Projective Geometry

Projective geometry studies geometric properties that are invariant under projective transformations. It extends Euclidean geometry by adding “points at infinity.”

Homogeneous Coordinates

In $n$-dimensional projective space $\mathbb{P}^n$, a point is represented by $(n+1)$ coordinates $[x_0 : x_1 : \dots : x_n]$, where not all are zero. Two sets of coordinates represent the same point if they are scalar multiples of each other: $[x] = [\lambda x] \quad \text{for } \lambda \neq 0$

For the projective line $\mathbb{P}^1$, the point $[1 : 0]$ represents the point at infinity.

def to_homogeneous(x, y):
  return [x, y, 1]

def from_homogeneous(coords):
  x, y, w = coords
  if w == 0:
      return "Point at infinity"
  return [x/w, y/w]

# A standard point
p1 = to_homogeneous(5, 10)
print(f"Homogeneous: {p1} -> Euclidean: {from_homogeneous(p1)}")

# A scaled homogeneous point
p2 = [10, 20, 2] # Same as [5, 10, 1]
print(f"Scaled Homogeneous: {p2} -> Euclidean: {from_homogeneous(p2)}")

Duality

One of the most profound features of projective geometry is the principle of duality: for any theorem in two-dimensional projective geometry, there is a dual theorem obtained by interchanging the roles of “points” and “lines.”

Cross-Ratio

The cross-ratio is the fundamental invariant of projective geometry. For four collinear points $A, B, C, D$, the cross-ratio is defined as: $(A, B; C, D) = \frac{AC \cdot BD}{BC \cdot AD}$ where $AC$ denotes the signed distance from $A$ to $C$.

def cross_ratio(a, b, c, d):
  return ((c - a) * (d - b)) / ((c - b) * (d - a))

print(f"Cross ratio of 0, 1, 2, 5: {cross_ratio(0, 1, 2, 5):.2f}")
print(f"Cross ratio of 10, 11, 12, 15: {cross_ratio(10, 11, 12, 15):.2f}")
# Note how translation preserves the ratio!

Perspective and Projections

Projective geometry provides the mathematical foundation for computer vision and 3D graphics, where 3D scenes are projected onto 2D image planes (cameras).