Transformation of coordinates (Projective; Affine; Metric)

• Cartesian transformation of coordinates
  
• Homogeneous coordinates - transformation
  
• Translation of the coordinate system.
  
• Rotation of a orthonormal coordinate system.
  
• Equation of a line after a coordinate transformation
  
• Projective, Affine, Metric transformations of coordinates
  
• Transformation of coordinates compared with Point transformations
  
  • Example of a point transformation
    
• Properties of transformations of points
  
  • Projective properties
    
  • Affine properties
    
  • Metric properties
    
  • Metric geometry
    
  • Affine geometry
    
  • Projective geometry
    
• Projective theorems - examples
  
  • Theorem of Ceva for concurrent lines
    
  • Theorem of Menelaus for collinear points
    
  • Theorem of Pappus-Pascal
    
  • Theorem of Desargues
    
• Affine theorems - examples
  
  • Menelaus's theorem in the affine plane
    
  • Theorem of Ceva in the affine plane
    
  • Concurrent lines in a triangle
    

Cartesian transformation of coordinates

 
        OP = OO' + O'P
<=>
        P = O' + x' O'E1' + y' O'E2'
<=>
        P = O' + x'(E1' - O') + y' (E2' - O')
<=>
        with coordinates this becomes

        (x,y) = (xo,yo) + x'((a1,b1) - (xo,yo)) + y'((a2,b2) - (xo,yo))
<=>
        x = xo + (a1 - xo)x' + (a2 - xo)y'
        y = yo + (b1 - yo)x' + (b2 - yo)y'

        with matrix notation this becomes
<=>
        [x]   [(a1 - xo)  (a2 - xo)] [x']   [xo]
            =                             +
        [y]   [(b1 - yo)  (b2 - yo)] [y']   [yo]

Homogeneous coordinates - transformation

• P is a regular point
  From previous formulas we can write

   
          [x/z]   [(a1 - xo)  (a2 - xo)] [x'/z']   [xo]
                =                                +
          [y/z]   [(b1 - yo)  (b2 - yo)] [y'/z']   [yo]
  
  <=>
          [x/z]   [(a1 - xo)  (a2 - xo)   xo] [x'/z']
          [y/z] = [(b1 - yo)  (b2 - yo)]  yo].[y'/z']
          [ 1 ]   [    0          0       1] [  1  ]
  
  <=>
          [ x ]   [(a1 - xo)  (a2 - xo)   xo] [ x' ]
          [ y ] = [(b1 - yo)  (b2 - yo)]  yo].[ y' ]
          [ z ]   [    0          0       1] [ z' ]
  
                  [(a1 - xo)  (a2 - xo)   xo]
  Denote M =      [(b1 - yo)  (b2 - yo)]  yo]   then
                  [    0          0       1]
  
          [ x ]     [ x' ]
          [ y ] = M.[ y' ]
          [ z ]     [ z' ]
  

  The 3x3 matrix M is called the transformation matrix.
  ((a₁ - x_o), (b₁ - y_o)) are the cartesian coordinates of E1' in the old coordinate system.
  ((a₂ - x_o), (b₂ - y_o)) are the cartesian coordinates of E2' in the old coordinate system.
  Since these two vectors are linear independent the determinant of the transformation matrix M is not zero and so the transformation matrix M is not singular.
  
• P is a ideal point
  It can be proved that the same formulas hold.

Translation of the coordinate system.

 
        [1   0   xo]
    M = [0   1   yo]
        [0   0    1]

Rotation of a orthonormal coordinate system.

 
        [cos(t)   -sin(t)   0]
    M = [sin(t)    cos(t)   0]
        [  0         0      1]

Equation of a line after a coordinate transformation

 
        line a has equation u x + v y + w z = 0

 
                  [x]
        [u  v  w].[y] = 0
                  [z]

 
                      [x']
        [u  v  w]. M. [y'] = 0
                      [z']

 
                   [x']
        [u' v' w'].[y'] = 0
                   [z']
<=>

        u' x' + v' y' + w' z'= 0

 
        [u  v  w]. M = [u'   v'   w']

Projective, Affine, Metric transformations of coordinates

Then all the points have new coordinates.
Such a linear permutation is called a projective transformation of coordinates.
It can be proved that the transformation formulas can be written in the form

 
        [x]     [a  b  c] [x']
        [y] =   [d  e  f].[y']
        [z]     [g  h  i] [z']

If we take out of this set of transformations, just the ones with the property that z = 0 is invariant for the transformation, then we say that the transformation is an affine transformation of coordinates.
In this case, the special points with homogeneous coordinates (x,y,0) get new coordinates with the same property. These points are the ideal points of the affine plane.
The formulas are

 
        [x]     [a  b  c] [x']
        [y] =   [d  e  f].[y']
        [z]     [0  0  i] [z']

If we take out of this set of affine transformations, just the ones that allows an invariant formula for the distance of two points, then we say that the transformations are metric transformation of coordinates..
It can be proved that each metric transformation is the composition of a finite number of translations, rotations and reflections in a line through the origin.

The metric transformations of coordinates are a subset of the affine transformations of coordinates.
The affine transformations of coordinates are a subset of the projective transformations of coordinates.

Transformation of coordinates compared with Point transformations

But you can also see it differently.

We transform de points into other points with a transformation T ^-1 and we don't change the coordinate system. Such transformation is a point transformation.

Example:

If we rotate the coordinate system, about point O, by an angle t then the fixed point P(x,y,z) get different coordinates P(x',y',z').

 
        [ x ]     [ x' ]
        [ y ] = M.[ y' ]
        [ z ]     [ z' ]

with
        [cos(t)   -sin(t)   0]
    M = [sin(t)    cos(t)   0]
        [  0         0      1]

 
        [ x ]       [ x' ]
        [ y ] = M-1.[ y' ]
        [ z ]       [ z' ]
with
        [cos(t)   -sin(t)   0]
    M = [sin(t)    cos(t)   0]
        [  0         0      1]

 
        [ x' ]       [ x ]
        [ y' ] =  M  [ y ]
        [ z' ]       [ z ]
met
        [cos(t)   -sin(t)   0]
    M = [sin(t)    cos(t)   0]
        [  0         0      1]

The transformation of coordinates X = M X' is equivalent with the point transformation X' = M X.

Example of a point transformation

We rotate point P by an angle t.

 
   [x']   [cos(t)   -sin(t)   0] [2]
   [y'] = [sin(t)    cos(t)   0] [3]
   [1 ]   [  0         0      1] [1]
<=>
    x' = 2 cos(t) - 3 sin(t)
    y' = 2 sin(t) + 3 cos(t)

  The image point Q is on 2 x + 3 y + 1 = 0
<=>
   4 cos(t) - 6 sin(t) + 6 sin(t) + 9 cos(t) +1 = 0
<=>
   cos(t) = -1/13
<=>
   cos(t) = cos(1.65)
<=>
   t = ± 1.65 + 2 k pi

Properties of transformations of points

Now, we take the formulas

 
        [x']      [x]
        [y'] = M .[y]   with M = a regular 3 x 3 matrix.
        [z']      [z]

Projective properties

Affine properties

 
                [a  b  c]
                [d  e  f]
                [0  0  i]

Metric properties

The projective properties are a subset of the affine properties.
The affine properties are a subset of the metric properties.

Metric geometry

By metric axes is meant an orthonormal basis in the plane.
In this case, two points (0,0,1) and (1,0,1) can be chosen arbitrarily. With these points, the coordinate system is completely determined.

Affine geometry

By affine axes is meant a general basis in the plane.
In this case, three points (0,0,1) (1,0,1) and (0,1,1) can be chosen arbitrarily but not collinear. With these points, the coordinate system is completely determined.

Projective geometry

By projective axes is meant that we can choose four points.
First three non-collinear points: (0,0,1) (0,1,0) and (1,0,0).
These points are the base points of the base triangle. Then we choose arbitrarily a unit point (1,1,1) not on the sides of the base triangle. With these points, the coordinate system is completely determined.

By choosing the axis in a smart way, many problems become easy to solve.

Because all this seems strange without examples, we'll give now four examples of a projective theorem and then three examples of a affine theorem.

Projective theorems - examples

Theorem of Ceva for concurrent lines

We'll search for the condition in order that l₁, l₂ and l₃ are concurrent.

In order to have a simple solution, we choose
A as point (1,0,0) ; B as point (0,1,0) ; C as point (0,0,1)
Then line a has equation x = 0; b has equation y = 0; c has equation z = 0.
A variable line l₁ through point C has equation x + k y = 0.
A variable line l₂ through point A has equation y + l z = 0.
A variable line l₃ through point B has equation z + m x = 0.
k,l and m are non-homogeneous parameters.
Well, l₁,l₂ and l₃ are concurrent if and only if

 
        | 1  k  0 |
        | 0  1  l | = 0
        | m  0  1 |
<=>
        1 + k l m = 0
<=>
        k l m = -1

Theorem of Menelaus for collinear points

We'll search for the condition in order that L₁, L₂ and L₃ are collinear.

In order to have a simple solution, we choose
A as point (1,0,0) ; B as point (0,1,0) ; C as point (0,0,1)
Then, L₁(1,k,0) L₂(0,1,l) L₃(m,0,1)
Well,

 
        L1, L2 and L3 are collinear
<=>
        | 1  k  0 |
        | 0  1  l | = 0
        | m  0  1 |
<=>
        1 + k l m = 0
<=>
        k l m = -1

Theorem of Pappus-Pascal

Proof:
Denote S as the intersection point of d1 and d2.
We choose:

 
        S(1,0,0) ; A4(0,1,0) ; A1(0,0,1)
Then
        d1: y = 0  and d2: z = 0

        A5(1,0,l)  A3(1,0,l')  A2(1,m,0)  A6(1,m',0)

        A1A2: m x - y = 0
        A4A5: l x - z = 0

                =>      P(1,m,l)

        A1A6: m' x - y = 0
        A3A4: l' x - z = 0

                =>      R(1,m',l')

        A5A6: -l m' x + l y + m' z = 0
        A2A3: -l'm  x + l'y + m  z = 0

                =>      Q(lm-l'm' , lmm' - l'mm' , ll'm - ll'm')

And now P,Q,R are collinear because

        |  1            m               l       |
        |  1            m'              l'      |  = 0
        |lm-l'm'    lmm' - l'mm'   ll'm - ll'm' |

Theorem of Desargues

Proof:

Choose: A(1,0,0) ; B(0,1,0) ; C(0,0,1) ; S(1,1,1)
Then:
A' on line SA => A'(1+l,1,1)
B' on line BS => B'(1,1+m,1)
C' on line SC => C'(1,1,1+n)

K is the intersection point of BC and B'C'.
Line BC has equation x = 0. So, the first coordinate of K is 0.
Since K is on B'C', the coordinates of K are a linear combination of (1,1+m,1) and (1,1,1+n). Since the first coordinate of K is 0, coordinates of K are (0,m,-n).

Similarly, we find L(l,0,-n) and M(l,-m,0).
And now K,L,M are collinear because

 
        | 0    m   -n |
        | l    0   -n | = 0
        | l   -m    0 |

Affine theorems - examples

Menelaus's theorem in the affine plane

We'll prove the following property about dividing ratios.
(A B L₁).(B C L₂).(C A L₃) = 1 <=> L₁, L₂, L₃ are collinear

Proof:
There are only affine properties in this problem, so we can choose the three points with simple coordinates.
Choose A(0,0,1) ; B(1,0,1) ; C(0,1,1)
Denote the dividing ratios: (A B L₁) = k ; (B C L₂) = l ; (C A L₃) = m
Then, the homogeneous coordinates of L₁, L₂ and L₃ are
L₁(-k,0,1-k) ; L₂(1,-l,1-l) ; L₃(0,1,1-m)

 
        L1, L2, L3 are collinear
<=>
        | -k   0   1-k |
        |  1   -l  1-l | = 0
        |  0   1   1-m |
<=>
        ...
<=>
        k l m = 1

Theorem of Ceva in the affine plane

We'll prove the following property about dividing ratios.
(A B L₁).(B C L₂).(C A L₃) = - 1 <=> l₁, l₂, l₃ are concurrent.

Proof:
There are only affine properties in this problem, so we can choose the three points with simple coordinates.
Choose A(0,0,1) ; B(1,0,1) ; C(0,1,1)
Denote the dividing ratios: (A B L₁) = k ; (B C L₂) = l ; (C A L₃) = m
Then, the homogeneous coordinates of the points L₁, L₂ and L₃ are
L₁(-k,0,1-k) ; L₂(1,-l,1-l) ; L₃(0,1,1-m)
Calculating the homogeneous coordinates of the lines l₁, l₂ and l₃, you'll find

 
        l1 (1-k , -k, k )
        l2 ( l  , 1 , 0 )
        l3 (-1  ,m-1, 1 )

        l1, l2, l3 are concurrent
<=>
        | 1-k    -k    k |
        |  l      1    0 | = 0
        | -1    m-1    1 |
<=>
        ...
<=>
        klm = -1

Concurrent lines in a triangle

Proof:
Say that the median line from A hits BC in point A'.
Since B'C' is parallel to BC we have

 
        B'A   C'A
        --- = ---
        B'B   C'C

<=>
        B'A   C'C
        --- . --- = 1
        B'B   C'A
<=>
        (A B B').(C A C') = 1

 
        A'B
        --- = -1
        A'C
<=>

        (B C A') = -1

 
        (A B B').(B C A').(C A C') = -1