Projective Geometry: Invariance and Duality

Projective geometry is a branch of mathematics that investigates properties of geometric figures that remain unchanged under central projection (perspective). Historically arising from the study of perspective in Renaissance art, it has evolved into a fundamental framework for algebraic geometry, computer vision, and theoretical physics. Unlike Euclidean geometry, projective geometry does not possess a notion of distance, angle, or parallelism, making it more fundamental in many topological and algebraic contexts.

1. The Projective Plane ${\mathbb{P}}^2(\mathbb{R})$

The real projective plane ${\mathbb{P}}^2$ is the completion of the Euclidean plane ${\mathbb{R}}^2$ by the addition of a “line at infinity.” Formally, it is the set of all lines passing through the origin in ${\mathbb{R}}^3$.

Homogeneous Coordinates

A point in ${\mathbb{P}}^2$ is represented by a triple $[x:y:w]$, where at least one coordinate is non-zero. The notation $[x:y:w]$ denotes an equivalence class under the relation: $[x:y:w]\sim [\lambda x:\lambda y:\lambda w]\text{for any }\lambda \in \mathbb{R}\setminus \{0\}$

For points where $w\ne 0$, we can map them back to the Euclidean plane: $[x:y:w]\to \left(\frac{x}{w},\frac{y}{w}\right)$ Conversely, an affine point $(x,y)\in {\mathbb{R}}^2$ is mapped to $[x:y:1]$.

The Point at Infinity

When $w=0$, the point $[x:y:0]$ represents a “direction” in the affine plane. This is interpreted as the point where all parallel lines with the same slope meet. The collection of all such points $[x:y:0]$ forms the line at infinity.

2. Axiomatic Foundations and Duality

Projective geometry can be defined purely through incidence axioms. One of the most striking results is the Principle of Duality.

Incidence Axioms

1. Any two distinct points lie on exactly one line.
2. Any two distinct lines intersect at exactly one point.
3. There exist four points, no three of which are collinear.

The second axiom is what distinguishes projective geometry from Euclidean geometry, where parallel lines never meet. In ${\mathbb{P}}^2$, “parallel” lines meet at a point on the line at infinity.

Duality

The axioms of the projective plane are symmetric with respect to “point” and “line.” Consequently, for any theorem proven in projective geometry, a dual theorem exists, obtained by swapping the words “point” and “line” and reversing incidence relations.

3. Projective Transformations (Homographies)

A projective transformation, or homography, is a bijection $f:{\mathbb{P}}^2\to {\mathbb{P}}^2$ that maps lines to lines. In homogeneous coordinates, a homography is represented by a non-singular $3\times 3$ matrix $M$ acting on the coordinate vectors: ${\mathbf{x}}^{\prime }=\mathbf{Mx}$ Since coordinates are homogeneous, $M$ and $kM$ represent the same transformation. These transformations form the Projective Linear Group $PGL(3,\mathbb{R})$.

4. The Cross-Ratio: The Fundamental Invariant

While distance and ratio of segments are not invariant under projection, the cross-ratio of four collinear points is.

Let $A,B,C,D$ be four distinct collinear points. If we assign them coordinates $a,b,c,d$ on the line (possibly including $\infty$), the cross-ratio $(A,B;C,D)$ is defined as: $(A,B;C,D)=\frac{(c-a)(d-b)}{(c-b)(d-a)}$

Significance

• If $(A,B;C,D)=\lambda$, then any homography will preserve this value.
• The cross-ratio allows for a coordinatization of the projective line ${\mathbb{P}}^1$.
• It is the foundation for defining “projective distance” in non-Euclidean geometries.

5. Harmonic Conjugates

A special case of the cross-ratio occurs when $(A,B;C,D)=-1$. In this case, the points $C$ and $D$ are said to be harmonic conjugates with respect to $A$ and $B$.

Geometrically, if $A,B,C$ are given, $D$ can be constructed using a complete quadrilateral. This relationship is central to the projectivity of conics and the poles/polars theory.

6. Famous Theorems

Desargues’ Theorem

Two triangles are in perspective from a point if and only if they are in perspective from a line.

• Triangles $△ABC$ and $△A^{\prime }{B}^{\prime }{C}^{\prime }$ are in perspective from point $O$ if $A{A}^{\prime },B{B}^{\prime },C{C}^{\prime }$ meet at $O$.
• They are in perspective from line $L$ if the intersections of corresponding sides $(AB\cap A^{\prime }{B}^{\prime },BC\cap B^{\prime }{C}^{\prime },CA\cap C^{\prime }{A}^{\prime })$ are collinear.

Pappus’s Hexagon Theorem

If three points $A,B,C$ lie on one line and $A^{\prime },B^{\prime },C^{\prime }$ lie on another, then the intersection points of the cross-pairs $(A{B}^{\prime },A^{\prime }B),(B{C}^{\prime },B^{\prime }C),(A{C}^{\prime },A^{\prime }C)$ are collinear.

7. Conics in Projective Space

In Euclidean geometry, we distinguish between ellipses, parabolas, and hyperbolas based on their eccentricity or relationship to the line at infinity. In projective geometry, these distinctions vanish.

Projective Equivalence

A conic is the set of points $\mathbf{x}\in {\mathbb{P}}^2$ satisfying a quadratic form: ${\mathbf{x}}^{T}Q\mathbf{x}=0$ where $Q$ is a symmetric $3\times 3$ matrix. Since all non-degenerate symmetric $3\times 3$ matrices over $\mathbb{R}$ are congruent to the identity matrix (up to signature), every non-degenerate conic in ${\mathbb{P}}^2$ is projectively equivalent to a circle.

In affine terms, a parabola is simply a conic tangent to the line at infinity, while a hyperbola intersects the line at infinity at two real points.

8. Computational Implementation

The following Python script calculates the cross-ratio of four points on a line or applies a homography matrix to a set of homogeneous points.

import numpy as np

def calculate_cross_ratio(a, b, c, d):
    """
    Calculates the cross-ratio (A, B; C, D).
    Uses the formula: ((c-a)*(d-b)) / ((c-b)*(d-a))
    """
    return ((c - a) * (d - b)) / ((c - b) * (d - a))

def apply_homography(H, point):
    """
    Applies a 3x3 homography matrix H to a 2D point (x, y).
    """
    # Convert to homogeneous [x, y, 1]
    p_homo = np.array([point[0], point[1], 1.0])
    
    # Transform
    res_homo = np.dot(H, p_homo)
    
    # Normalize back to affine (project)
    if res_homo[2] == 0:
        return (float('inf'), float('inf'))  # Point at infinity
    
    return (res_homo[0] / res_homo[2], res_homo[1] / res_homo[2])

# Example Usage
# Define points on a line
A, B, C, D = 0, 10, 2, 5
cr = calculate_cross_ratio(A, B, C, D)
print(f"Cross-ratio (A,B; C,D): {cr}")

# Apply a perspective shift (Homography)
# Example: rotate 45 degrees around Z then shift
theta = np.radians(45)
H = np.array([
    [np.cos(theta), -np.sin(theta), 5],
    [np.sin(theta),  np.cos(theta), 2],
    [0.1,           0,             1]  # Perspective component in last row
])

p = (1, 1)
p_transformed = apply_homography(H, p)
print(f"Point {p} transformed to {p_transformed}")

Summary

Projective geometry provides the “infinite background” that completes affine geometry. By removing the concepts of distance and angle, it reveals deeper structural properties like duality and projective invariance. This framework is essential for understanding how 3D space is mapped onto 2D planes, forming the cornerstone of modern imaging and computer vision.