2 Projective Geometry and Transformations of 2D 1

2.1 Planar geometry

The classical approach of Euclid studied geometry from a “geometric” or coordinate-free viewpoint. Since Descartes, it has been seen that geometry may be algebraicized.

In the algebraic approach, geometric entities are described in terms of coordinatesand algebraic entities. A significant advantage of the algebraic approach to geometryis that results derived in this way may more easily be used to derive algorithms andpractical computational methods.

2.2 The 2D projective plane

2.2.1 Points and lines

Homogeneous representation of lines. A line in the plane is represented by anequation such as ax + by + c = 0 or the vector (a, b, c)⊤.

Since (a, b, c)⊤ and k(a, b, c)⊤ (k ∕= 0) represent the same line, an equivalence classof vectors is known as a homogeneous vector. The set of equivalence classes of vectorsin R3 − (0, 0, 0)⊤ forms the projective space P2.

Homogeneous representation of points. Since (x, y, 1)⊤ and k(x, y, 1)⊤ (k ∕= 0) rep-resent the same point, points as homogeneous vectors are also elements of P2.

We distinguish between the homogeneous coordinates x = (x1, x2, x3)⊤ of a point

and the inhomogeneous coordinates (x, y)⊤.

Result 2.1. The point x lies on the line l if and only if x⊤l = 0.

Degrees of freedom (dof). In order to specify a point (or a line), two values must beprovided. Namely it has two degrees of freedom.

Result 2.2. The intersection of two lines l and l′ is the point x = l × l′.Given x = l × l′, we have

x⊤l = l⊤x = l⊤l × l′ = 0

Result 2.4. The line through two points x and x′ is l = x × x′.Given l = x × x′, we have

x⊤l = x⊤x × x′ = 0

1The notes in this document are taken by Wufei Ma from Multiple View Geometry in Computer Visionby Richard Hartley and Andrew Zisserman. I added some comments and calculations to those stepsthat are not quite “obvious” to me.

1

π

l

xO

x 1

x

x 3

2

idealpoint

Figure 1: A model of the projective plane.

2.2.2 Ideal points and the line at infinity

Intersection of parallel lines. Consider two lines l = (a, b, c)⊤ and l′ = (a, b, c′)⊤.Using Result 2.2, the intersection is

l × l′ = (bc′ − cb, ca − ac′, ab − ab) = (c′ − c)(b, a, 0)⊤

Ideal points and the line at infinity. One may augment R2 with last coordinatex3 = 0. The resulting space is the set of all homogeneous 3-vectors, namely theprojective space P2. The points with last coordinate x3 = 0 are known as ideal points,or points at infinity.

The set of ideal points, given by (x1, x2, 0)⊤, lies on a single line, the line at infinity,denoted by l∞ = (0, 0, 1)⊤.

In the projective plane P2, one may state without qualification that two distinctlines meet in a single point.

The study of the geometry of P2 is known as projective geometry.

A model for the projective plane. A fruitful way of thinking of P2 is as a set of raysin R3, as shown in Figure 1. A ray in R3, k(x1, x2, x3)

⊤, represent a single point inP2. The lines in P2 are planes passing through the origin.

Result 2.6. Duality principle. To any theorem of 2-dimensional projective geometrythere corresponds to a dual theorem, which may be derived by interchanging theroles of points and lines in the original theorem.

2.2.3 Conics and dual conics

A conic is a curve described by a second-degree equation in the plane. In Euclideangeometry conics are of three main types: hyperbola, ellipse, and parabola (apartfrom degenerate conics).

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The equation of a conic in inhomogeneous coordinates is

ax2 + bxy + cy2 + dx + ey + f = 0

“Homogenizing” by the replacements: x → x1/x3 and y → x2/x3 gives

ax21 + bx1x2 + cx2

2 + dx1x3 + ex2x3 + f x23 = 0

or in matrix form

x⊤Cx = 0

where

C =

!

"a b/2 d/2

b/2 c e/2d/2 e/2 f

#

$

The conic has five degress of freedom.

Five points define a conic. Each point (xi, yi) on the conic gives a constraint

ax2i + bxiyi + cy2

i + dxi + eyi + f = 0

which can be written as%x2

i xiyi y2i xi yi 1

&C = 0

Stacking the constraints we obtain

AC = 0

and the conic is the null vector of the matrix A. With five points, A is a 5 × 6 matrix.This shows that a conic is determined uniquely (up to scale) by five points.

Result 2.7. The line l tangent to C at a point x on C is given by l = Cx.To show that the point x is on l, we have

x⊤l = x⊤Cx = 0

To show that the line l is tangent to C, assume that l intersects with C at anotherpoint y (besides x). We have

y⊤l = y⊤Cx = 0

x⊤Cx = 0

Since C = C⊤, it follows that

(y + αx)⊤C(y + αx) = y⊤Cy + 2y⊤Cx + α2x⊤Cx = 0, ∀α

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Figure 2: Conic envelope: the lines l satisfying l⊤C∗l = 0.

This means the whole line l is on the conic C, and C is degenerate.

Dual Conics. The conic defined above is known as a point conic. By duality principle,there is a conic that defines an equation on lines, known as a dual conic, or line conic.

The dual conic is given by C∗, which is the adjoint matrix of C.From Result 2.7, we have l = Cx, so x = C−1l. Since x⊤Cx = 0, we have

(C−1l)⊤C(C−1l) = 0

which is

l⊤C−⊤

l = l⊤C∗l = 0

Dual conics are also known as conic envelopes, as illustrated in Figure 2.

Cofactor and adjoing matrix

Given a square matrix M, the cofactor matrix M∗ is given by

M∗ij = (−1)i+j det(M̂ij)

where M̂ij is the (i, j) minor of M. The adjoint matrix adj(M) is the transpose ofM∗. It is known that

adj(M) = det(M)M−1

Degenerate conics. Degenerate line conics include two lines (rank 2), and a repeatedline (rank 1). For example, C = lm⊤ + ml⊤. Degenerate point conics include twopoints (rank 2), and a repeated point (rank 1). For example, C = xy⊤ + yx⊤.

2.3 Projective transformations

Definition 2.9. A projectivity is an invertible mapping h from P2 to itself such thatthree points x1, x2, and x3 lie on the same line if and only if h(x1), h(x2), and h(x3) do.A projectivity is also called a collinearation, a projective transformation, or a homography.

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Theorem 2.10. A mapping h : P2 → P2 is a projectivity if and only if there exists anon-singular 3 × 3 matrix H such that for any point in P2 represented by a vector xit is true that h(x) = Hx.

Definition 2.11. Projective transformation. A planar projective transformation is alinear transformation on homogeneous 3-vector represented by a non-singular 3 × 3matrix:

x′ = Hx

Note that the matrix H is a homogeneous matrix, and it follows that the projectivetransformation has eight degrees of freedom.

2.3.1 Transformations of lines and conics

Transformation of lines. Under the point transformation x′ = Hx, we have

0 = l⊤x = l⊤H−1Hx = l⊤H−1Hx = (H−⊤l)⊤Hx

Hence a line transforms as

l′ = H−⊤l

Points transform according to H, whereas lines transform according to H−1. Onesays that points transform contravariantly and lines transform covariantly.

Transformation of conics. Under the point transform x′ = Hx, we have

0 = x⊤Cx = x′⊤H−⊤CH−1x′

Result 2.13. Under a point transformation x′ = Hx, a conic C transforms to C′ =H−⊤CH−1.

The presence of H−1 in this equation may be express by saying that a conic trans-forms covariantly.

Result 2.14. Under a point transform x′ = Hx, a dual conic C∗ transforms to C∗′ =HC∗H⊤.

2.4 A hierarchy of transformations

The projective transformations form a group, known as the projective linear group.The group of invertible n × n matrices with real elements is the (real) general

linear group on n dimensions, or GL(n).The projective linear group is identified by a scalar multiplier, giving PL(n) (this

is a quotient group of GL(n)).

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2.4.1 Class I: Isometries

Isometries are transformations of the plane R2 that preserve Euclidean distance, rep-resented as

'

(x′

y′

1

)

* =

!

"ε cos θ − sin θ txε sin θ cos θ ty

0 0 1

#

$

'

(xy1

)

*

where ε = ±1.If ε = 1, the isometry is orientation-preserving and is a Euclidean transformation.If ε = −1, the isometry is orientation-reversing.A planar Euclidean transformation can be written more precisely as

x′ = HEx =

+R t0⊤ 1

,x

A planar Euclidean transformation has three degrees of freedom and can be com-puted from two point correspondences.

2.4.2 Class II: Similarity transformations

A similarity transformation (or similarity) is an isometry composed with an isotropicscaling, also known as equi-form transformation. With no reflection, the similarityhas matrix representation

'

(x′

y′

1

)

* =

!

"s cos θ −s sin θ txs sin θ −s cos θ ty

0 0 1

#

$

'

(xy1

)

*

which can be written in block form as

x′ = HSx =

+sR t0⊤ 1

,x

A planar similarity has four degrees of freedom and can be computed from two pointcorrespondences.

Metric structure. The description metric structure implies that the structure is definedup to a similarity.

2.4.3 Class III: Affine transformation

An affine transformation (or affinity) is a non-singular linear transformation followedby a translation, given by

'

(x′

y′

1

)

* =

!

"a11 a12 txa21 a22 ty0 0 1

#

$

'

(xy1

)

*

6

or in block form

x′ = HAx =

+A t0⊤ 1

,x

A planar affinity has six degrees of freedom and can be computed from three pointcorrespondences.

With SVD, A can be decomposed as

A = UDV⊤ = UV⊤VDV⊤ = R(θ)R(−φ)DR(φ)

Hence the distortion of A can be seen as a rotation by R(θ) followed by a deformationR(−φ)DR(φ).

Invariants. Three important invariants are

◦ parallel lines

◦ ratio of lengths of parallel line segments

◦ ratio of areas

2.4.4 Class IV: Projective transformations

A projective transformation is given by a general non-singular linear transformationof homogeneous coordinates, represented as

'

(x′1x′2x′3

)

* =

!

"h11 h12 h13h21 h22 h23h31 h32 h33

#

$

'

(x1x2x3

)

*

or in the block form

x′ = HPx =

+A tv⊤ v

,x

A projective transformation has eight degrees of freedom and can be computed fromfour point correspondences.

Invariants. The most fundamental projective invariant is cross ratio.

2.4.5 Summary and comparison

For a given projective transformation, area scaling varies with position; and the ori-entation of a transformed line depends on both the orientation and position of thesource line.

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Group Invariant propertiesProjective (8 dof) Concurrency; collinearity; order of contact: intersection (1 pt);

tangency (2 pt); inflections (3 pt); tangent discontinuities andcusps; cross ratio

Alline (6 dof) Parallelism, ratio of areas, ratio of lengths on collinear or par-allel lines, linear combination of vectors. The line at infinity,l∞

Similarity (4 dof) Ratio of lengths, angle. The circular points, I, J.Euclidean (3 dof) Length, area.

Table 1: Geometric properties invariants to planar transformations. A transforma-tion lower in the table inherits the invariants of those above.

2.4.6 Decomposition of a projective transformation

A projective transformation can be decomposed as

H = HSHAHP =

+sR t0⊤ 1

, +K 00⊤ 1

, +I 0

v⊤ v

,=

+A tv⊤ v

,

where A = sRK+ tv⊤, and K an upper-triangular matrix normalized as det(K) = 1.

This decomposition is valid provided v ∕= 0 and s is unique if s is chosen positive.

2.4.7 The number of invariants

Result 2.16. The number of functionality independent invariants is equal to, orgreater than, the number of degrees of freedom of the configuration less the numberof degrees of freedom of the transformation.

The 2D transformation groups and their invariant properties are summarized in Ta-ble 1.

2.5 The projective geometry of 1D

We use x̄ to represent the homogeneous vector (x1, x2)⊤ and a projective transfor-

mation of a line is represented by

x̄′ = H2×2x̄

and has three degrees of freedom.

The cross ratio. Given four points x̄i, the cross ratio is defined as

Cross(x̄1, x̄2, x̄3, x̄4) =|x̄1x̄2||x̄3x̄4||x̄1x̄3||x̄2x̄4|

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Figure 3: Concurrent lines.

Figure 4: Removing perspective distortion.

where

|x̄ix̄j| = det+

xi1 xj1xi2 xj2

,

Concurrent lines. A configuration of concurrent lines is dual to collinear points ona line, as shown in Figure 3 .

2.6 Topology of the projective plane

There is a two-to-one correspondence between the unit sphere S2 in R3 and theprojective plane P2.

In the language of topology, the sphere S2 is a 2-sheeted covering space of P2.

Skipped: Page 47

Topology and orientation of P1 and P2.

2.7 Recovery of affine and metric properties from images

The goal is to remove projective distortion such that similarity properties (angles,ratios of lengths) could be measured on the original plane, as shown in Figure 4.

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The projective transformation can be completely removed by explicitly computingthe transformation given 4 reference points.

In the following, it is shown that the projective distortion may be removed once theimage of l∞ is specified, and the affine distortion removed once the image of thecircular points is specified. Then the remaining distortion is a similarity.

2.7.1 The line at infinity

Result 2.17. The line at infinity, l∞, is a fixed line under the projective transformationH if and only if H is an affinity.

2.7.2 Recovery of affine properties from images

Once the imaged line at infinity is identified in an image of a plane, we can transformthe identified l∞ to its canonical position (0, 0, 1)⊤. Affine measurements can bemade directly from the rectified image.

If the imaged line at affinity is l = (l1, l2, l3)⊤, then provided l3 ∕= 0, a suitableprojective point transformation which will map l back to l∞ is

H = HA

!

"1 0 00 1 0l1 l2 l3

#

$

since

H−⊤l =

!

"1 0 −l1/l30 1 −l2/l30 0 1/l3

#

$

'

(l1l2l3

)

* =

'

(001

)

*

Thus we have shown how to recover affine properties by specifying a line (2 dof).

Now we show that if affine properties are known, we can also determine the line atinfinity. More specifically, we can find the vanishing point using the cross ratio asfollows:

◦ Identify three collinear points a, b, c and the length ratio d(a, b) : d(b, c) = a : bis known.

◦ Measure the distance ratio in the image: d(a′, b′) : d(b′, c′) = a′ : b′.

◦ Represent points a, b, and c with homogeneous 2-vectors (0, 1)⊤, (a, 1)⊤, and(a + b, 1)⊤. Similarly, a′, b′, and c′ have coordinates 0, a′, and a′ + b′.

◦ Compute a 1D projective transformation H2×2 mapping a → a′, b → b′, andc → c′.

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ov/

b/c /

lba/

baa c

Figure 5: Determine the vanishing point given a known length ratio.

◦ The image of the point at infinity (1, 0)⊤ under H2×2 is the vanishing point onthe line 〈a′, b′, c′〉.

The vanishing point can also be computed from a purely geometric approach (inFigure 5:

◦ Given three collinear points a′, b′, and c′ corresponding to collinear worldpoints with interval ratio a : b.

◦ Draw a line through a′ and mark off points a = a′, b, and c from the ratio a : b.

◦ Join bb′ and cc′ and intersect at o.

◦ The line through o parallel to l meets the line a′c′ in the vanishing point v′.

2.7.3 The circular points and their dual

Under any similarity transformation, there are two points on l∞ which are fixed,which are the circular points (or absolute points) I and J with coordinates

I = (1, i, 0)⊤, J = (1,−i, 0)⊤

where

I′ = HSI =

!

"s cos θ −s sin θ txs sin θ s cos θ ty

0 0 1

#

$

'

(1i0

)

*

=

!

"s(cos θ − i sin θ)s(cos θ − i sin θ)i

0

#

$ = se−iθ

'

(1i0

)

*

Result 2.21. The circular points, I and J, are fixed points under the projective trans-formation H if and only if H is a similarity.

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The name “circular points” arises because every circle intersects I∞ at the circularpoints.

It will be shown that identifying the circular points (or their dual) allows the recoveryof similarity properties (angles, ratios of lengths).

The conic dual to the circular points. The conic

C∗∞ = IJ⊤ + JI⊤

is dual to the circular points.

Result 2.22. The dual conic C∗∞ is fixed under the projective transformation H if and

only if H is a similarity.

Two properties of C∗∞ in any projective frame:

◦ C∗∞ has 4 degrees of freedom (symmetric, det C∗

∞ = 0).

◦ l∞ is the null vector of C∗∞.

2.7.4 Angles on the projective plane

For the lines l and m, the angle between the two lines is

cos θ =l⊤C∗

∞m-(l⊤C∗

∞l)(m⊤C∗∞m)

Result 2.23. Once the conic C∗∞ is identified on the projective plane then Euclidean

angles may be measured by the equation above.In a Euclidean coordinate system, we have

cos θ =l⊤C∗

∞m-(l⊤C∗

∞l)(m⊤C∗∞m)

=l1m1 + l2m2

(l21 + l2

2)(m21 + m2

2)

Notice that

l⊤C∗∞m = (H−⊤l)⊤HC∗

∞H⊤(H−⊤m)

= l⊤H−1HC∗∞H⊤H−⊤m

= l⊤C∗∞m

Hence the equation for cos θ is invariant under point transformation H.

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Result 2.24. Lines l and m are orthogonal if l⊤C∗∞m = 0.

Geometrically, l and m are conjugate with respect to C∗∞.

Length ratios. Length ratios can also be measured once C∗∞ is identified:

d(b, c) : d(a, c) = sin α : sin β

2.7.5 Recovery of metric properties from images

Suppose the circular points are identified in an image, then the image is rectified bya projective transformation H that maps the imaged circular points to their canonicalposition on l∞.

The transformation between the world plane and the rectified image is then asimilarity.

Metric rectification using C∗∞. The dual conic C∗

∞ packages all the information re-quired for a metric rectification of both the projective and affine transformations.

Result 2.25. Once the conic C∗∞ is identified on the projective plane then projective

distortion may be rectified up to a similarity.Given a point transformation H, we have

C∗′∞ = (HPHAHS)C∗

∞(HPHAHS)⊤

= (HPHA)C∗∞(HAHP)

⊤

=

+K 0

v⊤K v

, +I 00 0

, +K⊤ K⊤v0⊤ v

,

=

+K 0

v⊤K 0

, +K⊤ K⊤v0⊤ v

,

=

+KK⊤ KK⊤v

v⊤KK⊤ v⊤KK⊤v

,

It is clear that the projective (v) and affine (K) are determined from the image ofC∗

∞.

A suitable rectifying homography may be obtained from the identified C∗∞ in an

image using the SVD:

C∗′∞ = UC∗

∞U⊤

Then the rectifying projectivity is H = U up to a similarity.

Example 2.26. Metric rectification I. Given an image that has been affinely rectified,we can determine a metric rectification by obtaining 2 degrees of freedom of the

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l x

y

C

Figure 6: The pole-polar relationships.

circular points from two imaged right angles on the world plane.

.l′1 l′2 l′3

/ +KK⊤ 00⊤ 0

,'

(m′

1m′

2m′

3

)

* = 0

so K can be determined up to scale by Cholesky decomposition.

Alternatively, the two constraints may be obtained from an imaged circle or twoknown length ratios.

Example 2.27. Metric rectificatin II. Given an original perspective image, we canobtain C∗

∞ from 5 constraints given by 5 pairs of orthogonal lines on the world plane.

Stratification. The two-step approach to remove affine and projective distortions istermed stratified.

2.8 More properties of conics

2.8.1 The pole-polar relationship

A point x and a conic C define a line l = Cx. The line l is called the polar of x withrespect to C, and the point x is the pole of l with respect to C. See Figure 6.

Definition 2.29. A correlation is an invertible mapping from points of P2 to lines ofP2. It is represented by a 3 × 3 non-singular matrix A as l = Ax.

Conjugate points. If the point y is on the line l = Cx then y⊤l = y⊤Cx = 0.If x is on the polar of y, then y is on the polar of x.

2.8.2 Classification of conics

Projective normal form for a conic. Since C is a symmetric matrix, it has real eigen-values and can be decomposed as a C = U⊤DU, where U is orthogonal and D isdiagonal. By applying a projective transformation U, we have C′ = D.

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Diagonal Equation Conic type(1, 1, 1) x2 + y2 + w2 = 0 Improper conics (no real points)(1, 1,−1) x2 + y2 − w2 = 0 Circle(1, 1, 0) x2 + y2 = 0 (0, 0, 1)⊤

(1,−1, 0) x2 − y2 = 0 x = ±y(1, 0, 0) x2 = 0 x = 0 counted twice

Table 2: Projective transformation of point conics.

e

ee

1

2

3

e

ee

1

2

3

x x /

H

Figure 7: Fixed points and lines of a plane projective transformation.

Further, D can be transformed to diag(ε1, ε2, ε3), where εi = ±1, 0. Various typesof conics may now be enumerated, as shown in Table 2.

Affine classification of conics. The classification of non-degenerate conics, hyper-bola, ellipse, and parabola depends on the relation of l∞ to the conic.

A conic is an (a) ellipse, (b) a parabola, or (c) a hyperbola according to whether it(a) has no real intersections, (b) is tangent to, or (c) has 2 real intersections with l∞.

2.9 Fixed points and lines

The key idea is that an eigenvector corresponds to a fixed point of the transformation,since

He = λe

Often the eigenvector and eigenvalue have physical or geometric significance incomputer vision applications.

A Euclidean matrix. The two ideal fixed points are the complex conjugate pair ofcircular points, I and J, with eigenvalues e±iθ. The third eigenvector, which has uniteigenvalue, is called the pole.

A similarity matrix. The two fixed points are again the circular points. The eigen-values are 1, se±iθ.

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An affine matrix. The two ideal fixed points can be real or complex conjugates, butthe fixed line l∞ through these points is real in either case.

2.10 Closure

2.10.1 Literature

2.10.2 Notes and exercises

Last Updated: Jun 01, 2020

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