Section 9.5 Desargues' Theorem
¶An important result relating the two notions of perspectivity introduced in Section 9.4 is that they are actually equivalent. This result is known as Desargues' Theorem.
Theorem 9.5.1.
Two projective triangles are perspective with respect to a point if and only if they are perspective with respect to a line.
Desargues' Theorem shouldn't come as a surprise, given our understanding of the duality between points and lines in projective geometry. A proof of Desargues' Theorem is beyond the scope of this book. However, we point out the remarkable fact that this proof requires three dimensions, even though the result could be stated in two dimensions.
In lieu of a proof, Figure 9.5.2 shows two triangles constructed so as to be perspective with respect to a point, showing that they are also perspective with respect to a line.
A good question about a geometry is whether it is Desarguean, that is, whether Desargues' Theorem holds. We already know that single elliptic geometry must be Desarguean, since the Klein disk can be used as a model of projective geometry. This result is illustrated in Figure 9.5.3. Similarly, hyperbolic geometry is Desarguean (when suitably interpreted), as illustrated in Figure 9.5.4.