General form for the rotation of a function.how to perform a rotation around a point which itself is rotating?A problem with definitions of rotation/reflection matrix/operatorWhat is the rotation axis and rotation angle of the composition of two rotation matrix in $mathbbR^3$Rotation Matrices - Rotating a point on a graphFor a general plane, what is the parametric equation for a circle laying in the planeShow that T is a rotation through an angle, and find the angleDetermining Rotation MatricesMatrix Representation of Rotation in $mathbbR^3$Let T1 be the linear transformation corresponding to a counterclockwise rotation of 120 degreesTransformation matrix for rotation about arbitrary axis

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General form for the rotation of a function.


how to perform a rotation around a point which itself is rotating?A problem with definitions of rotation/reflection matrix/operatorWhat is the rotation axis and rotation angle of the composition of two rotation matrix in $mathbbR^3$Rotation Matrices - Rotating a point on a graphFor a general plane, what is the parametric equation for a circle laying in the planeShow that T is a rotation through an angle, and find the angleDetermining Rotation MatricesMatrix Representation of Rotation in $mathbbR^3$Let T1 be the linear transformation corresponding to a counterclockwise rotation of 120 degreesTransformation matrix for rotation about arbitrary axis













3












$begingroup$


When rotating linear functions, I would approach the task geometrically (find invariant point etc.), yet I tried using a matrix which worked nicely.



This was what I did to rotate $y=2x+1$ by $fracpi4$ about the origin.



$$beginpmatrix
frac1sqrt2 &frac-1sqrt2 \
frac1sqrt2&frac1sqrt2
endpmatrix
beginpmatrix
x\ 2x+1endpmatrix
=
frac1sqrt2beginpmatrix
1-x\ 3x+1
endpmatrix$$



$$x'=frac1sqrt2(1-x); ,; y'=frac1sqrt2(3x+1)$$



So
$$y'=2sqrt2-3x'$$



I have also tried this on non-linear functions and it worked.



When I generalise this I obtain:



$$x'=xcos(theta )-ysin(theta); ,; y'=xcos(theta )+ysin(theta)$$



The problem is I can only seem to eliminate one of either $x$ or $y$. Is it possible to eliminate both?



Other related questions I have are:
Can you use matrices to rotate a function around an arbitrary point, not the origin?
Can this concept by applied to 3-dimensions? What about parametric functions?
Can anyone recommend any material for pursing matrix transformations of functions (possibly of more than one variable)?



For reference I am currently a secondary school student in the UK.



Thanks










share|cite|improve this question











$endgroup$




bumped to the homepage by Community yesterday


This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.














  • $begingroup$
    In regards to your question about rotating about a point other than the origin, you most likely will need to shift point of interest to origin, rotate, and shift back to point of interest.
    $endgroup$
    – jameselmore
    Dec 19 '14 at 17:45










  • $begingroup$
    Do you mean a substitution to shift the axes?
    $endgroup$
    – Gridley Quayle
    Dec 19 '14 at 17:48










  • $begingroup$
    Precisely! Could be simpler than rotating everything, tracking your now rotated point of interest and shifting back
    $endgroup$
    – jameselmore
    Dec 19 '14 at 18:08











  • $begingroup$
    So to rotate around $(a,b)$ I would use the substitutions $x=u-a$ and $y=v-b$?
    $endgroup$
    – Gridley Quayle
    Dec 19 '14 at 18:15










  • $begingroup$
    Correct you would make the following subs: $x=u-a$ and $y=v-b$, perform the rotation in the way you mention above (which is correct) and then shift back
    $endgroup$
    – jameselmore
    Dec 19 '14 at 18:19
















3












$begingroup$


When rotating linear functions, I would approach the task geometrically (find invariant point etc.), yet I tried using a matrix which worked nicely.



This was what I did to rotate $y=2x+1$ by $fracpi4$ about the origin.



$$beginpmatrix
frac1sqrt2 &frac-1sqrt2 \
frac1sqrt2&frac1sqrt2
endpmatrix
beginpmatrix
x\ 2x+1endpmatrix
=
frac1sqrt2beginpmatrix
1-x\ 3x+1
endpmatrix$$



$$x'=frac1sqrt2(1-x); ,; y'=frac1sqrt2(3x+1)$$



So
$$y'=2sqrt2-3x'$$



I have also tried this on non-linear functions and it worked.



When I generalise this I obtain:



$$x'=xcos(theta )-ysin(theta); ,; y'=xcos(theta )+ysin(theta)$$



The problem is I can only seem to eliminate one of either $x$ or $y$. Is it possible to eliminate both?



Other related questions I have are:
Can you use matrices to rotate a function around an arbitrary point, not the origin?
Can this concept by applied to 3-dimensions? What about parametric functions?
Can anyone recommend any material for pursing matrix transformations of functions (possibly of more than one variable)?



For reference I am currently a secondary school student in the UK.



Thanks










share|cite|improve this question











$endgroup$




bumped to the homepage by Community yesterday


This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.














  • $begingroup$
    In regards to your question about rotating about a point other than the origin, you most likely will need to shift point of interest to origin, rotate, and shift back to point of interest.
    $endgroup$
    – jameselmore
    Dec 19 '14 at 17:45










  • $begingroup$
    Do you mean a substitution to shift the axes?
    $endgroup$
    – Gridley Quayle
    Dec 19 '14 at 17:48










  • $begingroup$
    Precisely! Could be simpler than rotating everything, tracking your now rotated point of interest and shifting back
    $endgroup$
    – jameselmore
    Dec 19 '14 at 18:08











  • $begingroup$
    So to rotate around $(a,b)$ I would use the substitutions $x=u-a$ and $y=v-b$?
    $endgroup$
    – Gridley Quayle
    Dec 19 '14 at 18:15










  • $begingroup$
    Correct you would make the following subs: $x=u-a$ and $y=v-b$, perform the rotation in the way you mention above (which is correct) and then shift back
    $endgroup$
    – jameselmore
    Dec 19 '14 at 18:19














3












3








3





$begingroup$


When rotating linear functions, I would approach the task geometrically (find invariant point etc.), yet I tried using a matrix which worked nicely.



This was what I did to rotate $y=2x+1$ by $fracpi4$ about the origin.



$$beginpmatrix
frac1sqrt2 &frac-1sqrt2 \
frac1sqrt2&frac1sqrt2
endpmatrix
beginpmatrix
x\ 2x+1endpmatrix
=
frac1sqrt2beginpmatrix
1-x\ 3x+1
endpmatrix$$



$$x'=frac1sqrt2(1-x); ,; y'=frac1sqrt2(3x+1)$$



So
$$y'=2sqrt2-3x'$$



I have also tried this on non-linear functions and it worked.



When I generalise this I obtain:



$$x'=xcos(theta )-ysin(theta); ,; y'=xcos(theta )+ysin(theta)$$



The problem is I can only seem to eliminate one of either $x$ or $y$. Is it possible to eliminate both?



Other related questions I have are:
Can you use matrices to rotate a function around an arbitrary point, not the origin?
Can this concept by applied to 3-dimensions? What about parametric functions?
Can anyone recommend any material for pursing matrix transformations of functions (possibly of more than one variable)?



For reference I am currently a secondary school student in the UK.



Thanks










share|cite|improve this question











$endgroup$




When rotating linear functions, I would approach the task geometrically (find invariant point etc.), yet I tried using a matrix which worked nicely.



This was what I did to rotate $y=2x+1$ by $fracpi4$ about the origin.



$$beginpmatrix
frac1sqrt2 &frac-1sqrt2 \
frac1sqrt2&frac1sqrt2
endpmatrix
beginpmatrix
x\ 2x+1endpmatrix
=
frac1sqrt2beginpmatrix
1-x\ 3x+1
endpmatrix$$



$$x'=frac1sqrt2(1-x); ,; y'=frac1sqrt2(3x+1)$$



So
$$y'=2sqrt2-3x'$$



I have also tried this on non-linear functions and it worked.



When I generalise this I obtain:



$$x'=xcos(theta )-ysin(theta); ,; y'=xcos(theta )+ysin(theta)$$



The problem is I can only seem to eliminate one of either $x$ or $y$. Is it possible to eliminate both?



Other related questions I have are:
Can you use matrices to rotate a function around an arbitrary point, not the origin?
Can this concept by applied to 3-dimensions? What about parametric functions?
Can anyone recommend any material for pursing matrix transformations of functions (possibly of more than one variable)?



For reference I am currently a secondary school student in the UK.



Thanks







linear-algebra geometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 21 '14 at 22:41







Gridley Quayle

















asked Dec 19 '14 at 17:40









Gridley QuayleGridley Quayle

8991714




8991714





bumped to the homepage by Community yesterday


This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.







bumped to the homepage by Community yesterday


This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.













  • $begingroup$
    In regards to your question about rotating about a point other than the origin, you most likely will need to shift point of interest to origin, rotate, and shift back to point of interest.
    $endgroup$
    – jameselmore
    Dec 19 '14 at 17:45










  • $begingroup$
    Do you mean a substitution to shift the axes?
    $endgroup$
    – Gridley Quayle
    Dec 19 '14 at 17:48










  • $begingroup$
    Precisely! Could be simpler than rotating everything, tracking your now rotated point of interest and shifting back
    $endgroup$
    – jameselmore
    Dec 19 '14 at 18:08











  • $begingroup$
    So to rotate around $(a,b)$ I would use the substitutions $x=u-a$ and $y=v-b$?
    $endgroup$
    – Gridley Quayle
    Dec 19 '14 at 18:15










  • $begingroup$
    Correct you would make the following subs: $x=u-a$ and $y=v-b$, perform the rotation in the way you mention above (which is correct) and then shift back
    $endgroup$
    – jameselmore
    Dec 19 '14 at 18:19

















  • $begingroup$
    In regards to your question about rotating about a point other than the origin, you most likely will need to shift point of interest to origin, rotate, and shift back to point of interest.
    $endgroup$
    – jameselmore
    Dec 19 '14 at 17:45










  • $begingroup$
    Do you mean a substitution to shift the axes?
    $endgroup$
    – Gridley Quayle
    Dec 19 '14 at 17:48










  • $begingroup$
    Precisely! Could be simpler than rotating everything, tracking your now rotated point of interest and shifting back
    $endgroup$
    – jameselmore
    Dec 19 '14 at 18:08











  • $begingroup$
    So to rotate around $(a,b)$ I would use the substitutions $x=u-a$ and $y=v-b$?
    $endgroup$
    – Gridley Quayle
    Dec 19 '14 at 18:15










  • $begingroup$
    Correct you would make the following subs: $x=u-a$ and $y=v-b$, perform the rotation in the way you mention above (which is correct) and then shift back
    $endgroup$
    – jameselmore
    Dec 19 '14 at 18:19
















$begingroup$
In regards to your question about rotating about a point other than the origin, you most likely will need to shift point of interest to origin, rotate, and shift back to point of interest.
$endgroup$
– jameselmore
Dec 19 '14 at 17:45




$begingroup$
In regards to your question about rotating about a point other than the origin, you most likely will need to shift point of interest to origin, rotate, and shift back to point of interest.
$endgroup$
– jameselmore
Dec 19 '14 at 17:45












$begingroup$
Do you mean a substitution to shift the axes?
$endgroup$
– Gridley Quayle
Dec 19 '14 at 17:48




$begingroup$
Do you mean a substitution to shift the axes?
$endgroup$
– Gridley Quayle
Dec 19 '14 at 17:48












$begingroup$
Precisely! Could be simpler than rotating everything, tracking your now rotated point of interest and shifting back
$endgroup$
– jameselmore
Dec 19 '14 at 18:08





$begingroup$
Precisely! Could be simpler than rotating everything, tracking your now rotated point of interest and shifting back
$endgroup$
– jameselmore
Dec 19 '14 at 18:08













$begingroup$
So to rotate around $(a,b)$ I would use the substitutions $x=u-a$ and $y=v-b$?
$endgroup$
– Gridley Quayle
Dec 19 '14 at 18:15




$begingroup$
So to rotate around $(a,b)$ I would use the substitutions $x=u-a$ and $y=v-b$?
$endgroup$
– Gridley Quayle
Dec 19 '14 at 18:15












$begingroup$
Correct you would make the following subs: $x=u-a$ and $y=v-b$, perform the rotation in the way you mention above (which is correct) and then shift back
$endgroup$
– jameselmore
Dec 19 '14 at 18:19





$begingroup$
Correct you would make the following subs: $x=u-a$ and $y=v-b$, perform the rotation in the way you mention above (which is correct) and then shift back
$endgroup$
– jameselmore
Dec 19 '14 at 18:19











1 Answer
1






active

oldest

votes


















0












$begingroup$

You cannot eliminate one variable as the angle that the the function makes at a point is dependent on both x and y.



What you want to look at is called a rotation matrix.
http://en.wikipedia.org/wiki/Rotation_matrix



When you rotate a function about a point, you are causing a series of rotations about various axis. See the "In three dimensions" section of the Wikipedia article linked above, and pay attention to the "Rotation matrix from axis and angle" subsection. You can continue multiplying a function by rotation matrices to get your desired result.






share|cite|improve this answer









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    active

    oldest

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    0












    $begingroup$

    You cannot eliminate one variable as the angle that the the function makes at a point is dependent on both x and y.



    What you want to look at is called a rotation matrix.
    http://en.wikipedia.org/wiki/Rotation_matrix



    When you rotate a function about a point, you are causing a series of rotations about various axis. See the "In three dimensions" section of the Wikipedia article linked above, and pay attention to the "Rotation matrix from axis and angle" subsection. You can continue multiplying a function by rotation matrices to get your desired result.






    share|cite|improve this answer









    $endgroup$

















      0












      $begingroup$

      You cannot eliminate one variable as the angle that the the function makes at a point is dependent on both x and y.



      What you want to look at is called a rotation matrix.
      http://en.wikipedia.org/wiki/Rotation_matrix



      When you rotate a function about a point, you are causing a series of rotations about various axis. See the "In three dimensions" section of the Wikipedia article linked above, and pay attention to the "Rotation matrix from axis and angle" subsection. You can continue multiplying a function by rotation matrices to get your desired result.






      share|cite|improve this answer









      $endgroup$















        0












        0








        0





        $begingroup$

        You cannot eliminate one variable as the angle that the the function makes at a point is dependent on both x and y.



        What you want to look at is called a rotation matrix.
        http://en.wikipedia.org/wiki/Rotation_matrix



        When you rotate a function about a point, you are causing a series of rotations about various axis. See the "In three dimensions" section of the Wikipedia article linked above, and pay attention to the "Rotation matrix from axis and angle" subsection. You can continue multiplying a function by rotation matrices to get your desired result.






        share|cite|improve this answer









        $endgroup$



        You cannot eliminate one variable as the angle that the the function makes at a point is dependent on both x and y.



        What you want to look at is called a rotation matrix.
        http://en.wikipedia.org/wiki/Rotation_matrix



        When you rotate a function about a point, you are causing a series of rotations about various axis. See the "In three dimensions" section of the Wikipedia article linked above, and pay attention to the "Rotation matrix from axis and angle" subsection. You can continue multiplying a function by rotation matrices to get your desired result.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 21 '14 at 23:34









        philnphiln

        11




        11



























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