Projective Geometry - Main Theorem

TheoremDesargues' Theorem (Projective Form)

Let $\triangle ABC$ and $\triangle A'B'C'$ be two triangles in a projective plane. If the lines $AA'$ , $BB'$ , $CC'$ are concurrent at a point $O$ , then the three points:

$P = AB \cap A'B'$
$Q = BC \cap B'C'$
$R = CA \cap C'A'$

are collinear (lie on a line).

Conversely, if $P$ , $Q$ , $R$ are collinear, then $AA'$ , $BB'$ , $CC'$ are concurrent.

Desargues' theorem is fundamental to projective geometry and reveals the deep relationship between perspectivity and collinearity. The configuration has a beautiful self-dual structure: by duality, we can interchange the roles of points and lines.

The theorem holds automatically in projective 3-space (it can be proven by noting that the configuration lies in two intersecting planes), but it's an independent axiom for projective planes. Planes satisfying Desargues' theorem are called Desarguesian and can be coordinatized by division rings.

DefinitionPerspective Triangles

Two triangles are perspective from a point if lines through corresponding vertices are concurrent. They are perspective from a line if intersections of corresponding sides are collinear.

Desargues' theorem states: triangles perspective from a point are perspective from a line (and vice versa by the converse).

The proof in 3-dimensional projective space is elegant. Place the two triangles in distinct planes. The lines $AA'$ , $BB'$ , $CC'$ meet at point $O$ . The intersection points $P$ , $Q$ , $R$ lie on both planes (they're on corresponding sides), hence lie on the intersection line of the two planes.

ExampleApplications in Perspective Drawing

Desargues' theorem underlies the mathematical theory of perspective in art. When an artist draws a 3D scene on a 2D canvas, parallel lines appear to converge at vanishing points. The relationship between the original scene and the drawing is governed by a projective transformation, and Desargues' theorem ensures geometric consistency.

In computer graphics, perspective projection matrices use projective transformations to map 3D scenes onto 2D screens while preserving straight lines and cross-ratios.

The Desargues configuration consists of 10 points and 10 lines, with each point on 3 lines and each line through 3 points. This configuration is self-dual and appears throughout projective geometry and incidence geometry.

Remark

The validity of Desargues' theorem characterizes the coordinatizability of a projective plane:

Desarguesian planes can be coordinatized by division rings (skew fields)
If the plane is also Pappian (Pappus' theorem holds), it can be coordinatized by a field
Non-Desarguesian planes exist but are more exotic

This connection between geometry and algebra is a cornerstone of modern geometry.

Desargues' theorem generalizes to higher dimensions. In projective $n$ -space, if two $k$ -simplices are perspective from a point, they're perspective from a hyperplane. The theorem's validity depends on the ambient dimension and the intrinsic dimension of the configuration.