Which geometric properties are preserved by projective transformations?
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, projective transformations are mappings that preserve specific fundamental relationships between geometric figures while disregarding metric properties like distance and angle magnitude.12
The following properties are preserved (invariant) under projective transformations:
- Incidence: Points are mapped to points, lines to lines, and planes to planes. If two elements are incident (e.g., a point lies on a line), their images remain incident.23
- Collinearity: If three or more points lie on a common line, their projections will also be collinear.2
- Concurrency: If three or more lines meet at a common point, their projections will also meet at a common point.2
- Cross Ratio: For four distinct collinear points, the cross ratio is invariant under projection. This is a fundamental property that allows mathematicians to characterize projections without relying on Euclidean measurements of length.12
Because projective geometry focuses on these generic properties, it provides a unified framework for understanding figures such as conic sections, where circles, ellipses, parabolas, and hyperbolas are treated as projective images of one another.2