Section 9.4 Perspectivity
¶One of the applications of projective geometry is to capture three-dimensional images in two dimensions, whether as photographs, sketches, or paintings. Imagine a photograph of railroad tracks running off into the distance. Those lines are parallel – but they meet in the distance. Projective geometry incorporates this concept of objects being “in perspective”, and makes it precise.
Two triangles are perspective with respect to a point \(P\) if the lines connecting corresponding vertices meet at a single point, namely \(P\text{.}\) Perspectivity with respect to a point is illustrated in Figure 9.4.1.
Analogously, two triangles are perspective with respect to a line \(\ell\) if the points of intersection of corresponding sides lie on a single line, namely \(\ell\text{.}\) Perspectivity with respect to a line is illustrated in Figure 9.4.2.
These two forms of perspectivity are dual to each other, in the sense that one is obtained from the other by interchanging points and lines. Should you have difficulty visualizing statements about perspectivity with respect to a line, you may want to imagine the corresponding, dual statement about perspectivity with respect to a point, which is often easier to do.
Both forms of perspectivity hold for Euclidean geometry, since Euclidean triangles are also projective triangles. However, there are some special cases to consider, namely those involving ideal points and ideal lines.