Explore perpendicularity, polarity, and duality on a sphere in this 32-minute video lecture on Universal Hyperbolic Geometry. Delve into the concepts of great circles, poles, and polar lines in spherical and elliptic geometry. Discover the fascinating relationship between triangles and their polar counterparts, including the supplementary nature of angles and sides. Examine the extension of Apollonius' duality from circles to spheres, and learn about the point-plane duality in three-dimensional projective space. Gain valuable insights into these fundamental geometric concepts through clear explanations and visual demonstrations.