Coordinate transformation parameterization

[throneoo]

Homework Statement

Suppose two observers O and O', whose positions coincide , each sets up a set of 2D cartesian coordinates (x,y) and (x',y') respectively to describe the position of a certain object at a fixed point . Derive a set of formulae for one observer to convert the other observer's coordinates into his own.

Homework Equations

The Attempt at a Solution

Assuming linearity ,
x'=ax+by
y'=cx+dy

since the distance between that object and either observer is the same ,

d^2=x^2+y^2=x'^2+y'^2

0=(a^2+c^2-1)x^2+(b^2+d^2-1)y^2+2xy(ab+cd)

then , since the formulae must hold for all (x,y) ,

a^2+c^2-1=0
b^2+d^2-1=0
ab+cd=0

the standard parameterization gives
a=cosA ; b=sinB ; c=sinA ; d=cosB

so sin(A+B)=0
which has distinct solutions A=-B and A+B=pi
the first solution is relatively easy to understand , as it just yields a standard rotation of coordinate axes .

however, with A=pi-B ,

I get

x'=-xcos(A)+ysinA

y'=xsinA+ycosA

When I try to picture it , it's a transformation where the original x-axis 'rotates' about the original y-axis by 180 degrees , and the whole coordinate system rotates about the common origin by A radian in the clockwise direction . This is where I get uncomfortable and unsure of whether I'm right

 

[mfb]

It is a mirrored coordinate system. You get it from the regular solution by the substitution x' -> -x'.

 

[throneoo]

thanks. that's a helpful insight