How does projective geometry differ from Euclidean geometry?
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 differs from Euclidean geometry primarily in its treatment of measurements and the nature of space. While Euclidean geometry relies on concepts such as distance, angle, congruence, and similarity, projective geometry ignores these metrics to focus on properties that remain invariant under projection, such as collinearity and the cross ratio.123
Key distinctions include:
- Treatment of Parallel Lines: In Euclidean geometry, parallel lines never intersect. In projective geometry, parallel lines are considered to meet at a "point at infinity," a concept introduced to simplify geometric theorems and unify configurations.13
- Invariants: Projective geometry studies properties that are preserved when a figure is projected onto another surface, such as lines mapping to lines. Conversely, Euclidean properties like length and angle are typically distorted during projection and are not considered projective invariants.12
- Scope: By postulating points at infinity, projective geometry creates a framework where theoremsâsuch as Desarguesâs theoremâcan be stated more simply without needing to account for special cases involving parallel lines.3