4.6 Investigation into the Uniqueness of a Projectivity between Pencils of Points
 Teach to the problems, not to the text.
 —Kim Nebeuts, Return to Mathematical Circles (1988)

Theorem 4.10. If A, B, C and A', B', C' are distinct elements in pencils of points with distinct axes p and p', respectively, then there exists a projectivity such that ABC is projectively related to A'B'C'.

The theorem and its constructive proof, gave a procedure to determine a corresponding point D' on axis p' by following the perspectivities when a fourth point D on axis p was given. That is, let D be an element of axis p, then let D1 = DP · A'C and D' = D1 Q · p'. Is the point D' unique? Or, does the point D' depend on the choice of the point P?

Construct Projectivity Between Two Pencils of Points

Select the steps for the construction.
Drag the points A, B, and C on axes p.
Drag the points A', B', and C' on axes p'.
Select the steps for constructing D and D'.
Drag point D.

Timothy Peil, 6 February 2013, Created with GeoGebra

4.6 Definition of Perspectivity and ProjectivityBack to Fundamental Theorem of Projective Geometry 
Ch. 4 Projective TOC  Table of Contents
  Timothy Peil  Mathematics Dept.  MSU Moorhead
© Copyright 2013 - Timothy Peil