Projective Geometry - Examples and Constructions

Projective geometry provides elegant solutions to classical problems and unifies various geometric phenomena. Many constructions that require case analysis in Euclidean geometry become uniform in the projective setting.

ExamplePascal's Theorem

Pascal's Theorem states: If six points lie on a conic (including degenerate cases), then the three intersection points of opposite sides of the hexagon formed by these points are collinear.

Specifically, for hexagon $ABCDEF$ inscribed in a conic, let:

$P = AB \cap DE$
$Q = BC \cap EF$
$R = CD \cap FA$

Then $P$ , $Q$ , $R$ are collinear (lie on the Pascal line).

This theorem generalizes many special cases and demonstrates the power of projective methods.

The dual of Pascal's theorem is Brianchon's theorem: if a hexagon is circumscribed around a conic, then the three diagonals connecting opposite vertices are concurrent. This follows immediately from duality, showcasing the efficiency of projective reasoning.

DefinitionProjective Conics

A conic in $\mathbb{P}^2$ is the zero locus of a non-degenerate homogeneous quadratic polynomial:

ax^2 + by^2 + cz^2 + dxy + exz + fyz = 0

In projective space, all non-degenerate conics are equivalent under projective transformations. This unifies circles, ellipses, parabolas, and hyperbolas into a single type.

Projective space provides the natural setting for studying algebraic curves. A curve of degree $d$ has equation $F(x,y,z) = 0$ where $F$ is a homogeneous polynomial of degree $d$ . Bézout's theorem guarantees that two curves of degrees $m$ and $n$ with no common component intersect in exactly $mn$ points (counting multiplicity and including points at infinity).

ExamplePappus' Theorem

Given two lines with three points $A$ , $B$ , $C$ on one line and three points $D$ , $E$ , $F$ on the other, consider:

$P = AE \cap BD$
$Q = AF \cap CD$
$R = BF \cap CE$

Then $P$ , $Q$ , $R$ are collinear. This beautiful result holds in any projective plane and is fundamental to the theory of projective spaces.

The complete quadrilateral provides rich projective configurations. Given four lines in general position (no three concurrent), they form six vertices and three diagonal lines. The three diagonal points (intersections of opposite sides) form a diagonal triangle whose properties are projectively significant.

Remark

Projective geometry underlies much of classical algebraic geometry. Varieties are naturally studied in projective space because:

Polynomial equations extend naturally to homogeneous forms
Intersections are complete (no "missing" points at infinity)
Projective transformations form a large group of symmetries
Compactness properties simplify topology and dimension theory

Homogeneous coordinates enable elegant computational methods. To find the intersection of lines $[a_1:b_1:c_1]$ and $[a_2:b_2:c_2]$ , compute the cross product:

[a_1:b_1:c_1] \times [a_2:b_2:c_2] = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix}

This gives the homogeneous coordinates of the intersection point, with no special cases for parallel lines (which meet at infinity).