A question about rotation of a plane












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Suppose for any subspace $F$ of $mathbb{R}^d$(with the usual Euclidean norm), $pi_F$ denote the orthogonal projection onto $F$. Let $R$ be a rotation of $F$ and $F^prime = RF$. Prove that $forall p in mathbb{R}^d$,



$d(pi_F(p), pi_{F^prime}(p)) leq d(pi_F(p), Rpi_F(p))$.



This relation is used in this paper, equation 4.5.










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  • $begingroup$
    Are you sure that the inequality is correct? Take $P=(x,2x)$ in $mathbb{R}^2$, with $F$ the $x-$axis and $R$ the rotation of $pi/2$.
    $endgroup$
    – Emilio Novati
    Dec 2 '18 at 20:46
















4












$begingroup$


Suppose for any subspace $F$ of $mathbb{R}^d$(with the usual Euclidean norm), $pi_F$ denote the orthogonal projection onto $F$. Let $R$ be a rotation of $F$ and $F^prime = RF$. Prove that $forall p in mathbb{R}^d$,



$d(pi_F(p), pi_{F^prime}(p)) leq d(pi_F(p), Rpi_F(p))$.



This relation is used in this paper, equation 4.5.










share|cite|improve this question









$endgroup$












  • $begingroup$
    Are you sure that the inequality is correct? Take $P=(x,2x)$ in $mathbb{R}^2$, with $F$ the $x-$axis and $R$ the rotation of $pi/2$.
    $endgroup$
    – Emilio Novati
    Dec 2 '18 at 20:46














4












4








4


2



$begingroup$


Suppose for any subspace $F$ of $mathbb{R}^d$(with the usual Euclidean norm), $pi_F$ denote the orthogonal projection onto $F$. Let $R$ be a rotation of $F$ and $F^prime = RF$. Prove that $forall p in mathbb{R}^d$,



$d(pi_F(p), pi_{F^prime}(p)) leq d(pi_F(p), Rpi_F(p))$.



This relation is used in this paper, equation 4.5.










share|cite|improve this question









$endgroup$




Suppose for any subspace $F$ of $mathbb{R}^d$(with the usual Euclidean norm), $pi_F$ denote the orthogonal projection onto $F$. Let $R$ be a rotation of $F$ and $F^prime = RF$. Prove that $forall p in mathbb{R}^d$,



$d(pi_F(p), pi_{F^prime}(p)) leq d(pi_F(p), Rpi_F(p))$.



This relation is used in this paper, equation 4.5.







linear-algebra euclidean-geometry rotations






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asked Dec 2 '18 at 18:59









Sudipta RoySudipta Roy

29518




29518












  • $begingroup$
    Are you sure that the inequality is correct? Take $P=(x,2x)$ in $mathbb{R}^2$, with $F$ the $x-$axis and $R$ the rotation of $pi/2$.
    $endgroup$
    – Emilio Novati
    Dec 2 '18 at 20:46


















  • $begingroup$
    Are you sure that the inequality is correct? Take $P=(x,2x)$ in $mathbb{R}^2$, with $F$ the $x-$axis and $R$ the rotation of $pi/2$.
    $endgroup$
    – Emilio Novati
    Dec 2 '18 at 20:46
















$begingroup$
Are you sure that the inequality is correct? Take $P=(x,2x)$ in $mathbb{R}^2$, with $F$ the $x-$axis and $R$ the rotation of $pi/2$.
$endgroup$
– Emilio Novati
Dec 2 '18 at 20:46




$begingroup$
Are you sure that the inequality is correct? Take $P=(x,2x)$ in $mathbb{R}^2$, with $F$ the $x-$axis and $R$ the rotation of $pi/2$.
$endgroup$
– Emilio Novati
Dec 2 '18 at 20:46










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If I well understand the question the inequality is not true in general.



As suggested in my comment a simple counterexample can be done in $mathbb{R}^2$, as we can see in this figure.



enter image description here



Here the line $OC$ is the subspace $F$ and the line $OD$ is the subspace $F'=R_alpha F$. For a point $P$ we have the projections $E_1=pi_F(P)$ and $E_2=pi_{F'}(P)$, and the point $G=R_alpha(E_1)$.



We can see that $d(E_1E_2)=d(E_1Q)$ if $P$ is a point of the bisector of the angle $alpha$, otherwise we can have $d(E_1E_2)>d(E_1Q)$ or $d(E_1E_2)<d(E_1Q)$ depending on the position of $P$.






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    $begingroup$

    If I well understand the question the inequality is not true in general.



    As suggested in my comment a simple counterexample can be done in $mathbb{R}^2$, as we can see in this figure.



    enter image description here



    Here the line $OC$ is the subspace $F$ and the line $OD$ is the subspace $F'=R_alpha F$. For a point $P$ we have the projections $E_1=pi_F(P)$ and $E_2=pi_{F'}(P)$, and the point $G=R_alpha(E_1)$.



    We can see that $d(E_1E_2)=d(E_1Q)$ if $P$ is a point of the bisector of the angle $alpha$, otherwise we can have $d(E_1E_2)>d(E_1Q)$ or $d(E_1E_2)<d(E_1Q)$ depending on the position of $P$.






    share|cite|improve this answer









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      0












      $begingroup$

      If I well understand the question the inequality is not true in general.



      As suggested in my comment a simple counterexample can be done in $mathbb{R}^2$, as we can see in this figure.



      enter image description here



      Here the line $OC$ is the subspace $F$ and the line $OD$ is the subspace $F'=R_alpha F$. For a point $P$ we have the projections $E_1=pi_F(P)$ and $E_2=pi_{F'}(P)$, and the point $G=R_alpha(E_1)$.



      We can see that $d(E_1E_2)=d(E_1Q)$ if $P$ is a point of the bisector of the angle $alpha$, otherwise we can have $d(E_1E_2)>d(E_1Q)$ or $d(E_1E_2)<d(E_1Q)$ depending on the position of $P$.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        If I well understand the question the inequality is not true in general.



        As suggested in my comment a simple counterexample can be done in $mathbb{R}^2$, as we can see in this figure.



        enter image description here



        Here the line $OC$ is the subspace $F$ and the line $OD$ is the subspace $F'=R_alpha F$. For a point $P$ we have the projections $E_1=pi_F(P)$ and $E_2=pi_{F'}(P)$, and the point $G=R_alpha(E_1)$.



        We can see that $d(E_1E_2)=d(E_1Q)$ if $P$ is a point of the bisector of the angle $alpha$, otherwise we can have $d(E_1E_2)>d(E_1Q)$ or $d(E_1E_2)<d(E_1Q)$ depending on the position of $P$.






        share|cite|improve this answer









        $endgroup$



        If I well understand the question the inequality is not true in general.



        As suggested in my comment a simple counterexample can be done in $mathbb{R}^2$, as we can see in this figure.



        enter image description here



        Here the line $OC$ is the subspace $F$ and the line $OD$ is the subspace $F'=R_alpha F$. For a point $P$ we have the projections $E_1=pi_F(P)$ and $E_2=pi_{F'}(P)$, and the point $G=R_alpha(E_1)$.



        We can see that $d(E_1E_2)=d(E_1Q)$ if $P$ is a point of the bisector of the angle $alpha$, otherwise we can have $d(E_1E_2)>d(E_1Q)$ or $d(E_1E_2)<d(E_1Q)$ depending on the position of $P$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 14 '18 at 13:46









        Emilio NovatiEmilio Novati

        51.6k43473




        51.6k43473






























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