The study of incidence relations and invariants under projective transformations, extending affine geometry by adding points at infinity so that parallel lines meet, centering on cross-ratio, perspectivity, and duality between points and lines while ignoring metrical measures.

Principle

Organize geometry by invariance under the projective linear group: preserve incidence and cross-ratio, allow the compactification of affine space by a hyperplane at infinity, and treat dual statements (points ↔ lines) on equal footing.

Demonstration

In the real projective plane, two distinct lines always intersect in a unique point (possibly at infinity). A collineation maps straight lines to straight lines; the cross-ratio of four collinear points is invariant under projective maps and distinguishes projectively non-equivalent configurations.

Misapplication

Applying Euclidean notions of distance or assuming affine ratios on non-parallel intersecting lines remain meaningful; confusing projective equivalence with congruence leads to false identification of shapes that differ metrically.

Consequence

Correct use yields powerful classification of conics and perspective constructions, unifies Desargues and Pappus type results, and enables coordinate-free reasoning about perspective, harmonic sets and projective invariants across different models.

Reversal

Reversing to affine geometry removes the identification of parallelism via points at infinity and reintroduces distinct behavior of parallel lines; reversing to metric geometries reintroduces distances and angles absent in projective considerations.

Boundary

Applies to projective spaces over fields or division rings and to synthetic incidence structures; excludes metric notions, and many projective statements fail over degenerate rings or without sufficient field properties.

Semantic Tension

Tension exists with affine geometry (treatment of parallelism) and with metric geometry (presence of distances); also tension between synthetic projective axioms and coordinate (algebraic) projective models.

Synthesis

Projective geometry abstracts incidence and perspective by adjoining points at infinity and insisting on projective-linear invariants like cross-ratio and duality: it studies configurations up to projective transformations where parallelism ceases to be exceptional.