Скачать или смотреть Perpendicularity, polarity and duality on a sphere | Universal Hyperbolic Geometry 37

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.

Video Content:
00:00 Introduction great circles
7:38 Polar of a point
10:33 Poles/polars more symmetric in elliptic geometry
13:30 Polar triangle
17:29 The relationship between polar triangles
20:54 Duality
22:14 Point/plane duality induced by a sphere

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