Relative Coordinates of a point: I see it there !! Where can you see it?

It’s war!!!
‘O’ is an observer observing the target ‘T’.
‘G’ is the gun station where the artillery guns are located.

MAP

As the Map illustrates,

• ‘O‘ and ‘G‘ are facing α and β angels from north.
• According to the global coordinate system, ‘O‘ is at xo,yo and ‘G‘ is at xg,yg.
• ‘O’ uses metres in his coordinate system while ‘G’ uses decimetres.
• According to ‘O‘, Target ‘T‘ is at X_t,Y_t.

In order to fire the artillery guns ‘G‘ must convert the given target coordinate to relative to his own coordinate system.

To see in ‘G‘s angel ‘O‘ must rotate α – β anti-clock wise which causes the target to rotate α – β clock wise. By applying rotation to ‘T‘

To see in ‘G‘s position ‘O‘ must go to ‘G‘ which causes reverse translation to the target. By applying translation to the target

Because ‘O‘ is using a different unit system converting the units we get ‘T_G‘ the ‘T‘ s coordinate according to ‘G‘