rotation of spherical coordinate system
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 '18 at 3:42
Alicia
82
add a comment |
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 '18 at 3:42
Alicia
82
add a comment |
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 '18 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 '18 at 3:42
Alicia
82
asked Nov 19 '18 at 3:42
Alicia
82
asked Nov 19 '18 at 3:42
Alicia
82
asked Nov 19 '18 at 3:42
Alicia
82
asked Nov 19 '18 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 '18 at 4:34
Andrei
11.2k21026
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 '18 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 '18 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 '18 at 17:25
add a comment |
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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 '18 at 4:34
Andrei
11.2k21026
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 '18 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 '18 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 '18 at 17:25
add a comment |
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 '18 at 4:34
Andrei
11.2k21026
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 '18 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 '18 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 '18 at 17:25
add a comment |
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 '18 at 4:34
Andrei
11.2k21026
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 '18 at 4:34
Andrei
11.2k21026
answered Nov 19 '18 at 4:34
Andrei
11.2k21026
answered Nov 19 '18 at 4:34
Andrei
11.2k21026
answered Nov 19 '18 at 4:34
Andrei
11.2k21026
11.2k21026
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 '18 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 '18 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 '18 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 '18 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 '18 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 '18 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 '18 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 '18 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 '18 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 '18 at 16:57
I see, thank you a lot!
– Alicia
Nov 19 '18 at 17:25
I see, thank you a lot!
– Alicia
Nov 19 '18 at 17:25
add a comment |
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