Projectivity Between Two Pencils of Points p and p''

4.6 Dynamic Illustrations for Projectivities
Metrical geometry is thus a part of [projective] geometry, and [projective] geometry is all geometry.
—Arthur Cayley (1821–1895)

A one-to-one mapping between two pencils of points is called a projectivity if the mapping is a composition of finitely many elementary correspondences or perspectivities.

When a projectivity exists between two pencils, the pencils are said to be projectively related. Also, note that elementary correspondences and perspectivities themselves are projectivities.

Projectivity Between Two Pencils of Points with Axes p and p''

Drag the points A, B, and C on the axis p.
Drag the points forming the axes p, p' and p''.
Drag the center of perspectivity O and P.

Projectivity Between Two Pencils of Lines with Centers P and P'

Drag unlabeled points on axis o.
Drag centers P, P', and P''.
Drag points defining axes o and o'.

Projectivity Between a Pencil of Lines and a Pencil of Points

Drag unlabeled points on axis o.
Drag centers P and Q.
Drag points defining axes o and o'.

To better see the projectivities in each figure, we describe a path to follow beginning with one of the points/lines and following the "path of its projection." Click to display the following in the Definitions frame.

In Figure 1, from A follow the path with center O to axis p', from axis p' follow the corresponding path with center P to A".
In Figure 2, follow the path a from center P to axis o, from axis o follow the corresponding path from center P' to axis o' which leads to the corresponding path a" with center P".
In Figure 3, follow the path a from center P to axis o , then from axis o follow the corresponding path through center Q to A" on axis o'.

Timothy Peil, 4 February 2013, Created with GeoGebra