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
$begingroup$
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
linear-algebra geometry
asked Dec 19 '14 at 17:40
Gridley QuayleGridley Quayle
8991714
$endgroup$
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.
add a comment |
$begingroup$
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
linear-algebra geometry
asked Dec 19 '14 at 17:40
Gridley QuayleGridley Quayle
8991714
$endgroup$
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
add a comment |
$begingroup$
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
linear-algebra geometry
asked Dec 19 '14 at 17:40
Gridley QuayleGridley Quayle
8991714
$endgroup$
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
linear-algebra geometry
asked Dec 19 '14 at 17:40
Gridley QuayleGridley Quayle
8991714
asked Dec 19 '14 at 17:40
Gridley QuayleGridley Quayle
8991714
edited Dec 21 '14 at 22:41
Gridley Quayle
asked Dec 19 '14 at 17:40
Gridley QuayleGridley Quayle
8991714
asked Dec 19 '14 at 17:40
Gridley QuayleGridley Quayle
8991714
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered Dec 21 '14 at 23:34
philnphiln
11
$endgroup$
add a comment |
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1 Answer
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votes
$begingroup$
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.
answered Dec 21 '14 at 23:34
philnphiln
11
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 21 '14 at 23:34
philnphiln
11
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 21 '14 at 23:34
philnphiln
11
$endgroup$
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.
answered Dec 21 '14 at 23:34
philnphiln
11
answered Dec 21 '14 at 23:34
philnphiln
11
answered Dec 21 '14 at 23:34
philnphiln
11
answered Dec 21 '14 at 23:34
philnphiln
11
11
add a comment |
add a comment |
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$begingroup$
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