How can I rotate a coordinate system and write functions in the rotated system?

[Gackhammer]

Hey

So, I was wondering how to convert from one coordinate axes to another... in particular, where the new axes are y = x and y = -x, as seen by the picture below

I want it so that the Red dot in the new coordinate system will be ($\sqrt2$,$0$). Is there an easy way to do this? (My lookings on the internet have not come up with anything good yet)

EDIT: Ok, I just found the equations to rotate, but I was wondering how I can write functions in this rotated coordinate system (Im trying to relate this to my recurrences work/fixed point stuff). I am trying to rotate the coordinate system then find the zeros in the rotated coordinate system to find the fixed points of the function

 

[Vorde]

Find formulas to transform your old parameters (x & y) into your new ones, and then substitute one for the other in equations.

Are you sure a rotation will be enough for you? (A.K.A., what x,y point do you want to be mapped to your new point root(2),0 ?)

 

[HallsofIvy]

Vorde's point is that the distance from (0, 0) to (1, 0) is, of course, 1 while the distance from (0, 0) to your red point is $\sqrt{2}$. You will need an expansion as well as a rotation.

x'= x- y, y'= x+ y will map (1, 0) to (1, 1) and map (0, 1) to (-1, 1).
That is, essentially, a rotation by 45 degree, counter-clockwise and a multiplication by $\sqrt{2}$.

You could have gotten those relations more easily by seeing that they must be linear, of the form x'= ax+ by and y'= cx+dy, since you do not want one part of the plane "stretched" or "squeezed" more than another. Further, you want (1, 0) to be mapped to (1, 1) so 1= a(1)+ b(0) and 1= c(1)+ d(0) giving a= 1, c= 1. You also want (0, 1) mapped to (-1, 1) so -1= a(0)+ b(1) and 1= c(0)+ d(1) giving b= -1, d= 1: x'= x- y, y'= x+ y as I said.