Difference between revisions of "Talk:Extra:Intersecting planes"

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(Created page with "==Rotate <math>\mathbf{v}_{2b}</math> about <math>\mathbf{n}_{2}</math>== In general, rotation of a vector <math>\scriptstyle \mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x...")
 
 
(One intermediate revision by the same user not shown)
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==Rotate <math>\mathbf{v}_{2b}</math> about <math>\mathbf{n}_{1}</math>==
 +
In general, rotation of a vector <math>\scriptstyle
 +
\mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x & y  & z \end{bmatrix}</math>about an arbitrary unit-vector <math>\scriptstyle
 +
\mathbf{n} = \begin{bmatrix} u & v  & w \end{bmatrix}</math> gives ([https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle 1], [http://inside.mines.edu/fs_home/gmurray/ArbitraryAxisRotation/ 2]):
 +
::<math>
 +
\mathbf{v}_{\mathrm{end}} = \begin{bmatrix} u(ux +vy+wz)(1-\cos \theta) + x \cos \theta + (-wy+vz)\sin \theta \\
 +
v(ux+vy+wz)(1 - \cos \theta) + y \cos \theta + (wx-uz) \sin \theta\\
 +
w(ux+vy+wz)(1-\cos \theta) + z \cos \theta + (-vx+uy) \sin \theta \end{bmatrix}
 +
</math>
 +
In this particular case, we thus expect:
 +
::<math>
 +
\begin{alignat}{2}
 +
\mathbf{v}_{2} & = \begin{bmatrix} -y\sin \phi \\
 +
y \cos \phi \\
 +
z(1-\cos \phi) + z \cos \phi\end{bmatrix}
 +
\\
 +
  & = \begin{bmatrix} -q \cos \alpha\sin \phi \\
 +
q \cos \alpha \cos \phi \\
 +
q \sin \alpha(1-\cos \phi) + q \sin \alpha \cos \phi\end{bmatrix}
 +
\\
 +
  & = q \begin{bmatrix} - \cos \alpha\sin \phi \\
 +
\cos \alpha \cos \phi \\
 +
\sin \alpha\end{bmatrix}
 +
\end{alignat}
 +
</math>
 +
 +
 
==Rotate <math>\mathbf{v}_{2b}</math> about <math>\mathbf{n}_{2}</math>==
 
==Rotate <math>\mathbf{v}_{2b}</math> about <math>\mathbf{n}_{2}</math>==
  
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\cos\alpha \cos \phi \\  
 
\cos\alpha \cos \phi \\  
 
\sin\alpha \cos \phi\end{bmatrix}
 
\sin\alpha \cos \phi\end{bmatrix}
 +
\end{alignat}
 +
</math>
 +
 +
 +
==Generalized distance between two vectors==
 +
'''Warning: Errors below''' (this is just intermediate/working stuff)
 +
 +
Imagine [[reciprocal-space]] [[scattering]] that is a ring; more specifically a pseudo-toroid with Gaussian-like decay. The intensity overall is:
 +
::<math>
 +
I = \exp \left [ -(q_{rr}-q_0)^2/(2 \sigma_q^2) \right ] \exp \left [ -q_{rz}^2/(2 \sigma_q^2) \right ]
 +
</math>
 +
Where we use the subscript ''r'' to denote the reciprocal-space coordinate system, and <math>\scriptstyle q_{rr} = \sqrt{q_{rx}^2+q_{ry}^2}</math>. The plane of the [[detector]] (i.e. the [[Ewald sphere|Ewald plane]]) is denoted by ''d'':
 +
::<math>
 +
\mathbf{v}_{d} = \begin{bmatrix} q_{dx} & q_{dy}  & 0 \end{bmatrix}
 +
</math>
 +
We set the symmetry axis in realspace (detector coordinate system) to be the <math>\scriptstyle x</math>-axis. The reciprocal-space is tilted by <math>\scriptstyle \chi_0</math> (about the <math>\scriptstyle y</math>-axis), before the 'powder' rotation about the <math>\scriptstyle x</math>-axis (where <math>\scriptstyle \alpha</math> goes from <math>\scriptstyle -\pi</math> to <math>\scriptstyle +\pi</math>). Consider an initial vector:
 +
::<math>
 +
\mathbf{v}_{d0} = \begin{bmatrix} q \sin \phi & q \cos \phi  & 0 \end{bmatrix}
 +
</math>
 +
The 1st rotation (about <math>\scriptstyle y</math>-axis by <math>\scriptstyle \chi_0</math>) involves:
 +
::<math>
 +
\begin{alignat}{2}
 +
\mathbf{v}_{d1} & = R_y(\chi_0) \mathbf{v}_{d0} \\
 +
    & = \begin{bmatrix}
 +
\cos \chi_0 & 0 & \sin \chi_0 \\
 +
0 & 1 & 0 \\
 +
-\sin \chi_0 & 0 & \cos \chi_0 \\
 +
\end{bmatrix}
 +
\begin{bmatrix}
 +
q \sin \phi \\
 +
q \cos \phi \\
 +
0 \\
 +
\end{bmatrix} \\
 +
    & = \begin{bmatrix}
 +
q \sin \phi \cos \chi_0 \\
 +
q \cos \phi \\
 +
- q \sin \phi \sin \chi_0 \\
 +
\end{bmatrix}
 +
\end{alignat}
 +
</math>
 +
Consider a 2nd rotation around the vector (normal to the detector plane) ('''Warning: This is erroneous since the alpha rotation is just another phi rotation.'''):
 +
::<math>
 +
\begin{alignat}{2}
 +
\mathbf{n}_{d1} & = R_y(\chi_0) \mathbf{n}_{d0} \\
 +
    & = \begin{bmatrix}
 +
\cos \chi_0 & 0 & \sin \chi_0 \\
 +
0 & 1 & 0 \\
 +
-\sin \chi_0 & 0 & \cos \chi_0 \\
 +
\end{bmatrix}
 +
\begin{bmatrix}
 +
0 \\
 +
0 \\
 +
1 \\
 +
\end{bmatrix} \\
 +
    & = \begin{bmatrix}
 +
\sin \chi_0 \\
 +
0 \\
 +
\cos \chi_0 \\
 +
\end{bmatrix}
 +
\end{alignat}
 +
</math>
 +
So the second rotation yields:
 +
::<math>
 +
\begin{alignat}{2}
 +
\mathbf{v}_{d2} & =
 +
\begin{bmatrix} u(ux +wz)(1-\cos \alpha) + x \cos \alpha + -wy \sin \alpha \\
 +
y \cos \alpha + (wx-uz) \sin \alpha\\
 +
w(ux+wz)(1-\cos \alpha) + z \cos \alpha + uy \sin \alpha \end{bmatrix} \\
 +
    & =
 +
\begin{bmatrix} \sin \chi_0 (\sin \chi_0 q \sin \phi \cos \chi_0 - \cos \chi_0 q \sin \phi \sin \chi_0)(1-\cos \alpha) + q \sin \phi \cos \chi_0 \cos \alpha + - \cos \chi_0 q \cos \phi \sin \alpha \\
 +
q \cos \phi \cos \alpha + (\cos \chi_0 q \sin \phi \cos \chi_0 + \sin \chi_0 q \sin \phi \sin \chi_0) \sin \alpha\\
 +
\cos \chi_0(\sin \chi_0 q \sin \phi \cos \chi_0 - \cos \chi_0 q \sin \phi \sin \chi_0)(1-\cos \alpha) + - q \sin \phi \sin \chi_0 \cos \alpha + \sin \chi_0 q \cos \phi \sin \alpha \end{bmatrix} \\
 +
    & =
 +
q \begin{bmatrix} \sin^2 \chi_0 \sin \phi (\cos \chi_0 - \cos \chi_0 )(1-\cos \alpha) + \sin \phi \cos \chi_0 \cos \alpha - \cos \chi_0 \cos \phi \sin \alpha \\
 +
\cos \phi \cos \alpha + \sin \phi (\cos^2 \chi_0 + \sin^2 \chi_0 ) \sin \alpha\\
 +
\cos \chi_0 \sin \chi_0 \sin \phi (\cos \chi_0 - \cos \chi_0 )(1-\cos \alpha) - \sin \phi \sin \chi_0 \cos \alpha + \sin \chi_0 \cos \phi \sin \alpha \end{bmatrix} \\
 +
 +
    & =
 +
q \begin{bmatrix} \cos \chi_0 ( \sin \phi \cos \alpha - \cos \phi \sin \alpha ) \\
 +
\cos \phi \cos \alpha + \sin \phi  \sin \alpha\\
 +
-\sin \chi_0 ( \sin \phi \cos \alpha - \cos \phi \sin \alpha ) \end{bmatrix} \\
 +
 
\end{alignat}
 
\end{alignat}
 
</math>
 
</math>

Latest revision as of 17:26, 23 December 2015

Rotate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_{2b}} about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{n}_{1}}

In general, rotation of a vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x & y & z \end{bmatrix}} about an arbitrary unit-vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{n} = \begin{bmatrix} u & v & w \end{bmatrix}} gives (1, 2):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_{\mathrm{end}} = \begin{bmatrix} u(ux +vy+wz)(1-\cos \theta) + x \cos \theta + (-wy+vz)\sin \theta \\ v(ux+vy+wz)(1 - \cos \theta) + y \cos \theta + (wx-uz) \sin \theta\\ w(ux+vy+wz)(1-\cos \theta) + z \cos \theta + (-vx+uy) \sin \theta \end{bmatrix} }

In this particular case, we thus expect:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{v}_{2} & = \begin{bmatrix} -y\sin \phi \\ y \cos \phi \\ z(1-\cos \phi) + z \cos \phi\end{bmatrix} \\ & = \begin{bmatrix} -q \cos \alpha\sin \phi \\ q \cos \alpha \cos \phi \\ q \sin \alpha(1-\cos \phi) + q \sin \alpha \cos \phi\end{bmatrix} \\ & = q \begin{bmatrix} - \cos \alpha\sin \phi \\ \cos \alpha \cos \phi \\ \sin \alpha\end{bmatrix} \end{alignat} }


Rotate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_{2b}} about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{n}_{2}}

In general, rotation of a vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x & y & z \end{bmatrix}} about an arbitrary unit-vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{n} = \begin{bmatrix} u & v & w \end{bmatrix}} gives (1, 2):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_{\mathrm{end}} = \begin{bmatrix} u(ux +vy+wz)(1-\cos \theta) + x \cos \theta + (-wy+vz)\sin \theta \\ v(ux+vy+wz)(1 - \cos \theta) + y \cos \theta + (wx-uz) \sin \theta\\ w(ux+vy+wz)(1-\cos \theta) + z \cos \theta + (-vx+uy) \sin \theta \end{bmatrix} }

In this particular case, we thus expect:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{v}_{2} & = \begin{bmatrix} (-wy+vz)\sin \phi \\ v(vy+wz)(1 - \cos \phi) + y \cos \phi \\ w(vy+wz)(1-\cos \phi) + z \cos \phi\end{bmatrix} \\ & = \begin{bmatrix} (- \cos\alpha q \cos\alpha + -\sin\alpha q \sin\alpha)\sin \phi \\ -\sin\alpha(-\sin\alpha q \cos\alpha+\cos\alpha q \sin\alpha)(1 - \cos \phi) + q \cos\alpha \cos \phi \\ \cos\alpha(-\sin\alpha q \cos\alpha+ \cos\alpha q \sin\alpha)(1-\cos \phi) + q \sin\alpha \cos \phi\end{bmatrix} \\ & = q \begin{bmatrix} -(\cos^2 \alpha +\sin ^2 \alpha)\sin \phi \\ \sin^2 \alpha(\cos\alpha-\cos\alpha )(1 - \cos \phi) + \cos\alpha \cos \phi \\ \cos^2 \alpha(-\sin\alpha + \sin\alpha)(1-\cos \phi) + \sin\alpha \cos \phi\end{bmatrix} \\ & = q \begin{bmatrix} -\sin \phi \\ \cos\alpha \cos \phi \\ \sin\alpha \cos \phi\end{bmatrix} \end{alignat} }


Generalized distance between two vectors

Warning: Errors below (this is just intermediate/working stuff)

Imagine reciprocal-space scattering that is a ring; more specifically a pseudo-toroid with Gaussian-like decay. The intensity overall is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I = \exp \left [ -(q_{rr}-q_0)^2/(2 \sigma_q^2) \right ] \exp \left [ -q_{rz}^2/(2 \sigma_q^2) \right ] }

Where we use the subscript r to denote the reciprocal-space coordinate system, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle q_{rr} = \sqrt{q_{rx}^2+q_{ry}^2}} . The plane of the detector (i.e. the Ewald plane) is denoted by d:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_{d} = \begin{bmatrix} q_{dx} & q_{dy} & 0 \end{bmatrix} }

We set the symmetry axis in realspace (detector coordinate system) to be the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle x} -axis. The reciprocal-space is tilted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi_0} (about the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle y} -axis), before the 'powder' rotation about the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle x} -axis (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \alpha} goes from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle -\pi} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle +\pi} ). Consider an initial vector:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_{d0} = \begin{bmatrix} q \sin \phi & q \cos \phi & 0 \end{bmatrix} }

The 1st rotation (about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle y} -axis by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi_0} ) involves:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{v}_{d1} & = R_y(\chi_0) \mathbf{v}_{d0} \\ & = \begin{bmatrix} \cos \chi_0 & 0 & \sin \chi_0 \\ 0 & 1 & 0 \\ -\sin \chi_0 & 0 & \cos \chi_0 \\ \end{bmatrix} \begin{bmatrix} q \sin \phi \\ q \cos \phi \\ 0 \\ \end{bmatrix} \\ & = \begin{bmatrix} q \sin \phi \cos \chi_0 \\ q \cos \phi \\ - q \sin \phi \sin \chi_0 \\ \end{bmatrix} \end{alignat} }

Consider a 2nd rotation around the vector (normal to the detector plane) (Warning: This is erroneous since the alpha rotation is just another phi rotation.):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{n}_{d1} & = R_y(\chi_0) \mathbf{n}_{d0} \\ & = \begin{bmatrix} \cos \chi_0 & 0 & \sin \chi_0 \\ 0 & 1 & 0 \\ -\sin \chi_0 & 0 & \cos \chi_0 \\ \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \\ \end{bmatrix} \\ & = \begin{bmatrix} \sin \chi_0 \\ 0 \\ \cos \chi_0 \\ \end{bmatrix} \end{alignat} }

So the second rotation yields:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{v}_{d2} & = \begin{bmatrix} u(ux +wz)(1-\cos \alpha) + x \cos \alpha + -wy \sin \alpha \\ y \cos \alpha + (wx-uz) \sin \alpha\\ w(ux+wz)(1-\cos \alpha) + z \cos \alpha + uy \sin \alpha \end{bmatrix} \\ & = \begin{bmatrix} \sin \chi_0 (\sin \chi_0 q \sin \phi \cos \chi_0 - \cos \chi_0 q \sin \phi \sin \chi_0)(1-\cos \alpha) + q \sin \phi \cos \chi_0 \cos \alpha + - \cos \chi_0 q \cos \phi \sin \alpha \\ q \cos \phi \cos \alpha + (\cos \chi_0 q \sin \phi \cos \chi_0 + \sin \chi_0 q \sin \phi \sin \chi_0) \sin \alpha\\ \cos \chi_0(\sin \chi_0 q \sin \phi \cos \chi_0 - \cos \chi_0 q \sin \phi \sin \chi_0)(1-\cos \alpha) + - q \sin \phi \sin \chi_0 \cos \alpha + \sin \chi_0 q \cos \phi \sin \alpha \end{bmatrix} \\ & = q \begin{bmatrix} \sin^2 \chi_0 \sin \phi (\cos \chi_0 - \cos \chi_0 )(1-\cos \alpha) + \sin \phi \cos \chi_0 \cos \alpha - \cos \chi_0 \cos \phi \sin \alpha \\ \cos \phi \cos \alpha + \sin \phi (\cos^2 \chi_0 + \sin^2 \chi_0 ) \sin \alpha\\ \cos \chi_0 \sin \chi_0 \sin \phi (\cos \chi_0 - \cos \chi_0 )(1-\cos \alpha) - \sin \phi \sin \chi_0 \cos \alpha + \sin \chi_0 \cos \phi \sin \alpha \end{bmatrix} \\ & = q \begin{bmatrix} \cos \chi_0 ( \sin \phi \cos \alpha - \cos \phi \sin \alpha ) \\ \cos \phi \cos \alpha + \sin \phi \sin \alpha\\ -\sin \chi_0 ( \sin \phi \cos \alpha - \cos \phi \sin \alpha ) \end{bmatrix} \\ \end{alignat} }
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