Lecture 1.

3
Basic projective geometry

Thomas Opsahl

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Motivation

• Projective geometry is an alternative to

Euclidean geometry

• Many results, derivations and expressions

in computer vision are easiest described in
the projective framework
– The perspective camera model Projective representation versus Euclidean

 fu s cu  1 0 0 0 
u =  0 fv cv  0 1 0 0  x
  
 0 0 1  0 0 1 0 

“Equal up to scale”

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Motivation

• Projective geometry is an alternative to

Euclidean geometry

• Many results, derivations and expressions

in computer vision are easiest described in
the projective framework
– The perspective camera model Projective representation versus Euclidean
 fu s cu 
1 0 0  
cv  x
1
u=  0 fv

0 1 0   0 0 z
 1 
 x y 
f
 uz + s + cu
z
u= 
 f +c y 
 v
z
v

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Motivation
2
−1
• Projective geometry is an alternative to R t 
Euclidean geometry x x =   y
 0 1

• Many results, derivations and expressions

in computer vision are easiest described in LINEAR
the projective framework
– The perspective camera model
– Transformations R t 
2
y y =   x
 0 1

Projective representation versus Euclidean

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Motivation
2
• Projective geometry is an alternative to
Euclidean geometry x =x R −1 ( y − t )

• Many results, derivations and expressions

in computer vision are easiest described in NON LINEAR
the projective framework
– The perspective camera model
– Transformations
2
y =
y Rx + t

Projective representation versus Euclidean

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Points in the projective plane ℙ2
How to describe points in the plane?

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Points in the projective plane ℙ2
How to describe points in the plane?

Euclidean plane ℝ2
• Choose a 2D coordinate frame
• Points have 2 unique coordinates
 x
=
x   ∈ 2
y x  y
x
2

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Points in the projective plane ℙ2
How to describe points in the plane?

Euclidean plane ℝ2
x • Choose a 2D coordinate frame
2 • Points have 2 unique coordinates
w
 x
=
x   ∈ 2
y x  y
x
Projective plane  2
2 • Expand coordinate frame to 3D
y • Points have 3 homogeneous coordinates
 x 
w = 1 x  y  ∈  2
=
x  
 w 
where
x λ x ∀λ ∈  \ {0}
=
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Points in the projective plane ℙ2
Observations
1. Any point 𝐱𝐱 = 𝑥𝑥, 𝑦𝑦 𝑇𝑇 in the Euclidean
plane has a corresponding
x homogeneous point 𝐱𝐱� = 𝑥𝑥, 𝑦𝑦, 1 𝑇𝑇 in the
2 projective plane
w

y 2. Homogeneous points of the form

x 𝑥𝑥�, 𝑦𝑦�, 0 𝑇𝑇 does not have counterparts in
x the Euclidean plane
2
y They correspond to points at infinity

w = 1
x

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Points in the projective plane ℙ2
Observations
3. When we work with geometrical
problems in the plane, we can swap
x between the Euclidean representation
2 and the projective representation
w
2 → 2
y x
 x
x  x  y
 y 
   
2  1 
y

2 → 2
w = 1
x  x   x 
 y    w 
   
 y 

 w
 w  10
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Lines in the projective plane ℙ2
How to describe lines in the plane?

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Lines in the projective plane ℙ2
How to describe lines in the plane?

Euclidean plane ℝ2
• 3 parameters 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 ∈ ℝ
𝑙𝑙 = 𝑥𝑥, 𝑦𝑦 | 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0

y
l
x
2

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Lines in the projective plane ℙ2
How to describe lines in the plane?

Euclidean plane ℝ2
• 3 parameters 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 ∈ ℝ
2 𝑙𝑙 = 𝑥𝑥, 𝑦𝑦 | 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0
w

y Projective plane  2
l • Homogeneous vector 𝐥𝐥̃ = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 𝑇𝑇
x 𝑙𝑙 = 𝐱𝐱� ∈  2 | ̃𝐥𝐥𝑇𝑇 𝐱𝐱� = 0
2
y

w = 1 l
x

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Lines in the projective plane ℙ2
Observations
1. Points and lines in the projective plane
have the same representation, we say
that points and lines are dual objects in
2 2
w

y 2. All lines in the Euclidean plane have a

l corresponding line in the projective
x plane
2
y 3. The line 𝐥𝐥̃ = 0,0,1 𝑇𝑇 in the projective
plane does not have an Euclidean
w = 1 l counterpart
x This line consists entirely of ideal
points, and is know as the line at infinity

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Lines in the projective plane ℙ2
Properties of lines in the projective plane
1. In the projective plane, all lines
intersect, parallel lines intersect at
infinity
2
w
Two lines 𝐥𝐥̃1 and 𝐥𝐥̃2 intersect in the point
y 𝐱𝐱� = 𝐥𝐥̃1 ×𝐥𝐥̃2
l
x 2. The line passing through points 𝐱𝐱�1 and
2 𝐱𝐱� 2 is given by
y 𝐥𝐥̃ = 𝐱𝐱�1 ×𝐱𝐱� 2

w = 1 l
x

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Example 1
x1 = ( 2, 4 )
x2 = ( 5,13)

Determine the line passing through the two points 𝐱𝐱 𝟏𝟏 and 𝐱𝐱 𝟐𝟐

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 Matrix representation of
the cross product
Example 1 u × v  u∧ v
x1 = ( 2, 4 )
x2 = ( 5,13) where
∧
Determine the line passing through the two points 𝐱𝐱 𝟏𝟏 and 𝐱𝐱 𝟐𝟐  u1  def  0 −u3 u2 
u 
= u 0 −u1 
 2  3 
Homogeneous representation of the points u3   −u2 u1 0 
2 5
x 1 = 4 ∈  2 x 2 = 13 ∈  2
   
1   1 
Homogeneous representation of line
 0 −1 4   5   −9   −3
l = x × x = x ∧ x =  1 0 −2  13 =  3  =  1 
1 2 1 2       
 −4 2 0   1   6   2 
Equation of the line
−3 x + y + 2 = 0 ⇔ y = 3 x − 2
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Example 2
y= x − 2
y= −2 x + 3

At which point does these two lines intersect?

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Example 2
y= x − 2
y= −2 x + 3

At which point does these two lines intersect?

1  −2 
y =x − 2  l1 = −1 ∈  2 y=−2 x + 3  l2 =− 1 ∈  2
   
 −2   3 

Point of intersection

 0 2 −1  −2   −5  −5 
 −3   1.67 
x =l1 × l2 =l1∧ l2 = −2 0 −1  −1 = 1  ⇒ =x  ≈ 
     −
  
1 0.33 
 1 1 0   3   −3  −3 

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Example 3
y= x − 2
y= x + 3

At which point does these two lines intersect?

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Example 3
y= x − 2 Euclidean geometry
y= x + 3 Parallel lines never intersect!

At which point does these two lines intersect?

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Example 3
y= x − 2 Euclidean geometry
y= x + 3 Parallel lines never intersect!

At which point does these two lines intersect?

1 1
y =x − 2  l1 = −1 ∈  2 y =x + 3  l2 = −1 ∈  2
   
 −2   3 

Point of intersection

 0 2 −1  1   −5 Projective geometry

x =l1 × l2 =l1∧ l2 = −2 0 −1  −1 = −5
All lines intersect!
    
 1 1 0   3   0  Parallel lines intersect at infinity

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Example 4
Cameras can observe points that are
“infinitely” far away
 x
 fu s cu  1 0 0 0   
u =  0 cv  0 1 0 0   
y
fv
  z
 0 0 1  0 0 1 0   
0

In images of planar surfaces we can see

how the surface converges towards a line

Any two parallel lines in the plane will

appear to intersect on this line
Image: Flicker.com (Melita)

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Example 4
Cameras can observe points that are
“infinitely” far away
 x
 fu s cu  1 0 0 0   
u =  0 cv  0 1 0 0   
y
fv
  z
 0 0 1  0 0 1 0    The line at infinity
0

In images of planar surfaces we can see

how the surface converges towards a line

Any two parallel lines in the plane will

appear to intersect on this line
Image: Flicker.com (Melita)

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Example 4

Different directions correspond to different

points at infinity

The set of all infinite points constitute the

line at infinity

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Linear transformations of the projective plane ℙ2

• A linear transformation of ℙ2 can be represented by a invertible homogeneous 3 × 3 matrix

H : 2 → 2
x  Hx
where
H λ H ∀λ ∈  \ {0}
=

• Important groups of linear projective transformations

Homographies
• Each group is closed under
– Matrix multiplication
Affine Similarities Euclidean
– Matrix inverse

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Linear transformations of the projective plane ℙ2

Transformation Matrix #DoF Preserves Visualization

Euclidean R t 3 Lengths
0T 1
+ all below

Similarity  sR t  4 Angles
 0T 1 s∈ + all below
 
Affine  a11 a12 a13  6 Parallelism, line at
a a22 a23  infinity
 21  + all below
 0 0 1 
Homography  h11 h12 h13  8 Straight lines
h h22 h23 
 21 
 h31 h32 h33 

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Linear transformations of the projective plane ℙ2

• Several image operations correspond to a linear projective transformation

– Rotation
– Translation
– Resizing

R t
H= T
0 1

t=0

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Linear transformations of the projective plane ℙ2

• Several image operations correspond to a linear projective transformation

– Rotation
– Translation
– Resizing

R t
H= T
0 1

t≠0

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Linear transformations of the projective plane ℙ2

• Several image operations correspond to a linear projective transformation

– Rotation
– Translation
– Resizing

 sR t 
H= T 
 0 1 

t=0
s <1
R=I

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Linear transformations of the projective plane ℙ2

• Perspective imaging of a flat surface can be described by a homography

Hx surface = u image surface

image u x
c
u 𝑥𝑥 𝑧𝑧

v
Flat surface with a 2D
𝑦𝑦 coordinate frame

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Linear transformations of the projective plane ℙ2

• The central projection between two planes corresponds to a homography

Hx 1 = y 2
1
2

c x
y
𝑥𝑥 𝑧𝑧

𝑦𝑦

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Linear transformations of the projective plane ℙ2

• For images of a flat surface, a homography can be used to «change» the camera position

http://www.robots.ox.ac.uk/~vgg/hzbook.html http://www.robots.ox.ac.uk/~vgg/hzbook.html

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The projective space ℙ3

• The relationship between the Euclidean space ℝ3 and the projective space 3 is much like
the relationship between ℝ2 and  2
– We represent points in homogeneous coordinates
 x   x 
 y   y 
x  =
=  λ   ∀λ ∈  \ {0}
 z   z 
   
 w  w
– Points at infinity have 𝑤𝑤
�=0 3 → 3 3 → 3
x  x x  x
– We can transform between ℝ3 and  3  x  x   x 
 x    y   w 
 y   y
      y 
z  z   w 
 z       z 
1     w 
w
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Linear transformations of the projective space ℙ3
Transformation of 𝟑𝟑 Matrix #DoF Preserves

Euclidean  R t 6 Volumes, volume ratios, lengths

 T  + all below
0 1
Similarity  sR t  7 Angles
 0T 1 s ∈  + all below
 
Affine  a11 a12 a13 a14  12 Parallelism of planes,
a a a23 a24  The plane at infinity
 21 22 
 a31 a32 a33 a34  + all below
 
0 0 0 1
Homography  h11 h12 h13 h14  15 Intersection and tangency of
h h h23 h24  surfaces in contact, straight lines
 21 22 
 h31 h32 h33 h34 
 
 h41 h42 h43 h44 

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Summary

• Projective plane ℙ2 and space ℙ3

– Alternative representation of points
– Homogeneous coordinates
– Can swap between ℝ𝑛𝑛 and ℙ𝑛𝑛

• Linear projective transformations

– Homogeneous matrices
– Several groups

2 → 2 2 → 2
x  x x  x
Homographies
 x   x   x
 y    w   x  y
  
   y   y  
Affine Similarities Euclidean  w   w   1 
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Further reading

• Do you want to know more?

• Online book by Richard Szeliski: Computer Vision: Algorithms and Applications

http://szeliski.org/Book/drafts/SzeliskiBook_20100903_draft.pdf
– Chapter 2 is about “image formation” and covers some projective geometry, focusing on
transformations, in section 2.1.1-2.1.4

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