# Euclidean Isometries - Help Please!

• Dec 13th 2009, 01:58 PM
ReneePatt
Need examples of Euclidean isometries f and g such that $f\circ{g}$ $g\circ{f}$.
• Dec 13th 2009, 02:45 PM
Laurent
Quote:

Originally Posted by ReneePatt
Need examples of Euclidean isometries f and g such that $f\circ{g}$ $g\circ{f}$.

There are so many examples. For instance, look at what happens when you compose two rotations with different centers. Or two symmetries with respect to non-orthogonal lines.
• Dec 13th 2009, 03:02 PM
ReneePatt
Quote:

Originally Posted by Laurent
There are so many examples. For instance, look at what happens when you compose two rotations with different centers. Or two symmetries with respect to non-orthogonal lines.

That's my problem. I have no idea how to do this and the book doesn't tell me. I've been searching the internet for examples so that I can better understand. I'm a distant learning student so I have to pretty much teach myself all this stuff.
• Dec 13th 2009, 08:55 PM
Krahl
let A be the matrix [cos90 -sin90|sin90 cos90] a rotation of 90. let X=(x,y) vector.
let f(X)=AX so the action of f on X rotates it by 90.
and let g be a rotation with respect to the point (1,0) not the origin.
so what g does is
g(X)=A(x-1,y)+(1,0) (i.e. take 1 away from x then rotate w.r.t origin then replace x)

so (fog)(X)=f(g(x))=A(A(x-1,y)+(1,0))
and (gof)(X)=g(f(x))=A(AX-(1,0))+(1,0)

picking point (2,1)
fog(X)=(-1,0)
gof(X)=(-1,-2)
so check by putting values for the vector X and seeing these two arent always equal.
((x-1,y) and similar expressions are vectors)