### UHG37: Perpendicularity, polarity and duality on a sphere

This video discusses perpendicularity on a sphere, associating two poles to every great circle, and one polar line (great circle) to every point. This association is cleaner in elliptic geometry, where there is then a 1-1 correspondence between elliptic points (pairs of antipodal points on a sphere) and elliptic lines (great circles).We introduce the polar triangle of a triangle, and explain the supplementary relation between angles and sides in a triangle and sides and angles in the polar triangle.Then we extend the duality of Apollonius from the case of a circle to the case of a sphere; this associates to a point in three dimensional space (actually projective space) a plane. The dual of a line is another line.

Length:
31:54