Results 1 to 4 of 4

Math Help - isometry

  1. #1
    Newbie
    Joined
    Jan 2013
    From
    Portugal
    Posts
    4

    isometry

    Exercice:
    "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?

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,761
    Thanks
    1137

    Re: isometry

    Quote Originally Posted by Pipita View Post
    Exercice:
    "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?

    Thanks!
    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.

    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2013
    From
    Portugal
    Posts
    4

    Re: isometry

    Quote Originally Posted by romsek View Post
    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.

    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.

    yeah i know the definition of isometry but i really dont know how to start this proof...it seems it misses some info
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,932
    Thanks
    782

    Re: isometry

    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"?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. isometry
    Posted in the Geometry Forum
    Replies: 1
    Last Post: July 20th 2011, 12:49 AM
  2. Symmetry vs Isometry
    Posted in the Algebra Forum
    Replies: 1
    Last Post: December 9th 2008, 10:53 PM
  3. Opposite Isometry
    Posted in the Geometry Forum
    Replies: 2
    Last Post: November 19th 2008, 04:36 PM
  4. Isometry
    Posted in the Geometry Forum
    Replies: 6
    Last Post: January 23rd 2007, 06:27 PM
  5. Isometry of line
    Posted in the Geometry Forum
    Replies: 0
    Last Post: August 13th 2006, 04:08 PM

Search Tags


/mathhelpforum @mathhelpforum