How does a projective transformation map points lines and planes?
The sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.
In projective geometry, a projective transformation is a mapping of space that systematically preserves the fundamental structural relationships between geometric elements. Specifically, it functions as follows:1
- Points: Points are mapped onto points.1
- Lines: Lines are mapped onto lines. This ensures that if three points are collinear (share a common line), their projections will also be collinear.12
- Planes: Planes are mapped onto planes.1
- Incidence: Any two incident elements (such as a point lying on a line) are mapped into two incident elements. Similarly, if three lines meet at a common point, their projections will also meet at a common point.12
While projective transformations do not preserve distances or angles, they maintain these essential properties of collinearity and incidence, allowing for the study of geometric figures through their projected images.2