Lesson Transformations of coordinates

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In the following lesson we will deal with different kinds of the Transformation of axis and will deal with the issues related to them.

Transformations of coordinate

To transform from one coordinate system to another. Consider ( x, y ) & ( x%5B1%5D, y%5B1%5D ), these are coordinates of arbitrary point P in old and new coordinate systems correspondingly.

Translation of axes :

Let's move the coordinate system XOY in a plane so, that the axes OX and OY are parallel to themselves, and the origin of coordinates O moves to the point O%5B1%5D ( a, b ). We'll receive the new coordinate system X%5B1%5DO%5B1%5DY%5B1%5D



Now,

Pm=x and Pm%5B1%5D=x%5B1%5D

Pn=y and Pn%5B1%5D=y%5B1%5D

AO%5B1%5D=a and BO%5B1%5D=b

Coordinates of the point P in the new and old coordinate systems are tied by the equations:

Pm%5B1%5D=%28Pm-AO%5B1%5D%29

=>x%5B1%5D=%28x-a%29

and,

Pn%5B1%5D=%28Pn-BO%5B1%5D%29

=>y%5B1%5D=%28y-b%29

Another form of transformation is by rotating the axis about the origin or a fixed point.

Turning around origin of coordinates.

Let's turn the coordinate system XOY in a plane by an angle Theta in anticlockwise direction.



And,
Pm=x and Pm%5B1%5D=x%5B1%5D

Pn=y and Pn%5B1%5D=y%5B1%5D

Now coordinates of the point P in the new and old coordinate systems are tied by the equations:

x%5B1%5D=x%2AcosTheta+%2B+y%2AsinTheta ...................(1)

y%5B1%5D=+-+x%2AsinTheta+%2B+y%2AcosTheta ...................(2)

If the axis is rotated in the clockwise direction then the change coordinates will be obtained by replacing angle Theta by -Theta.

The change coordinates will be,

x%5B1%5D=x%2Acos%28-Theta%29+%2B+y%2Asin%28-Theta%29

or, x%5B1%5D=x%2AcosTheta+%2B%28-1%2Ay%2AsinTheta%29

or, x%5B1%5D=x%2AcosTheta+-+y%2AsinTheta ...............(3)

Now,

y%5B1%5D=+-+x%2AsinTheta+%2B+y%2AcosTheta

or, y%5B1%5D=-+x%2Asin%28-Theta%29+%2B+y%2Acos%28-Theta%29

or, y%5B1%5D=x%2AsinTheta+%2B+y%2AcosTheta ...............(4)

In the particular case when Theta++=+pi++ we'll receive a central symmetry relatively about the origin O :

Hence putting Theta++=+pi++ in equation 1 and 2, the coordinates will be,

x%5B1%5D=x%2Acos%28pi%29+-+y%2Asin%28pi%29

as cospi=-1 and sinpi=0

Hence, x%5B1%5D=-x

Similarly,

y%5B1%5D=x%2Asin%28pi%29+%2B+y%2Acos%28pi%29

or y%5B1%5D=-y

For a homothetic transformation with a center O (a,b) and a coefficient k%3C=0

The change coordinates will be,

x%5B1%5D-a+=+k%28x-a%29

y%5B1%5D-a+=+k%28y-b%29

For a Affine Transformation :

The change coordinates will be,

x%5B1%5D=ax%2Bby%2Bc

y%5B1%5D=dx%2Bey%2Bf

where, ae-bd%21=0
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