Parametrizing and Projecting a Sphere - Universal Hyperbolic Geometry

Explore stereographic and gnomonic projections of a sphere in this 39-minute video lecture on Universal Hyperbolic Geometry. Begin with a review of three-dimensional coordinate systems before delving into rational parametrization of a sphere, analogous to circle parametrization. Examine stereographic projection from the south pole through the equatorial plane and gnomonic projection from the sphere's center through a tangent plane. Discover how gnomonic projection aligns naturally with elliptic geometry, where antipodal points on a sphere are identified. Cover topics including spherical coordinates, algebraic foundations, and the relationship between these projections and elliptic geometry.

Introduction
Stereographic projection
Recall parametrization of a circle
Algebraic underpinnings
Parametrization formula for a sphere
Spherical co-ordinates
Gnomonic projection
Gnomonic projection works more naturally with elliptic geometry