"Show that a map F:R^3->R^3 is an isometry if and only if there is an orthogonal matrix M (3 by 3 real matrix) and a vector v in R^3 such that F(x)=Mx +v. Prove also that arc length, curvature and torsion are preserved under isometries."
I realy dont know how to start...could you help me starting?
An Isometry F is such that given y1=Fx1, y2=Fx2, then d(y1,y2) = d(x1,x2) where d() is the metric on your space. The usual Euclidean metric if not otherwise specified.
Originally Posted by Pipita
so suppose Fx = Mx + v, i.e. a potential scaling/rotation and a translation. It should be pretty clear that the translation has no effect on the length. So you just have to show what properties M must have to preserve lengths.
In the other direction suppose you have an isometry, then d(y1,y2)=d(x1,x2) and you can work out from there what linear transformations satisfy this and thus derive M and v.
The rest of it is just chasing the transformation through the process of deriving arc length, curvature, torsion.
Originally Posted by romsek
yeah i know the definition of isometry but i really dont know how to start this proof...it seems it misses some info
You are not giving a lot of information here. What "seems it misses some info"? Are you looking for a description for how to set up a proof? That is simple. You are asked to prove an if and only if statement. So, you assume that F(x) = Mx+v for some orthogonal matrix M and some vector v. Prove that F is an isometry. Next, suppose F is an isometry, and prove the existence of M and v. What exactly is the "missed info"?