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   
θ = tan-1(y/x)


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