Projective Geometry 14 Pappus's Hexagon Via Circle Projections

After giving formal definitions of perspectives and projections of conics, we focus upon a specific example on the circle (although these discussions hold equally for other conic sections). In particular, we suppose there are three pairs of corresponding points on a circle. In particular, one may form a projection by using any particular pair of points as centers, for two subsequent perspectives, and for a properly chosen intermediate line, this results in a projection of the circle to itself which puts the other two pairs of points into correspondence. More importantly, the `properly chosen intermediate line' turns out to be precisely Pascal's line, from his hexagon theorem. By elongating the circle, until it is a long ellipse, the configuration appears similar to that of Pappus's Hexagon theorem, and the intermediate line corresponds with Pappus's line. After noting these interesting connections, we examine the projection in question more practically, by using geogebra dynamics to visualize it as a mapping from the circle to itself.

Видео Projective Geometry 14 Pappus's Hexagon Via Circle Projections канала Richard Southwell