Planes, lines, quadrics in three-space — the plane at infinity, the absolute conic, and everything you need before cameras enter the picture.
Chapter 2 gave us the geometry of the image plane — a 2D projective world. But cameras live in 3D space, and so do the scenes they photograph. To understand how a camera projects 3D reality onto a 2D image, we need the geometry of three-dimensional projective space, IP3.
The good news: almost everything from 2D generalizes cleanly. Points become 4-vectors. Lines (which were dual to points in 2D) become more interesting — they now have 4 degrees of freedom and need special representations. The line at infinity becomes the plane at infinity. The circular points become the absolute conic.
The extra dimension also introduces genuinely new objects: quadric surfaces (the 3D analog of conics), twisted cubics, and Plucker coordinates for lines.
A set of points in 3D space and the planes they define. Rotate the view with the slider.
A point in IP3 is a homogeneous 4-vector X = (X1, X2, X3, X4)T. When X4 ≠ 0, it represents the Euclidean point (X1/X4, X2/X4, X3/X4). When X4 = 0, it is a point at infinity encoding a direction.
A plane is also a 4-vector π = (π1, π2, π3, π4)T. A point X lies on plane π if and only if πTX = 0. This is exactly the same incidence relation as points and lines in 2D, but now in 4D homogeneous space.
| Entity | Representation | DOF |
|---|---|---|
| Point | 4-vector X | 3 |
| Plane | 4-vector π | 3 |
| Line | Plucker matrix or 6-vector | 4 |
| Quadric | 4×4 symmetric matrix Q | 9 |
Lines in IP3 are more complex than in IP2. A line has 4 degrees of freedom (pick two points on it: 3 + 3 = 6 DOF for two points, minus 2 for position along the line). There is no single 4-vector representation that works.
Instead, we use Plucker coordinates. Given two points A and B on the line, the Plucker matrix is the 4×4 skew-symmetric matrix L = ABT − BAT. Its six independent entries form the Plucker vector (l12, l13, l14, l23, l42, l34).
A plane π contains the line L if and only if Lπ = 0. Two lines L and L' intersect if and only if their Plucker vectors satisfy a bilinear condition. The dual Plucker matrix L* represents the same line but in terms of the two planes that define it.
A quadric surface in IP3 is the 3D analog of a conic. It is defined by a 4×4 symmetric matrix Q. A point X lies on the quadric if XTQX = 0. Quadrics include ellipsoids, hyperboloids, paraboloids, cylinders, and cones.
A quadric has 9 degrees of freedom (10 independent entries in a 4×4 symmetric matrix, minus 1 for scale). Therefore 9 points in general position determine a unique quadric.
Toggle between different quadric types. All are described by XTQX = 0 for different matrices Q.
| Quadric Type | Signature of Q | Example |
|---|---|---|
| Ellipsoid | (+,+,+,−) | x²+y²+z²=1 |
| Hyperboloid (1 sheet) | (+,+,−,−) | x²+y²−z²=1 |
| Cone | (+,+,−,0) | x²+y²−z²=0 |
Just as in 2D, 3D transformations form a hierarchy. A projective transformation of IP3 is a 4×4 non-singular matrix H with 15 DOF. Specializing H step by step gives affine, similarity, and Euclidean transformations.
The plane at infinity, π∞, is the set of all points at infinity in IP3. In canonical coordinates it is π∞ = (0, 0, 0, 1)T. Every direction in 3D corresponds to a unique point on π∞, and parallel lines meet on it.
Two planes are parallel if and only if their intersection line lies on π∞. An affine transformation is precisely a projective transformation that fixes π∞. This is the 3D analog of fixing l∞ in 2D.
In 2D, the circular points (1, ±i, 0)T encode metric structure on the line at infinity. In 3D, their analog is the absolute conic Ω∞ — a conic on the plane at infinity that encodes the full Euclidean structure of 3D space.
The absolute conic is defined by the equations X12 + X22 + X32 = 0, X4 = 0. It consists entirely of complex points (no real point satisfies x2+y2+z2=0). Despite being "imaginary," it is the most important geometric object in metric vision.
The image of the absolute conic (IAC) will play a starring role in camera calibration (Chapter 8). When a camera photographs Ω∞, it projects to a conic ω in the image that encodes the calibration matrix K: specifically, ω = (KKT)−1.
Working with Ω∞ directly is awkward because it lives on π∞, which we may not know. The absolute dual quadric Q*∞ provides a more convenient representation that lives in the full 3D space.
Q*∞ is a degenerate dual quadric (a 4×4 matrix of rank 3) defined in canonical coordinates as:
It has three key properties that make it the workhorse of auto-calibration:
In auto-calibration (Chapter 19), we estimate Q*∞ from multiple images without any calibration pattern, then extract both the plane at infinity and the absolute conic from it. This gives us the full metric structure of the scene.
A twisted cubic is a curve in IP3 that is the 3D analog of a conic in IP2. Parametrically it is (θ3, θ2, θ, 1)T — a degree-3 curve that twists through space without lying in any plane.
Twisted cubics are determined by 6 points in general position. Any projective transformation maps a twisted cubic to another twisted cubic. They arise naturally in structure-from-motion: the set of camera centres consistent with 6 image correspondences traces a twisted cubic.
This chapter extended every 2D projective concept to 3D. The key new objects and their 2D analogs are:
| 2D (Chapter 2) | 3D (Chapter 3) | Role |
|---|---|---|
| Point (3-vector) | Point (4-vector) | Basic element |
| Line (3-vector) | Plane (4-vector) | Dual element |
| — | Line (Plucker, 4 DOF) | Self-dual element |
| Conic (3×3 sym) | Quadric (4×4 sym) | Degree-2 surface |
| l∞ | π∞ | Affine ↔ projective boundary |
| Circular points | Absolute conic Ω∞ | Metric structure |
| — | Absolute dual quadric Q*∞ | Auto-calibration target |