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Math Help - Identify the rigid motion T o f o T^(-1)

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    Identify the rigid motion T o f o T^(-1)

    If f is any rigid motion and T is any translation, identify the rigid motion T o f o T^(-1).

    Can I split the f into three reflections since there is a theorem that says "Every rigid motion is the composition of at most three relfections and every rigid motion is a translation, a rotation, or a glide relfection"?

    My intuition tells me that the final answer would be f by itself since T o T^(-1) will be cancelled out ... I just guess.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rainyice View Post
    If f is any rigid motion and T is any translation, identify the rigid motion T o f o T^(-1).

    Can I split the f into three reflections since there is a theorem that says "Every rigid motion is the composition of at most three relfections and every rigid motion is a translation, a rotation, or a glide relfection"?

    My intuition tells me that the final answer would be f by itself since T o T^(-1) will be cancelled out ... I just guess.
    In what form can every translation be written?
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  3. #3
    Senior Member Pinkk's Avatar
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    I have the same exact question actually. I know that every translation is the composition of two reflections with the two axes of reflection being parallel to each other...
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    I have the same exact question actually. I know that every translation is the composition of two reflections with the two axes of reflection being parallel to each other...
    If you don't mind me asking, what class is this for?? Some of your questions suggest you're in an introductory analysis course...but....what's with all the isometry/rigid motion questions? I mean, a couple are a nice supplement to a discussion of mappings but the majority of your questions are these.
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    Senior Member Pinkk's Avatar
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    Introduction to Modern Geometry. First portion of the class is examining Euclid's Postulates and Hilbert's Axioms and classifying/identifying/composing/etc rigid motions.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    Introduction to Modern Geometry. First portion of the class is examining Euclid's Postulates and Hilbert's Axioms and classifying/identifying/composing/etc rigid motions.
    That's really cool!
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  7. #7
    Senior Member Pinkk's Avatar
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    I'm still completely stuck on this.
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    Quote Originally Posted by Pinkk View Post
    I'm still completely stuck on this.
    Do you agree with my intuition that the outcome will be f by itself? I viewed this like: go up one block, then go right one block, and then go down one block. The displacement is one block to the right.
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  9. #9
    Senior Member Pinkk's Avatar
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    That's only if f is another translation.
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    Quote Originally Posted by Drexel28 View Post
    If you don't mind me asking, what class is this for?? Some of your questions suggest you're in an introductory analysis course...but....what's with all the isometry/rigid motion questions? I mean, a couple are a nice supplement to a discussion of mappings but the majority of your questions are these.
    do you have any good suggustion to do this problem?
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