rotation of spherical coordinate system
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If I rotate the coordinate system by spherical angles θ and ϕ, and the vector in the new system is v'=(x′,y′,z′), what is its coordinate (x,y,z) in the original coordinate system? Thank you in advance!
matrices coordinate-systems
asked Nov 19 at 3:42
Alicia
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If I rotate the coordinate system by spherical angles θ and ϕ, and the vector in the new system is v'=(x′,y′,z′), what is its coordinate (x,y,z) in the original coordinate system? Thank you in advance!
matrices coordinate-systems
asked Nov 19 at 3:42
Alicia
82
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
If I rotate the coordinate system by spherical angles θ and ϕ, and the vector in the new system is v'=(x′,y′,z′), what is its coordinate (x,y,z) in the original coordinate system? Thank you in advance!
matrices coordinate-systems
asked Nov 19 at 3:42
Alicia
82
If I rotate the coordinate system by spherical angles θ and ϕ, and the vector in the new system is v'=(x′,y′,z′), what is its coordinate (x,y,z) in the original coordinate system? Thank you in advance!
matrices coordinate-systems
matrices coordinate-systems
asked Nov 19 at 3:42
Alicia
82
asked Nov 19 at 3:42
Alicia
82
asked Nov 19 at 3:42
Alicia
82
asked Nov 19 at 3:42
Alicia
82
asked Nov 19 at 3:42
Alicia
82
82
add a comment |
add a comment |
1 Answer
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It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $theta$, then a rotation around $z$ axis by angle $phi$. You can then write $$textrm v'=R_z(phi)R_x(theta)textrm v$$
Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.
Then just multiplying with the inverses, in reverse order, you get
$$R_x^{-1}(theta)R_z^{-1}(phi)textrm v'=R_x^{-1}(theta)R_z^{-1}(phi)R_z(phi)R_x(theta)textrm v=textrm v$$
Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle:
$R^{-1}(alpha)=R^T(alpha)=R(-alpha)$
answered Nov 19 at 4:34
Andrei
10.4k21025
Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
– Alicia
Nov 19 at 15:58
You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
– Andrei
Nov 19 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 at 17:25
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $theta$, then a rotation around $z$ axis by angle $phi$. You can then write $$textrm v'=R_z(phi)R_x(theta)textrm v$$
Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.
Then just multiplying with the inverses, in reverse order, you get
$$R_x^{-1}(theta)R_z^{-1}(phi)textrm v'=R_x^{-1}(theta)R_z^{-1}(phi)R_z(phi)R_x(theta)textrm v=textrm v$$
Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle:
$R^{-1}(alpha)=R^T(alpha)=R(-alpha)$
answered Nov 19 at 4:34
Andrei
10.4k21025
Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
– Alicia
Nov 19 at 15:58
You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
– Andrei
Nov 19 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 at 17:25
add a comment |
up vote
0
down vote
accepted
It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $theta$, then a rotation around $z$ axis by angle $phi$. You can then write $$textrm v'=R_z(phi)R_x(theta)textrm v$$
Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.
Then just multiplying with the inverses, in reverse order, you get
$$R_x^{-1}(theta)R_z^{-1}(phi)textrm v'=R_x^{-1}(theta)R_z^{-1}(phi)R_z(phi)R_x(theta)textrm v=textrm v$$
Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle:
$R^{-1}(alpha)=R^T(alpha)=R(-alpha)$
answered Nov 19 at 4:34
Andrei
10.4k21025
Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
– Alicia
Nov 19 at 15:58
You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
– Andrei
Nov 19 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 at 17:25
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $theta$, then a rotation around $z$ axis by angle $phi$. You can then write $$textrm v'=R_z(phi)R_x(theta)textrm v$$
Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.
Then just multiplying with the inverses, in reverse order, you get
$$R_x^{-1}(theta)R_z^{-1}(phi)textrm v'=R_x^{-1}(theta)R_z^{-1}(phi)R_z(phi)R_x(theta)textrm v=textrm v$$
Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle:
$R^{-1}(alpha)=R^T(alpha)=R(-alpha)$
answered Nov 19 at 4:34
Andrei
10.4k21025
It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $theta$, then a rotation around $z$ axis by angle $phi$. You can then write $$textrm v'=R_z(phi)R_x(theta)textrm v$$
Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.
Then just multiplying with the inverses, in reverse order, you get
$$R_x^{-1}(theta)R_z^{-1}(phi)textrm v'=R_x^{-1}(theta)R_z^{-1}(phi)R_z(phi)R_x(theta)textrm v=textrm v$$
Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle:
$R^{-1}(alpha)=R^T(alpha)=R(-alpha)$
answered Nov 19 at 4:34
Andrei
10.4k21025
answered Nov 19 at 4:34
Andrei
10.4k21025
answered Nov 19 at 4:34
Andrei
10.4k21025
answered Nov 19 at 4:34
Andrei
10.4k21025
10.4k21025
Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
– Alicia
Nov 19 at 15:58
You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
– Andrei
Nov 19 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 at 17:25
add a comment |
Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
– Alicia
Nov 19 at 15:58
You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
– Andrei
Nov 19 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 at 17:25
Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
– Alicia
Nov 19 at 15:58
Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
– Alicia
Nov 19 at 15:58
You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
– Andrei
Nov 19 at 16:57
You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
– Andrei
Nov 19 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 at 17:25
I see, thank you a lot!
– Alicia
Nov 19 at 17:25
add a comment |
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