How does projective geometry study the properties of shapes under projection?
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.
Projective geometry studies the relationships between geometric figures and their images under projection, such as shadows cast by objects or perspective drawings. By focusing on properties that remain unchangedâor invariantâunder these mappings, the field enables a clearer understanding of generic geometric characteristics while ignoring measurements like distance and angle, which are typically not preserved.12
Key aspects of how this geometry functions include:
- Preservation of Incidence: Projective transformations map points to points, lines to lines, and planes to planes, while maintaining the incidence of these elements. For instance, if three points are collinear, their projections remain collinear.31
- Projective Invariants: Because distances and angles are distorted, the field identifies other invariants. A fundamental example is the cross ratio of four collinear points, which remains constant under projection.1
- Unification of Figures: By introducing points at infinity to represent the intersection of parallel lines, projective geometry unifies disparate shapes. For example, it treats all conic sectionsâcircles, ellipses, parabolas, and hyperbolasâas projective images of a circle, depending on the orientation of the projection plane.12
- Duality: This principle allows for the systematic interchange of "points" and "lines" in geometric proofs, effectively doubling the number of known theorems by showing that properties of points and lines are deeply interconnected.2