Hmmm...I try to develop a pattern how a matrix of affine transformation is built it because I always forget how to relate two frames ,it's what in math is known as changing the basis right?

Lots of books try to present the subject in a intuitive manner but for me it's easier if I take a math approach.

Let's consider the frames at the link :

http://anorganik.uni-tuebingen.de/klaus ... r/rotationAs I said in my previous post

*To get the coordinate of a vector V in a coordinate space described by the versors (i,j,k)*

must be made the scalar product (V*i,V*j,V*j) .This is the pattern or rule which help me to build the transformation matrix .

LEt's see how the things going:

We may say that the coordinates of point P in the frame(X',Y') are (x',y') and (x,y) in the frame(X,Y) respectively .

The tranformation is (X',Y') - > (X,Y) so I want to get the matrix M which relates the 2 pairs of coordinates

(x',y') *M= (x,y);

Accordingly to what I said x=OP*i (*-scalar product) where coordinates of i in the frame (X',Y') are i(cos @ ,-sin @)

The same goes for y: y=OP*j where j(sin@,cos@) where i and j are the versors of axis X respectively Y.

So x = OP*i => x = x'cos@-y'sin@ and y = x'*sin@+y'*cos@ . To have this eq's in a matrix form the versors i and j must be arranged as columns so:

(x',y')| cos @ sin@ | = (x,y) that's the relation everyone know it .

|-sin @ cos@|

My source of confusion was the fact most books present the relation like this:

(x,y) | cos @ sin@ | = (x',y')

|-sin @ cos@|

but with the axis like in figure from the link above .

So in my mind I switched the coordinates(frame of reference) also and everything blew up