1. Projective 2D geometry Appunti basati sulla parte iniziale del testo
di R.Hartley e A.Zisserman “Multiple view geometry”
Rielaborazione di materiale di M.Pollefeys

2. Projective 2D Geometry Points, lines & conics
Transformations & invariants
1D projective geometry and
the Cross-ratio

3. Homogeneous coordinates
Homogeneous representation of lines
equivalence class of vectors, any vector is representative
Set of all equivalence classes in R3(0,0,0)T forms P2
Homogeneous representation of points on
if and only if
The point x lies on the line l if and only if xTl=lTx=0
Homogeneous coordinates
Inhomogeneous coordinates
but only 2DOF

4. Points from lines and vice-versa
Intersections of lines
The intersection of two lines and is
Line joining two points
The line through two points and is
Line joining two points: parametric equation
A point on the line through two points x and x’
is y = x + q x’
Example

5. Ideal points and the line at infinity
Intersections of parallel lines
Example
tangent vector
normal direction
Ideal points
Line at infinity
Note that in P2 there is no distinction
between ideal points and others

6. A model for the projective plane
exactly one line through two points
exaclty one point at intersection of two lines

7. Duality Duality principle:
To any theorem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem

8. Conics Curve described by 2nd-degree equation in the plane
or homogenized
or in matrix form
with
5DOF:

9. Five points define a conic
For each point the conic passes through or
stacking constraints yields

10. Pole-polar relationship
The polar line l=Cx of the point x with respect to the conic C intersects the conic in two points. The two lines tangent to C at these points intersect at x

11. Polarity: cross ratio
Cross ratio of 4 colinear points y = x + q x’ (with i=1,..,4)
ratio of ratios i i
Harmonic 4-tuple of colinear points: such that CR=-1

12. Chasles’ theorem A B C X D
Conic = locus of constant cross-ratio towards 4 ref. points A B C X D

13. Polarity: conjugacy
Given a point y and a conic C, a point z is conjugated of y wrt C
if the 4-tuple (y,z,x’ ,x’’ ) is harmonic,
where x’ and x’’ are the intersection points between C and line (y,z).
Intersection points (y+q z) C (y +q z) = 0 is a 2-nd degree equation on q
y Cy + 2 q y Cz + q z Cz = 0 --> two solutions q’ and q’’.
Since y and z correspond to q = 0 and q = , harmonic -> q’+ q’’ = 0
Hence the two solutions sum up to zero: y Cz = 0, and also z Cy = 0
 the point z is on the line Cy, which is called the polar line of y wrt C
the polar line Cy is the locus of the points conjugate of y wrt C
Conic C is the locus of points belonging to their own polar line y Cy = 0 T T T 2 T T T T

14. Correlations and conjugate points
A correlation is an invertible mapping from points of P2 to lines of P2. It is represented by a 3x3 non-singular matrix A as l=Ax
Conjugate points with respect to C
(on each others polar)
Conjugate lines
with respect to C*
(through each others pole)

15. Polarity and tangent lines
Consider a line through a point y, tangent to a conic C:
let us determine the point z conjugate of y wrt C:
since the intersection points coincide, then q’ = q’’
 harmonic 4-tuple: also z coincides with the tangency point
 All points y on the tangent are conjugate to tangency point z wrt C
 polar line of a point z on the conic C is tangent to C through z
In addition: since there are two tangents to C through a point y not on C,
both tangency points are conjugate to y wrt C
polar line of y wrt C (=locus of the conjugate points)
= line through the two tangency points of lines through y

16. Tangent lines to conics
The line l tangent to C at point x on C is given by l=Cx l x C

17. Dual conics  l C C C l = 0  C = C = C
A line tangent to the conic C satisfies
In general (C full rank): in fact -1
Line l is the polar line of y : y = C l , but since y Cy = 0
 l C C C l = 0  C = C = C T T -T -1 * -T -1
Dual conics = line conics = conic envelopes

18. Degenerate conics
A conic is degenerate if matrix C is not of full rank
e.g. two lines (rank 2)
e.g. repeated line (rank 1)
Degenerate line conics: 2 points (rank 2), double point (rank1)
Note that for degenerate conics

19. Projective transformations
Definition:
A projectivity is an invertible mapping h from P2 to itself such that three points x1,x2,x3 lie on the same line if and only if h(x1),h(x2),h(x3) do.
A mapping h:P2P2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2 represented by a vector x it is true that h(x)=Hx
Theorem:
Definition: Projective transformation or
8DOF
projectivity=collineation=projective transformation=homography

20. Mapping between planes
central projection may be expressed by x’=Hx
(application of theorem)

21. Removing projective distortion
select four points in a plane with known coordinates
(linear in hij)
(2 constraints/point, 8DOF  4 points needed)
Remark: no calibration at all necessary,
better ways to compute (see later)

22. More examples

23. Transformation of lines and conics
For a point transformation
Transformation for lines
Transformation for conics
Transformation for dual conics

24. A hierarchy of transformations
Projective linear group
Affine group (last row (0,0,1))
Euclidean group (upper left 2x2 orthogonal)
Oriented Euclidean group (upper left 2x2 det 1)
Alternative, characterize transformation in terms of elements or quantities that are preserved or invariant
e.g. Euclidean transformations leave distances unchanged

25. Class I: Isometries orientation preserving: orientation reversing:
(iso=same, metric=measure)
orientation preserving:
orientation reversing:
3DOF (1 rotation, 2 translation)
special cases: pure rotation, pure translation
Invariants: length, angle, area

26. Class II: Similarities
(isometry + scale)
4DOF (1 scale, 1 rotation, 2 translation)
also know as equi-form (shape preserving)
metric structure = structure up to similarity (in literature)
Invariants: ratios of length, angle, ratios of areas, parallel lines

27. Class III: Affine transformations
where
6DOF (2 scale, 2 rotation, 2 translation)
non-isotropic scaling! (2DOF: scale ratio and orientation)
Invariants: parallel lines, ratios of parallel segment lengths, ratios of areas

28. Action of affinities and projectivities on line at infinity
Line at infinity stays at infinity,
but points move along line
Line at infinity becomes finite,
allows to observe vanishing points, horizon,

29. Class VI: Projective transformations
8DOF (2 scale, 2 rotation, 2 translation, 2 line at infinity)
Action: non-homogeneous over the plane
Invariants: cross-ratio of four points on a line
(ratio of ratio)

30. Decomposition of projective transformations
decomposition unique (if chosen s>0)
upper-triangular,
Example:

31. Overview transformations
Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio
Projective
8dof
Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids).
The line at infinity l∞
Affine
6dof
Ratios of lengths, angles.
The circular points I,J
Similarity
4dof
Euclidean
3dof
lengths, areas.

32. Number of invariants?
The number of functional invariants is equal to, or greater than, the number of degrees of freedom of the configuration less the number of degrees of freedom of the transformation
e.g. configuration of 4 points in general position has 8 dof (2/pt)
and so 4 similarity, 2 affinity and zero projective invariants

33. Projective geometry of 1D
3DOF (2x2-1)
The cross ratio
Invariant under projective transformations

34. Recovering metric and affine properties from images
Parallelism
Parallel length ratios
Angles
Length ratios

35. Note: not fixed pointwise
The line at infinity
The line at infinity l is a fixed line under a projective transformation H if and only if H is an affinity
Note: not fixed pointwise

36. Affine properties from images
projection
rectification
in fact, any point x on l’ is mapped to a point at the ∞ ∞

37. Affine rectification v1 l∞ v2 l1 l3 l2 l4

38. Distance ratios

39. The circular points
The circular points I, J are fixed points under the projective transformation H iff H is a similarity

40. The circular points “circular points” l∞
Intersection points between any circle and l∞
Algebraically, encodes orthogonal directions

41. Conic dual to the circular points
A line conic: set of lines through any of the circular points
The dual conic is fixed conic under the
projective transformation H iff H is a similarity
Note: has 4DOF
l∞ is the null vector

42. Angles
Euclidean:
Projective:
in fact, e.g.
(orthogonal)

43. Length ratios

44. Metric properties from images
Rectifying transformation from SVD

45. Why ?
Normally: SVD (Singular Value Decomposition)
with U and V orthogonal
But is symmetric 
and SVD is unique 
Observation : H=U
orthogonal (3x3): not a P2 isometry

46. Metric from affine
Once the image has been affinely rectified

47. Metric from affine

48. Metric from projective

49. Projective conic classification
Diagonal
Equation
Conic type
(1,1,1)
improper conic
(1,1,-1)
circle
(1,1,0)
single real point
(1,-1,0)
two lines
(1,0,0)
single line

50. Affine conic classification
ellipse
parabola
hyperbola

51. Fixed points and lines (eigenvectors H =fixed points)
(1=2  pointwise fixed line)
(eigenvectors H-T =fixed lines)