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How do I compose isometries algebraically?
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I've learned the general gist of isometries: reflections, translations, rotations, and glide reflections.
However, I've been taught this geometrically. So when I want to find what $textref_l(textref_m)$ (composition of two reflections at different lines $m$ and $l$) is for example, I can only draw certain shape like a triangle and see that it can be a rotation. However, apparently it can also be a translation (no idea how).
I'm asking if there are rules that can algebraically help me determine that and similar things like:
(a) a reflection in some line $l$ followed by a rotation about a point on $l$, (b) rotation about a point followed by a rotation about a different point. Are there algebraic rules of composition that can help me determine what type of isometries that these can represent?
group-theory geometry
asked Aug 3 at 20:44
Nebulae
435
add a comment |Â
up vote
0
down vote
favorite
I've learned the general gist of isometries: reflections, translations, rotations, and glide reflections.
However, I've been taught this geometrically. So when I want to find what $textref_l(textref_m)$ (composition of two reflections at different lines $m$ and $l$) is for example, I can only draw certain shape like a triangle and see that it can be a rotation. However, apparently it can also be a translation (no idea how).
I'm asking if there are rules that can algebraically help me determine that and similar things like:
(a) a reflection in some line $l$ followed by a rotation about a point on $l$, (b) rotation about a point followed by a rotation about a different point. Are there algebraic rules of composition that can help me determine what type of isometries that these can represent?
group-theory geometry
asked Aug 3 at 20:44
Nebulae
435
You assume isometries of the plane $BbbR^2$? We can use matrices.
– Dietrich Burde
Aug 3 at 20:55
@DietrichBurde Yes, sorry I should have said that. It's all in $mathbbR^2$. I'd love to know this matrices method.
– Nebulae
Aug 3 at 20:57
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I've learned the general gist of isometries: reflections, translations, rotations, and glide reflections.
However, I've been taught this geometrically. So when I want to find what $textref_l(textref_m)$ (composition of two reflections at different lines $m$ and $l$) is for example, I can only draw certain shape like a triangle and see that it can be a rotation. However, apparently it can also be a translation (no idea how).
I'm asking if there are rules that can algebraically help me determine that and similar things like:
(a) a reflection in some line $l$ followed by a rotation about a point on $l$, (b) rotation about a point followed by a rotation about a different point. Are there algebraic rules of composition that can help me determine what type of isometries that these can represent?
group-theory geometry
asked Aug 3 at 20:44
Nebulae
435
I've learned the general gist of isometries: reflections, translations, rotations, and glide reflections.
However, I've been taught this geometrically. So when I want to find what $textref_l(textref_m)$ (composition of two reflections at different lines $m$ and $l$) is for example, I can only draw certain shape like a triangle and see that it can be a rotation. However, apparently it can also be a translation (no idea how).
I'm asking if there are rules that can algebraically help me determine that and similar things like:
(a) a reflection in some line $l$ followed by a rotation about a point on $l$, (b) rotation about a point followed by a rotation about a different point. Are there algebraic rules of composition that can help me determine what type of isometries that these can represent?
group-theory geometry
asked Aug 3 at 20:44
Nebulae
435
asked Aug 3 at 20:44
Nebulae
435
asked Aug 3 at 20:44
Nebulae
435
asked Aug 3 at 20:44
Nebulae
435
435
You assume isometries of the plane $BbbR^2$? We can use matrices.
– Dietrich Burde
Aug 3 at 20:55
@DietrichBurde Yes, sorry I should have said that. It's all in $mathbbR^2$. I'd love to know this matrices method.
– Nebulae
Aug 3 at 20:57
add a comment |Â
You assume isometries of the plane $BbbR^2$? We can use matrices.
– Dietrich Burde
Aug 3 at 20:55
@DietrichBurde Yes, sorry I should have said that. It's all in $mathbbR^2$. I'd love to know this matrices method.
– Nebulae
Aug 3 at 20:57
You assume isometries of the plane $BbbR^2$? We can use matrices.
– Dietrich Burde
Aug 3 at 20:55
You assume isometries of the plane $BbbR^2$? We can use matrices.
– Dietrich Burde
Aug 3 at 20:55
@DietrichBurde Yes, sorry I should have said that. It's all in $mathbbR^2$. I'd love to know this matrices method.
– Nebulae
Aug 3 at 20:57
@DietrichBurde Yes, sorry I should have said that. It's all in $mathbbR^2$. I'd love to know this matrices method.
– Nebulae
Aug 3 at 20:57
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
accepted
The orientation perserving isometry group of the plane is $SO_2(mathbbR)rtimesmathbbR^2$ (the orthogonal group $SO_2(mathbbR)$ are the rotations and $mathbbR^2$ are the translations). This can be embedded in $SL_3(mathbbR)$ as follows:
$$
left(
beginarrayccc
1&0&0\
a&costheta&-sintheta\
b&sintheta&costheta\
endarray
right) textfirst rotation by theta text then translation by (a,b).
$$
This acts on a point $(x,y)$ of the plane by
$$
left(
beginarrayccc
1&0&0\
a&costheta&-sintheta\
b&sintheta&costheta\
endarray
right)
left(
beginarrayc
1\
x\
y\
endarray
right).
$$
For instance if $theta=0$ we get the translation
$$
(x,y)mapsto(a+x,b+y)
$$
or if $a=b=0$ we get the rotation
$$
(x,y)mapsto(xcostheta+ysintheta,xsintheta-ycostheta).
$$
Together we get a rotation by $theta$ followed by a translation by $(a,b)$:
$$
(x,y)mapsto(a+xcostheta+ysintheta,b+xsintheta-ycostheta).
$$
This also extends as you might think to the full isometry group $O_2(mathbbR)rtimesmathbbR^2$ by throwing in a reflection, e.g.
$$
left(
beginarrayccc
1&0&0\
0&-1&0\
0&0&1\
endarray
right), (x,y)mapsto(-x,y),
$$
and it also extends to higher dimension in the obvious way.
answered Aug 3 at 21:42
yoyo
6,3291525
Thank you very much for this. Regarding the reflection matrix M, does it act on the $(x,y)$ by plane via $M (1,x,y)^t$? where $t$ denotes the transpose, like the rotation-translation matrix you've given? So If I want to find translation(rotation(reflection)), what would it look like? I may need to read more about this, and this will be clarified. I appreciate your work, though, this is massively helpful!
– Nebulae
Aug 3 at 22:50
add a comment |Â
up vote
0
down vote
Yes, those formulas exist. The reflection of a point $(p,q)$ on the line $ax+by+c=0$ is$$left(fracp(a^2-b^2)-2b(aq+c)a^2+b^2,fracq(b^2-a^2)-2a(bp+c)a^2+b^2right).$$The rotation of $(p,q)$ with angle $theta$ around $(a,b)$ is$$bigl(a+(p-a)cos(theta )-(q-b)sin(theta),b+(p-a) sin (theta )+(q-b) cos (theta )bigr).$$You can use these formulas to compute the composition of rotations or of reflections.
answered Aug 3 at 21:09
José Carlos Santos
112k1696172
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
The orientation perserving isometry group of the plane is $SO_2(mathbbR)rtimesmathbbR^2$ (the orthogonal group $SO_2(mathbbR)$ are the rotations and $mathbbR^2$ are the translations). This can be embedded in $SL_3(mathbbR)$ as follows:
$$
left(
beginarrayccc
1&0&0\
a&costheta&-sintheta\
b&sintheta&costheta\
endarray
right) textfirst rotation by theta text then translation by (a,b).
$$
This acts on a point $(x,y)$ of the plane by
$$
left(
beginarrayccc
1&0&0\
a&costheta&-sintheta\
b&sintheta&costheta\
endarray
right)
left(
beginarrayc
1\
x\
y\
endarray
right).
$$
For instance if $theta=0$ we get the translation
$$
(x,y)mapsto(a+x,b+y)
$$
or if $a=b=0$ we get the rotation
$$
(x,y)mapsto(xcostheta+ysintheta,xsintheta-ycostheta).
$$
Together we get a rotation by $theta$ followed by a translation by $(a,b)$:
$$
(x,y)mapsto(a+xcostheta+ysintheta,b+xsintheta-ycostheta).
$$
This also extends as you might think to the full isometry group $O_2(mathbbR)rtimesmathbbR^2$ by throwing in a reflection, e.g.
$$
left(
beginarrayccc
1&0&0\
0&-1&0\
0&0&1\
endarray
right), (x,y)mapsto(-x,y),
$$
and it also extends to higher dimension in the obvious way.
answered Aug 3 at 21:42
yoyo
6,3291525
Thank you very much for this. Regarding the reflection matrix M, does it act on the $(x,y)$ by plane via $M (1,x,y)^t$? where $t$ denotes the transpose, like the rotation-translation matrix you've given? So If I want to find translation(rotation(reflection)), what would it look like? I may need to read more about this, and this will be clarified. I appreciate your work, though, this is massively helpful!
– Nebulae
Aug 3 at 22:50
add a comment |Â
up vote
2
down vote
accepted
The orientation perserving isometry group of the plane is $SO_2(mathbbR)rtimesmathbbR^2$ (the orthogonal group $SO_2(mathbbR)$ are the rotations and $mathbbR^2$ are the translations). This can be embedded in $SL_3(mathbbR)$ as follows:
$$
left(
beginarrayccc
1&0&0\
a&costheta&-sintheta\
b&sintheta&costheta\
endarray
right) textfirst rotation by theta text then translation by (a,b).
$$
This acts on a point $(x,y)$ of the plane by
$$
left(
beginarrayccc
1&0&0\
a&costheta&-sintheta\
b&sintheta&costheta\
endarray
right)
left(
beginarrayc
1\
x\
y\
endarray
right).
$$
For instance if $theta=0$ we get the translation
$$
(x,y)mapsto(a+x,b+y)
$$
or if $a=b=0$ we get the rotation
$$
(x,y)mapsto(xcostheta+ysintheta,xsintheta-ycostheta).
$$
Together we get a rotation by $theta$ followed by a translation by $(a,b)$:
$$
(x,y)mapsto(a+xcostheta+ysintheta,b+xsintheta-ycostheta).
$$
This also extends as you might think to the full isometry group $O_2(mathbbR)rtimesmathbbR^2$ by throwing in a reflection, e.g.
$$
left(
beginarrayccc
1&0&0\
0&-1&0\
0&0&1\
endarray
right), (x,y)mapsto(-x,y),
$$
and it also extends to higher dimension in the obvious way.
answered Aug 3 at 21:42
yoyo
6,3291525
Thank you very much for this. Regarding the reflection matrix M, does it act on the $(x,y)$ by plane via $M (1,x,y)^t$? where $t$ denotes the transpose, like the rotation-translation matrix you've given? So If I want to find translation(rotation(reflection)), what would it look like? I may need to read more about this, and this will be clarified. I appreciate your work, though, this is massively helpful!
– Nebulae
Aug 3 at 22:50
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
The orientation perserving isometry group of the plane is $SO_2(mathbbR)rtimesmathbbR^2$ (the orthogonal group $SO_2(mathbbR)$ are the rotations and $mathbbR^2$ are the translations). This can be embedded in $SL_3(mathbbR)$ as follows:
$$
left(
beginarrayccc
1&0&0\
a&costheta&-sintheta\
b&sintheta&costheta\
endarray
right) textfirst rotation by theta text then translation by (a,b).
$$
This acts on a point $(x,y)$ of the plane by
$$
left(
beginarrayccc
1&0&0\
a&costheta&-sintheta\
b&sintheta&costheta\
endarray
right)
left(
beginarrayc
1\
x\
y\
endarray
right).
$$
For instance if $theta=0$ we get the translation
$$
(x,y)mapsto(a+x,b+y)
$$
or if $a=b=0$ we get the rotation
$$
(x,y)mapsto(xcostheta+ysintheta,xsintheta-ycostheta).
$$
Together we get a rotation by $theta$ followed by a translation by $(a,b)$:
$$
(x,y)mapsto(a+xcostheta+ysintheta,b+xsintheta-ycostheta).
$$
This also extends as you might think to the full isometry group $O_2(mathbbR)rtimesmathbbR^2$ by throwing in a reflection, e.g.
$$
left(
beginarrayccc
1&0&0\
0&-1&0\
0&0&1\
endarray
right), (x,y)mapsto(-x,y),
$$
and it also extends to higher dimension in the obvious way.
answered Aug 3 at 21:42
yoyo
6,3291525
The orientation perserving isometry group of the plane is $SO_2(mathbbR)rtimesmathbbR^2$ (the orthogonal group $SO_2(mathbbR)$ are the rotations and $mathbbR^2$ are the translations). This can be embedded in $SL_3(mathbbR)$ as follows:
$$
left(
beginarrayccc
1&0&0\
a&costheta&-sintheta\
b&sintheta&costheta\
endarray
right) textfirst rotation by theta text then translation by (a,b).
$$
This acts on a point $(x,y)$ of the plane by
$$
left(
beginarrayccc
1&0&0\
a&costheta&-sintheta\
b&sintheta&costheta\
endarray
right)
left(
beginarrayc
1\
x\
y\
endarray
right).
$$
For instance if $theta=0$ we get the translation
$$
(x,y)mapsto(a+x,b+y)
$$
or if $a=b=0$ we get the rotation
$$
(x,y)mapsto(xcostheta+ysintheta,xsintheta-ycostheta).
$$
Together we get a rotation by $theta$ followed by a translation by $(a,b)$:
$$
(x,y)mapsto(a+xcostheta+ysintheta,b+xsintheta-ycostheta).
$$
This also extends as you might think to the full isometry group $O_2(mathbbR)rtimesmathbbR^2$ by throwing in a reflection, e.g.
$$
left(
beginarrayccc
1&0&0\
0&-1&0\
0&0&1\
endarray
right), (x,y)mapsto(-x,y),
$$
and it also extends to higher dimension in the obvious way.
answered Aug 3 at 21:42
yoyo
6,3291525
answered Aug 3 at 21:42
yoyo
6,3291525
answered Aug 3 at 21:42
yoyo
6,3291525
answered Aug 3 at 21:42
yoyo
6,3291525
6,3291525
Thank you very much for this. Regarding the reflection matrix M, does it act on the $(x,y)$ by plane via $M (1,x,y)^t$? where $t$ denotes the transpose, like the rotation-translation matrix you've given? So If I want to find translation(rotation(reflection)), what would it look like? I may need to read more about this, and this will be clarified. I appreciate your work, though, this is massively helpful!
– Nebulae
Aug 3 at 22:50
add a comment |Â
Thank you very much for this. Regarding the reflection matrix M, does it act on the $(x,y)$ by plane via $M (1,x,y)^t$? where $t$ denotes the transpose, like the rotation-translation matrix you've given? So If I want to find translation(rotation(reflection)), what would it look like? I may need to read more about this, and this will be clarified. I appreciate your work, though, this is massively helpful!
– Nebulae
Aug 3 at 22:50
Thank you very much for this. Regarding the reflection matrix M, does it act on the $(x,y)$ by plane via $M (1,x,y)^t$? where $t$ denotes the transpose, like the rotation-translation matrix you've given? So If I want to find translation(rotation(reflection)), what would it look like? I may need to read more about this, and this will be clarified. I appreciate your work, though, this is massively helpful!
– Nebulae
Aug 3 at 22:50
Thank you very much for this. Regarding the reflection matrix M, does it act on the $(x,y)$ by plane via $M (1,x,y)^t$? where $t$ denotes the transpose, like the rotation-translation matrix you've given? So If I want to find translation(rotation(reflection)), what would it look like? I may need to read more about this, and this will be clarified. I appreciate your work, though, this is massively helpful!
– Nebulae
Aug 3 at 22:50
add a comment |Â
up vote
0
down vote
Yes, those formulas exist. The reflection of a point $(p,q)$ on the line $ax+by+c=0$ is$$left(fracp(a^2-b^2)-2b(aq+c)a^2+b^2,fracq(b^2-a^2)-2a(bp+c)a^2+b^2right).$$The rotation of $(p,q)$ with angle $theta$ around $(a,b)$ is$$bigl(a+(p-a)cos(theta )-(q-b)sin(theta),b+(p-a) sin (theta )+(q-b) cos (theta )bigr).$$You can use these formulas to compute the composition of rotations or of reflections.
answered Aug 3 at 21:09
José Carlos Santos
112k1696172
add a comment |Â
up vote
0
down vote
Yes, those formulas exist. The reflection of a point $(p,q)$ on the line $ax+by+c=0$ is$$left(fracp(a^2-b^2)-2b(aq+c)a^2+b^2,fracq(b^2-a^2)-2a(bp+c)a^2+b^2right).$$The rotation of $(p,q)$ with angle $theta$ around $(a,b)$ is$$bigl(a+(p-a)cos(theta )-(q-b)sin(theta),b+(p-a) sin (theta )+(q-b) cos (theta )bigr).$$You can use these formulas to compute the composition of rotations or of reflections.
answered Aug 3 at 21:09
José Carlos Santos
112k1696172
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Yes, those formulas exist. The reflection of a point $(p,q)$ on the line $ax+by+c=0$ is$$left(fracp(a^2-b^2)-2b(aq+c)a^2+b^2,fracq(b^2-a^2)-2a(bp+c)a^2+b^2right).$$The rotation of $(p,q)$ with angle $theta$ around $(a,b)$ is$$bigl(a+(p-a)cos(theta )-(q-b)sin(theta),b+(p-a) sin (theta )+(q-b) cos (theta )bigr).$$You can use these formulas to compute the composition of rotations or of reflections.
answered Aug 3 at 21:09
José Carlos Santos
112k1696172
Yes, those formulas exist. The reflection of a point $(p,q)$ on the line $ax+by+c=0$ is$$left(fracp(a^2-b^2)-2b(aq+c)a^2+b^2,fracq(b^2-a^2)-2a(bp+c)a^2+b^2right).$$The rotation of $(p,q)$ with angle $theta$ around $(a,b)$ is$$bigl(a+(p-a)cos(theta )-(q-b)sin(theta),b+(p-a) sin (theta )+(q-b) cos (theta )bigr).$$You can use these formulas to compute the composition of rotations or of reflections.
answered Aug 3 at 21:09
José Carlos Santos
112k1696172
answered Aug 3 at 21:09
José Carlos Santos
112k1696172
answered Aug 3 at 21:09
José Carlos Santos
112k1696172
answered Aug 3 at 21:09
José Carlos Santos
112k1696172
112k1696172
add a comment |Â
add a comment |Â
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You assume isometries of the plane $BbbR^2$? We can use matrices.
– Dietrich Burde
Aug 3 at 20:55
@DietrichBurde Yes, sorry I should have said that. It's all in $mathbbR^2$. I'd love to know this matrices method.
– Nebulae
Aug 3 at 20:57