Geometry
Projective Geometry
Цели урока
- Understand the projective plane P^2 and the meaning of homogeneous coordinates
- Master homographies and their 8 degrees of freedom
- Compute and apply the cross-ratio as a projective invariant
- Know Desargues' theorem, Pascal's theorem, and the duality principle
Предварительные знания
- Previous geometry lesson
- Geometric transformations
- Introduction to projective geometry
Why are all types of conics (ellipse, parabola, hyperbola) the same object in projective geometry, and where does the distinction arise in the Euclidean world?
- Google Street View: stitching 20 million panoramic images per day via RANSAC homography estimation
- Tesla Autopilot: SLAM navigation uses the cross-ratio to verify point matches across frames without knowing the camera matrix
- 3D rendering: perspective projection of 3D scenes uses the matrix P = K[R|t] of size 3x4
- Error-correcting codes: projective spaces over finite fields F_q are the geometry of AG codes
From Renaissance Painters to Computer Vision
Gerard Desargues (1591-1661), architect and mathematician, published his treatise in 1639. Descartes was skeptical; most colleagues did not follow. Blaise Pascal, aged 16, read Desargues and in a few months proved his inscribed hexagon theorem. Poncelet, imprisoned in Russia after the Napoleonic wars, reconstructed all of projective geometry from memory without a single book. Hilbert formalized the axiomatics around 1900. In the 2000s, OpenCV and COLMAP turned 17th-century theorems into algorithms processing billions of images daily.
The Projective Plane and Homogeneous Coordinates
2012, Google Street View. The panorama stitching algorithm processes 20 million images per day. Each alignment of two frames reduces to finding a 3x3 matrix - a homography. Projective geometry is what makes seamless Google Maps possible.
The projective plane P^2 is the set of lines through the origin in R^3. A point (X:Y:Z) is defined up to a nonzero scalar: (X:Y:Z) = (lambda*X : lambda*Y : lambda*Z) for any lambda != 0.
Parallel lines in Euclidean geometry never meet. In projective geometry they always meet - at a point at infinity. This makes perspective mathematically consistent: railway tracks converge on the horizon, and that is not an illusion but a theorem.
A homography has 8 degrees of freedom. Finding it requires at least 4 point correspondences in general position - which is why RANSAC in OpenCV iterates over quadruples of points.
How many degrees of freedom does a homography P^2 -> P^2 have?
A 3x3 matrix has 9 entries. Two matrices differing by a nonzero scalar define the same transformation, leaving 8 degrees of freedom. Four point correspondences in general position determine it uniquely.
Cross-Ratio and Projective Invariants
What survives a homography? Lengths - no. Angles - no. Ratios of lengths - no either. But there is one function of four points that survives every projective transformation. The cross-ratio - the only complete invariant of four points on a projective line.
Cross-ratio in SLAM
Invariant for point-match verification
In the SLAM system used by Tesla Autopilot, multiple cameras capture road markings from different angles. A homography changes point coordinates. The cross-ratio of four marking points stays constant. This lets the system verify matches even under strong perspective distortion, without needing to know the camera matrix.
The cross-ratio is defined only for four distinct points in general position. When two points coincide, it takes the degenerate values 0, 1, or infinity.
Which value of the cross-ratio corresponds to harmonic division?
Harmonic division means cross-ratio = -1. Points c and d divide segment ab internally and externally in the same ratio.
Desargues, Pascal and the Duality Principle
1648. Gerard Desargues publishes a theorem that most contemporaries do not understand. Descartes is dismissive. Pascal, age 16, reads Desargues and in a few months proves his hexagon theorem. Poncelet in Russian captivity rediscovers the whole subject without books. 350 years later - the foundation of computer graphics.
| Statement | Dual |
|---|---|
| Two points determine a line | Two lines determine a point |
| Three points are collinear | Three lines are concurrent |
| Pascal's theorem (inscribed hexagon) | Brianchon's theorem (circumscribed hexagon) |
| A point lies on a line | A line passes through a point |
The duality principle is one of the most structural ideas in mathematics. Points and lines in P^2 are completely interchangeable. This is not an axiom convention - it follows from the fact that P^2 is self-dual as a categorical object.
What does the duality principle in the projective plane state?
The axioms of P^2 are symmetric under swapping 'point' and 'line'. Any proved theorem automatically yields a second theorem after this substitution.
Conics in the Projective Plane
In Euclidean geometry there are three types of conics: ellipse, parabola, hyperbola. In projective geometry - only one. Any two nondegenerate conics are projectively equivalent. The type of conic is an artifact of choosing an affine chart, not an intrinsic property.
OpenCV exploits this in camera calibration by circular patterns: 5 points on a circle (which is a general conic in projective coordinates) uniquely determine the scene geometry. Checkerboard is the special case; circular dots are more accurate.
Pascal's theorem holds for any conic: six points on a conic determine a hexagon whose three pairs of opposite sides meet at collinear points. This is used algorithmically to test whether six points lie on a common conic.
How many points in general position uniquely determine a conic in P^2?
A conic is determined by a symmetric 3x3 matrix (6 entries minus 1 for scale = 5 parameters). Each point imposes one linear equation, so 5 points in general position uniquely determine the conic.
Connections to Other Topics
Projective geometry underlies computer vision, algebraic geometry, and coding theory.
- Coordinate Geometry — Projective coordinates extend affine ones to include points at infinity
- Geometry in Computer Science — Homogeneous coordinates and projective transforms are the basis of 3D graphics pipelines
- Algebraic Curves: Bezout's Theorem and Genus — Projective plane is the natural stage for algebraic curves and Bezout's theorem
Итоги
- P^2 = lines through 0 in R^3; point (X:Y:Z) defined up to nonzero scale
- Homography H in GL(3): 8 degrees of freedom, found from 4 point pairs in general position
- Cross-ratio (a,b;c,d) - the only complete projective invariant of 4 points on P^1
- Desargues: perspectivity from a point is equivalent to perspectivity from a line
- Duality principle: swapping 'point' and 'line' preserves truth of theorems in P^2
- Conic in P^2 has 5 degrees of freedom; 5 points in general position determine it uniquely
Вопросы для размышления
- Why do parallel lines never meet in Euclidean geometry but always meet at a point at infinity in projective geometry?
- How does the cross-ratio allow verification of point matches between frames without knowing the camera matrix?
- What does the duality principle in P^2 have in common with duality in linear algebra (dual vector space V*)?