Projective Geometry & Homogeneous Coordinates

Motivation

Two problems keep showing up in camera geometry that ordinary Euclidean coordinates handle badly:

1. Parallel lines "meet" under perspective. Stand on a straight road and look down it — the two edges, parallel in the world, visibly converge to a single point on the horizon in the photo. In plain (x, y) coordinates, parallel lines have no intersection at all; the algebra has a hole exactly where the picture doesn't. 2. Perspective projection is a division, not a matrix multiply. u = f·x/z is not linear in (x, y, z) — you can't write it as a single matrix times a vector, which makes chaining camera transforms (rotate, then translate, then project) painful.

Homogeneous coordinates fix both at once: add one extra coordinate, and every point at infinity becomes an ordinary point, and every perspective divide becomes "multiply by a matrix, then divide once at the very end."

The core trick

Represent a 2D point (x, y) by a 3-vector x̃ = (x, y, 1). That's a point in P², the projective plane. The rule that makes it work: scale doesn't matter. The vectors (x, y, 1), (2x, 2y, 2), and (−5x, −5y, −5) all represent the same point. Formally, x̃ ≅ ỹ ("equal up to scale") means x̃ = λ ỹ for some nonzero λ. Every symbol ≅ in this series is exactly this: "same object, don't care which representative vector you're holding."

To go back to ordinary coordinates, divide by the last entry: (x, y, w) → (x/w, y/w). This is the "divide once at the end" from above — and it's exactly what u = f·x/z is doing in the pinhole model.

Now the trick that solves problem 1 above: what if w = 0? (x, y, 0) has no Euclidean counterpart — dividing by zero is undefined — but it's a perfectly good vector, so it's a perfectly good point of P². These are the points at infinity (or ideal points), one for every direction (x, y) in the plane. Two parallel lines, which never meet in the Euclidean plane, meet exactly at the ideal point in their shared direction. Nothing is undefined anymore — the hole is gone.

Interactive

Left: a point (x, y) you can drag in the image plane w = 1 (gray), plotted in 3D as (x, y, w)-space. The line through the origin and that point is its entire equivalence class — every point on it is the same projective point. Drag the scale λ slider to slide along that ray and watch the recovered Euclidean point not move. Drag w toward 0 to see the point recede toward infinity in the image plane.

Notice: changing λ moves the plotted 3D dot a lot, but the readout's recovered Euclidean point never changes — that's exactly the "don't care which representative" rule. Changing w directly (holding x, y fixed) is different: that's a genuinely different homogeneous vector (x, y, w), so its Euclidean point (x/w, y/w) races off toward infinity as w → 0.

The other half of P²

A 2D line ax + by + c = 0 is represented the same way as a point: as a homogeneous 3-vector l = (a, b, c), also only defined up to scale. A point lies on a line exactly when their dot product vanishes:

Because points and lines are both just 3-vectors related by the same incidence equation, P² has a beautiful symmetry called duality: any true statement about points and lines stays true if you swap the words "point" ↔ "line" and the operation "intersection" ↔ "join." Two facts fall out of this for free, both computed with a single cross product:

Why the cross product works: l = x̃₁ × x̃₂ is by construction orthogonal to both x̃₁ and x̃₂, i.e. l · x̃₁ = l · x̃₂ = 0 — exactly the incidence condition, satisfied for both points at once. This one identity is the entire computational engine behind vanishing points, epipolar lines as l = Fx₁ in Part 6, and much of what looks like magic later in this series.

And it explains ideal points cleanly: two parallel lines l₁, l₂ (same (a, b) direction, different c) cross to give a point with w = 0 — an ideal point, in the direction perpendicular to (a, b), i.e. along the lines themselves. All ideal points together form one line, the line at infinity l∞ = (0, 0, 1), which every ordinary line meets at its own vanishing direction and no other.

Interactive

Drag the two colored points to define a line through them (computed as l = x̃₁ × x̃₂). A second, independent line is fixed for comparison; drag its angle slider to rotate it. The readout shows both lines' coefficients and their intersection x̃ = l₁ × l₂, including its raw w — watch w shrink toward zero exactly as the two lines become parallel, and the intersection point fly off-canvas toward infinity.

Drag the two orange handles anywhere — the blue line through them updates live. Click "Snap parallel" to make the fixed line exactly parallel to the draggable one and watch the intersection's w hit (numerically) zero: same direction, no finite meeting point, an ideal point on the line at infinity.

Just enough to recognize it later

Conics. A conic (ellipse, parabola, circle, pair of lines, ...) is a symmetric 3×3 matrix C with x̃ᵀCx̃ = 0; the absolute conic is the special one that camera calibration recovers. Conics & the absolute conic works it out in full.

P³ and camera projection. The same language generalizes to 3D points, 4-vector planes, and a camera as a 3×4 matrix P with x̃ = PX̃. 3D projective space & Plücker lines takes that up.

The transformation hierarchy. Not every transform of P² is equally destructive — Euclidean, similarity, affine, and projective maps each add degrees of freedom and give up an invariant. The projective transformation hierarchy is the full stratification, including cross-ratio.

Recap