Extra:Intersecting planes - Revision history

KevinYager: /* a */

2015-12-23T22:16:51Z

‎a

KevinYager: /* Generalized intersection */

2015-12-23T22:01:18Z

‎Generalized intersection

KevinYager: /* a */

2015-12-23T21:52:01Z

‎a

KevinYager: /* a */

2015-12-23T21:50:36Z

‎a

KevinYager: /* Generalized intersection */

2015-12-23T21:48:28Z

‎Generalized intersection

KevinYager: /* Generalized intersection */

2015-12-23T21:48:12Z

‎Generalized intersection

KevinYager: /* Generalized intersection */

2015-12-23T21:46:15Z

‎Generalized intersection

KevinYager: /* Generalized intersection */

2015-12-23T21:40:18Z

‎Generalized intersection

KevinYager: /* Generalized intersection */

2015-12-23T21:36:54Z

‎Generalized intersection

KevinYager: /* Generalized intersection */

2015-12-23T21:28:45Z

‎Generalized intersection

@@ Line 180: / Line 180: @@
 </math>
 More specifically:
 ::<math>

@@ Line 180: / Line 180: @@
 </math>
+===a===
 More specifically:
 ::<math>

@@ Line 180: / Line 180: @@
 </math>
-====a====
 More specifically:
 ::<math>
@@ Line 196: / Line 196: @@
      & =
 q \begin{bmatrix}
-\cos \chi_0 \sin \phi(\cos^2 \chi_0 +\sin^2 \chi_0 )(1-\cos \alpha) + \sin \phi \cos \chi_0 \cos \alpha +\sin \chi_0 \cos \phi \sin \alpha \\
+\cos \chi_0 \sin \phi(1-\cos \alpha) + \sin \phi \cos \chi_0 \cos \alpha +\sin \chi_0 \cos \phi \sin \alpha \\
-\cos \phi \cos \alpha + \sin \phi (-\sin \chi_0  \cos \chi_0 + \cos \chi_0 \sin \chi_0 ) \sin \alpha\\
+\cos \phi \cos \alpha\\
--\sin \chi_0 \sin \phi ( \cos^2 \chi_0  +\sin^2 \chi_0 )(1-\cos \alpha) - \sin \phi \sin \chi_0 \cos \alpha + \cos \chi_0 \cos \phi \sin \alpha \end{bmatrix} \\
+-\sin \chi_0 \sin \phi (1-\cos \alpha) - \sin \phi \sin \chi_0 \cos \alpha + \cos \chi_0 \cos \phi \sin \alpha \end{bmatrix} \\
 \end{alignat}
 </math>
 This vector describes the possible positions of the intersecting detector-plane, in the sample's reciprocal-space.

@@ Line 192: / Line 192: @@
 q \begin{bmatrix}
 \cos \chi_0 \sin \phi(\cos^2 \chi_0 +\sin^2 \chi_0 )(1-\cos \alpha) + \sin \phi \cos \chi_0 \cos \alpha +\sin \chi_0 \cos \phi \sin \alpha \\
-\cos \phi \cos \alpha + (-\sin \chi_0 \sin \phi \cos \chi_0 + \cos \chi_0 \sin \phi \sin \chi_0 ) \sin \alpha\\
+\cos \phi \cos \alpha + \sin \phi (-\sin \chi_0  \cos \chi_0 + \cos \chi_0 \sin \chi_0 ) \sin \alpha\\
--\sin \chi_0 ( \cos \chi_0 \sin \phi \cos \chi_0 +\sin \chi_0 \sin \phi \sin \chi_0 )(1-\cos \alpha) - \sin \phi \sin \chi_0 \cos \alpha + \cos \chi_0 \cos \phi \sin \alpha \end{bmatrix} \\
+-\sin \chi_0 \sin \phi ( \cos^2 \chi_0  +\sin^2 \chi_0 )(1-\cos \alpha) - \sin \phi \sin \chi_0 \cos \alpha + \cos \chi_0 \cos \phi \sin \alpha \end{bmatrix} \\
 \end{alignat}
 </math>
 This vector describes the possible positions of the intersecting detector-plane, in the sample's reciprocal-space.

@@ Line 180: / Line 180: @@
 </math>
 More specifically:
 ::<math>

@@ Line 182: / Line 182: @@
 More specifically:
 ::<math>
-\mathbf{v}_{d2} =
+\begin{alignat}{2}
-\begin{bmatrix} u(ux +wz)(1-\cos \alpha) + x \cos \alpha + -wy \sin \alpha \\
+\mathbf{v}_{d2} & =
-y \cos \alpha + (wx-uz) \sin \alpha\\
+\begin{bmatrix}
-w(ux+wz)(1-\cos \alpha) + z \cos \alpha + uy \sin \alpha \end{bmatrix}
+\cos \chi_0 (\cos \chi_0 q \sin \phi \cos \chi_0 +\sin \chi_0 q \sin \phi \sin \chi_0)(1-\cos \alpha) + q \sin \phi \cos \chi_0 \cos \alpha +\sin \chi_0 q \cos \phi \sin \alpha \\
 </math>
 This vector describes the possible positions of the intersecting detector-plane, in the sample's reciprocal-space.

@@ Line 134: / Line 134: @@
 ::<math>
 \begin{alignat}{2}
-\mathbf{u}_{d1} & = R_y(\chi_0) \mathbf{n}_{d0} \\
+\mathbf{u}_{d1} & = R_y(\chi_0) \mathbf{u}_{d0} \\
      & = \begin{bmatrix}
 \cos \chi_0 & 0 & \sin \chi_0 \\
@@ Line 174: / Line 174: @@
 This reduces the problem to:
 ::<math>
+\mathbf{v}_{\mathrm{end}} =
 </math>
 More specifically:
 ::<math>
+\mathbf{v}_{d2} =
 </math>
 This vector describes the possible positions of the intersecting detector-plane, in the sample's reciprocal-space.

@@ Line 153: / Line 153: @@
 </math>
-===more===
+The second rotation can again be thought of in general terms as <math>\scriptstyle \mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x & y  & z \end{bmatrix}</math> rotated by <math>\scriptstyle \theta</math> about an unit-vector <math>\scriptstyle \mathbf{u} = \begin{bmatrix} u & v  & w \end{bmatrix}</math>.
-So the second rotation yields:
+::<math>
 ::<math>
 \begin{alignat}{2}
-\mathbf{v}_{d2} & =
+\mathbf{v}_{\mathrm{start}} &
-\begin{bmatrix} u(ux +wz)(1-\cos \alpha) + x \cos \alpha + -wy \sin \alpha \\
+= \begin{bmatrix} x \\ y  \\ z \end{bmatrix}
-y \cos \alpha + (wx-uz) \sin \alpha\\
+= \mathbf{v}_{d1}
-w(ux+wz)(1-\cos \alpha) + z \cos \alpha + uy \sin \alpha \end{bmatrix} \\
+= \begin{bmatrix} q \sin \phi \cos \chi_0 \\ q \cos \phi  \\ -q \sin \phi \sin \chi_0 \end{bmatrix} \\
-    & =
+\mathbf{u} &
-\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 \\
+= \begin{bmatrix} u \\ v  \\ w \end{bmatrix}
-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\\
+= \mathbf{u}_{d1}
-\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} \\
+= \begin{bmatrix} \cos \chi_0 \\ 0  \\ -\sin \chi_0 \end{bmatrix}
 \end{alignat}
 </math>
 This vector describes the possible positions of the intersecting detector-plane, in the sample's reciprocal-space.