Transformation of Coordinates Involving Pure Translation

{

x = x' + x0 y = y' + y0

or

{

x' = x - x0 y' = y - y0

where (x, y) are old coordinates [i.e. coordinates relative to xy system], (x',y') are new coordinates [relative to x'y' system] and (x₀, y₀) are the coordinates of the new origin 0' relative to the old xy coordinate system.

Transformation of Coordinates Involving Rotation

{

x = x' cosα - y' sinα y = x' sinα + y' cosα

or

{

x' = x cosα + y sinα y' = y cosα - x sinα

where the origins of the old [xy] and new [x'y'] coordinate systems are the same but the x' axis makes an angle α with the positive x axis.

Transformation of Coordinates Involving Translation and Rotation

{

x = x' cosα - y' sinα + x0 y = x' sinα + y' cosα + y0

or

{

x' = (x - x0)cosα + (y - y0)sinα y' = (y – y0)cosα - (x - x0)sinα

where the new origin O' of x'y' coordinate system has coordinates (x₀, y₀) relative to the old xy coordinate system and the x' axis makes an angle α with the positive x axis.

Polar Coordinates(r, θ)

A point P can be located by rectangular coordinates (x, y) or polar coordinates (r, θ). The transformation between these coordinates

  

    x = r cosθ

    y = r sinθ

   

 or   

 

     r = Ö(x² + y²)

     θ = tan^-1(y/x)