Question

Using just Axioms P1–P4, prove that every projectivity can be written as the composition of at most three perspectivities. [Hint: there are two cases —either the projectivity maps a line to a different line or it maps a line to itself. Use the construction from the proof of Theorem 9.6.]

Answer 1

To prove that every projectivity can be written as the composition of at most three perspectivities, we consider two cases: when the projectivity maps a line to a different line and when it maps a line to itself. Construction from the proof of Theorem 9.6 is used. In the first case, three perspectivities are used and in the second case, two perspectivities are used.

Step-by-step explanation:

To prove that every projectivity can be written as the composition of at most three perspectivities using Axioms P1-P4, we consider two cases. In the first case, if the projectivity maps a line to a different line, we can use the construction from the proof of Theorem 9.6. We can use three perspectivities to map the given line to a line through the vanishing point, then to a line parallel to the vanishing line, and finally to the desired line. In the second case, if the projectivity maps a line to itself, we can use two perspectivities to map the line to a line through the vanishing point and then back to the original line.