface extends infinitely in every direction.

Case 2: Given a line and a point not on the line, there are infinitely MANY lines through the point parallel to the given line