Transformation Of Coordinates

Before discussing the transformation of coordinates we will study what transformation is?
Moving the shape to a different Position but the shape, size and area, line length and angle of that shape is same than this process is known as transformation, so when we move this shape then its coordinate also gets changed and this change is called transformation.
Now we will see coordinate transformation process.
We will study coordinate Transformations by Cartesian method.
A Cartesian coordinates system is totally defined by its origin and vectors along x-axis and y-axis.
Suppose we take first origin ‘O’ and unit vector OA1 and OA2 and we take a Point ‘M’ which has coordinates (r, s) relative to that coordinate system.
Then we have new coordinates with the origin O’ and unit vectors OA1 and OA2 and point ‘M’ has coordinates (r’, s’) which is related to the new coordinate system.
Then the transformation formula between (r, s) and (r’, s’).
OM= OO' + O'M
=> M = O’ + r’ O'M1' + s' O'M2'
=> M = O' + r'(M1' - O') + s' (P2' - O')
=> With coordinates this becomes
        (r, s) = (ro, so) + r'((a1, b1) - (ro, so)) + s'((a2, b2) - (ro, so))
=> r = so + (a1 - ro) r' + (a2 - ro) s'
      s = so + (b1 - so) r' + (b2 - so) s'
 With matrix notation this becomes
        [r]   [(a1 - ro) (a2 - ro)] [r']   [ro]
        =   +[s]   [(b1 - so) (b2 - so)] [s']   [so]
Now we will see properties of transformation.
Some of the properties of transformation are given below.
1.    Projective properties;
2.    Affine properties;
3.    Metric properties;
4.    Affine Geometry;
5.    Projective geometry.
Some theorem is also defined for the transformation which are:
Theorem of ceva for concurrent lines;
Theorem of Menelaus for Collinear Points;
Theorem of pappus – pascal;
And theorem of Desargues;
These all are the theorem of transformations.