Homogeneous coordinates, points, lines, conics, and the hierarchy from projective down to Euclidean — all in the plane.
You photograph a tiled floor. The square tiles look like trapezoids. The grout lines that were parallel now converge toward a point on the horizon. Angles are distorted. Distances are meaningless. What geometric properties survive this visual mangling?
The answer is: straightness (lines remain lines), incidence (if a point was on a line, it stays on a line), and cross-ratio (a specific ratio of ratios along a line). These are the invariants of projective geometry.
This chapter works entirely in the 2D projective plane — IP2. Think of it as the image plane with points at infinity added. Everything here generalizes to 3D in Chapter 3, but 2D is easier to visualize and draws nicer pictures.
Two sets of parallel lines converge to vanishing points on the line at infinity (the horizon).
In homogeneous coordinates, a point in IP2 is a 3-vector x = (x1, x2, x3)T. A line is also a 3-vector l = (a, b, c)T, representing ax1 + bx2 + cx3 = 0.
A point x lies on a line l if and only if lTx = 0. This is beautifully symmetric — you can't tell from the equation whether you're testing a point against a line or a line against a point.
Click to place points. The line through each consecutive pair is computed via the cross product.
Here is something remarkable: every statement about points and lines in projective geometry has a dual statement obtained by swapping "point" and "line." If "two points define a unique line," then the dual is "two lines define a unique point." Both are true in IP2.
This works because points and lines are both represented by 3-vectors, and the incidence relation lTx = 0 is symmetric in form. A theorem proved for points automatically gives a free theorem for lines.
| Statement | Dual Statement |
|---|---|
| Two points define a line | Two lines define a point |
| Points on a line are collinear | Lines through a point are concurrent |
| Line through x, y: l = x × y | Point on l, m: x = l × m |
A conic in IP2 is defined by a 3×3 symmetric matrix C. A point x lies on the conic if xTCx = 0. Conics include ellipses, parabolas, and hyperbolas — but in projective geometry these are all the same object, just seen from different coordinate frames.
A conic has 5 degrees of freedom (6 entries in a symmetric 3×3 matrix, minus 1 for scale). Therefore 5 points in general position determine a unique conic.
Drag any of the 5 control points to reshape the conic.
A projective transformation (or homography) of IP2 is an invertible linear map of homogeneous coordinates: x' = Hx, where H is a non-singular 3×3 matrix. It has 8 degrees of freedom (9 entries minus 1 for scale).
This is the most general transformation that maps lines to lines. It can model any perspective view of a plane: the distortion you see when looking at a tiled floor from an angle.
Drag the four corner handles of the unit square to see how the projective transformation warps the interior grid.
An affine transformation is a special projective transformation that keeps the line at infinity fixed. In matrix form, the last row is (0, 0, 1):
Affine transformations preserve parallelism and ratios of lengths on parallel lines. They can stretch, shear, rotate, and translate — but they cannot create the "vanishing point" effect of perspective. In image terms: an affine map is what you get when the camera is very far from the scene (telephoto lens).
Continuing down the hierarchy, a similarity transformation (4 DOF) preserves angles and ratios of distances. It's rotation + uniform scaling + translation. A Euclidean transformation (3 DOF) preserves absolute distances — it's just rotation + translation.
The key geometric objects that distinguish these levels are the circular points: the two complex points (1, ±i, 0)T on the line at infinity. Every circle passes through them. Identifying the circular points gives you the full Euclidean structure of the plane.
| Transform | DOF | Preserves |
|---|---|---|
| Euclidean | 3 | Distances, angles |
| Similarity | 4 | Angles, distance ratios |
| Affine | 6 | Parallelism, area ratios |
| Projective | 8 | Cross-ratio, incidence |
All the transformations form a nested hierarchy. Each level is a subgroup of the one above it, obtained by fixing additional geometric structure:
See how each transformation level warps an original shape. Select a level to see what gets distorted and what is preserved.
Given a perspective image of a plane (a projective view), can we recover metric properties like angles and length ratios? Yes — if we can identify the right geometric structures in the image.
Recovering affine properties: Find the vanishing line (image of the line at infinity). Two sets of parallel lines give two vanishing points; the line through them is the vanishing line. Once identified, we can rectify parallelism.
Recovering metric properties: Find the image of the circular points on the vanishing line. Two known angles or two known length ratios give enough constraints. Once identified, we can rectify angles and shapes.
This chapter established the foundation: homogeneous coordinates, the duality of points and lines, conics, and the transformation hierarchy from projective down to Euclidean. Every concept here extends to 3D in the next chapter.
| 2D Concept | 3D Analog (Ch 3) |
|---|---|
| Line at infinity l∞ | Plane at infinity π∞ |
| Circular points (1, ±i, 0) | Absolute conic Ω∞ |
| Conic (3×3 matrix) | Quadric (4×4 matrix) |
| Cross product x × y | Null-space methods |