Projective Geometry - Key Properties

Projective geometry possesses remarkable symmetry and duality properties that distinguish it from affine and Euclidean geometries. These properties make projective theorems particularly elegant and powerful.

DefinitionPrinciple of Duality

In projective plane geometry, the principle of duality states that every theorem remains valid when the words "point" and "line" are exchanged (and "incidence" relations are reversed).

For example:

Primal: Two distinct points determine a unique line
Dual: Two distinct lines determine a unique point (intersection)

This duality extends to higher dimensions, exchanging $k$ -dimensional and $(n-k)$ -dimensional subspaces in $\mathbb{P}^n$ .

The duality arises from the symmetry in homogeneous coordinates: a point $[x:y:z]$ lies on line $[a:b:c]$ if and only if $ax + by + cz = 0$ , a condition symmetric in point and line coordinates. This algebraic symmetry reflects a profound geometric duality.

Cross-ratio is the fundamental projective invariant. Unlike affine or Euclidean geometry, projective transformations don't preserve distances or ratios of lengths. However, they preserve the cross-ratio of four collinear points.

DefinitionCross-Ratio

For four collinear points $A, B, C, D$ in projective space, their cross-ratio is:

(A,B;C,D) = \frac{AC \cdot BD}{AD \cdot BC}

where products denote signed distances (in an affine chart).

In homogeneous coordinates, if $A = [a], B = [b], C = [c], D = [d]$ :

(A,B;C,D) = \frac{\det(a,c) \cdot \det(b,d)}{\det(a,d) \cdot \det(b,c)}

Cross-ratios are preserved by projective transformations, making them the analogue of distance in Euclidean geometry. Four points are said to form a harmonic range if their cross-ratio equals $-1$ , a particularly important configuration.

ExampleHarmonic Division

If point $D$ divides segment $AB$ internally in ratio $m:n$ and point $C$ divides it externally in the same ratio, then $(A,B;C,D) = -1$ . Points $C$ and $D$ are called harmonic conjugates with respect to $A$ and $B$ .

This relationship is projectively invariant and arises naturally in many geometric constructions.

The fundamental theorem of projective geometry states that any bijection between projective spaces (of dimension $\geq 2$ ) that preserves collinearity is a projective transformation. This characterizes projective transformations by their most basic geometric property.

Remark

Projective space is complete: every two distinct lines meet, and every sequence of points has compactness properties. This completeness eliminates exceptions and makes theorems cleaner. For instance, Bézout's theorem states that two plane curves of degrees $m$ and $n$ intersect in exactly $mn$ points (counting multiplicity) in the projective plane, with no exceptions for "parallel" curves.

Incidence relations in projective geometry are particularly regular. In $\mathbb{P}^2$ :

Two distinct points determine a unique line
Two distinct lines determine a unique point
Three non-collinear points (or three non-concurrent lines) determine the plane uniquely

This uniformity contrasts with affine geometry where parallel lines don't intersect.