Perspectivity Between Two Pencils of Points

4.2 Investigate Perspectivity and Projectivity
If I have seen further than others, it is because I have stood on the shoulders of giants.
—Isaac Newton (1642–1727)

A set of points incident with a line is called a pencil of points (or range of points) and the line is called the axis.
Example. A set of points {A, B, C, D} on a line l is called a pencil of points with axis l.

A one-to-one mapping between two pencils of points is called a perspectivity if the lines incident with the corresponding points of the two pencils are concurrent. The point where the lines intersect is called the center of the perspectivity.
A one-to-one mapping between two pencils of points is called a projectivity if the mapping is a composition of finitely many perspectivities.

The diagram is a perspectivity between two pencils of points with axes p and p' and center O; ABC is perspectively related to A'B'C' through O.

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

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

The diagram is a projectivity between two pencils of points with axes p and p'' consisting of the product of two perspectivities with centers P and Q; ABC is projectively related to A''B''C'', ABC is perspectively related to A'B'C' through P and A'B'C' is perspectively related to A''B''C'' through Q.

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 P and Q.

Timothy Peil, 4 February 2013, Created with GeoGebra