Talk:Geometry:TSAXS 3D - Revision history

KevinYager: /* Compute q_y */

2019-04-15T20:29:50Z

‎Compute q_y

KevinYager: /* Compute q_y */

2019-04-15T20:25:58Z

‎Compute q_y

KevinYager: /* Compute q_y */

2019-04-15T20:23:07Z

‎Compute q_y

KevinYager: /* Compute q_y */

2019-04-15T20:20:36Z

‎Compute q_y

KevinYager: /* Compute q_y */

2019-04-15T20:12:23Z

‎Compute q_y

KevinYager: /* Compute q_y */

2019-04-15T20:12:09Z

‎Compute q_y

KevinYager at 20:11, 15 April 2019

2019-04-15T20:11:53Z

KevinYager at 17:09, 13 January 2016

2016-01-13T17:09:58Z

KevinYager: /* Working results 1 */

2016-01-13T17:04:35Z

‎Working results 1

KevinYager: /* Working results 1 */

2016-01-13T16:47:48Z

‎Working results 1

@@ Line 12: / Line 12: @@
 \frac{q_x}{k} & = \sin \theta_f \cos \alpha_f \\
 \theta_f & = \sin^{-1} \left[ \frac{q_x}{k} \frac{1}{\cos \alpha_f} \right] \\
-q_y & = \cos \theta_f \cos \alpha_f - 1 \\
+\frac{q_y}{k} & = \cos \theta_f \cos \alpha_f - 1 \\
- & = \cos \left( \sin^{-1} \left[ \frac{q_x}{k} \frac{1}{\cos \alpha_f} \right] \right ) \cos \left ( \sin^{-1} \left[ \frac{q_z}{k} \right] \right ) - 1 \\
+q_y & = k \left ( \cos \left( \sin^{-1} \left[ \frac{q_x}{k} \frac{1}{\cos \alpha_f} \right] \right ) \cos \left ( \sin^{-1} \left[ \frac{q_z}{k} \right] \right ) - 1 \right )\\
-  & = \sqrt{ 1 - \left[ \frac{q_x}{k} \frac{1}{\cos \alpha_f} \right]^2 } \sqrt{ 1 - \left[ \frac{q_z}{k} \right]^2 } - 1
+  & = k \left ( \sqrt{ 1 - \left[ \frac{q_x}{k} \frac{1}{\cos \alpha_f} \right]^2 } \sqrt{ 1 - \left[ \frac{q_z}{k} \right]^2 } - 1 \right )
 \end{alignat}
 </math>
@@ Line 20: / Line 20: @@
 :<math>
 \begin{alignat}{2}
-q_y & = \sqrt{ 1 - \left[ \frac{q_x}{k} \frac{1}{\sqrt{1-[q_z/k]^2}} \right]^2 } \sqrt{ 1 - \left[ \frac{q_z}{k} \right]^2 } - 1 \\
+q_y & = k \left ( \sqrt{ 1 - \left[ \frac{q_x}{k} \frac{1}{\sqrt{1-[q_z/k]^2}} \right]^2 } \sqrt{ 1 - \left[ \frac{q_z}{k} \right]^2 } - 1 \right ) \\
-     & = \sqrt{ 1 - \frac{q_x^2}{k^2 (1-q_z^2/k^2) } } \sqrt{ 1 - \frac{q_z^2}{k^2} } - 1
+     & = k \sqrt{ 1 - \frac{q_x^2}{k^2 (1-q_z^2/k^2) } } \sqrt{ 1 - \frac{q_z^2}{k^2} } - k
 \end{alignat}
 </math>

@@ Line 20: / Line 20: @@
 :<math>
 \begin{alignat}{2}
-q_y & = \sqrt{ 1 - \left[ \frac{q_x}{k} \frac{1}{\sqrt{1-[q_z/k]^2}} \right]^2 } \sqrt{ 1 - \left[ \frac{q_z}{k} \right]^2 } - 1
+q_y & = \sqrt{ 1 - \left[ \frac{q_x}{k} \frac{1}{\sqrt{1-[q_z/k]^2}} \right]^2 } \sqrt{ 1 - \left[ \frac{q_z}{k} \right]^2 } - 1 \\
 \end{alignat}
 </math>

@@ Line 15: / Line 15: @@
   & = \cos \left( \sin^{-1} \left[ \frac{q_x}{k} \frac{1}{\cos \alpha_f} \right] \right ) \cos \left ( \sin^{-1} \left[ \frac{q_z}{k} \right] \right ) - 1 \\
   & = \sqrt{ 1 - \left[ \frac{q_x}{k} \frac{1}{\cos \alpha_f} \right]^2 } \sqrt{ 1 - \left[ \frac{q_z}{k} \right]^2 } - 1
 \end{alignat}
 </math>

@@ Line 3: / Line 3: @@
 \begin{alignat}{2}
 \mathbf{q} & = \begin{bmatrix} q_x \\ q_y \\ q_z \end{bmatrix} \\
-     & = \frac{2 \pi}{\lambda} \begin{bmatrix} \sin \theta_f \cos \alpha_f  \\ \cos \theta_f \cos \alpha_f - 1 \\ \sin \alpha_f \end{bmatrix}
+     & = k \begin{bmatrix} \sin \theta_f \cos \alpha_f  \\ \cos \theta_f \cos \alpha_f - 1 \\ \sin \alpha_f \end{bmatrix}
 \end{alignat}
 </math>
 So:
 :<math>
 </math>

@@ Line 1: / Line 1: @@
-===Compute <math>q_y</math>===
+====Compute <math>q_y</math>====
 :<math>
 \begin{alignat}{2}

← Older revision		Revision as of 20:11, 15 April 2019
Line 1:		Line 1:
		+	===Compute <math>q_y</math>===
		+	:<math>
		+	\begin{alignat}{2}
		+	\mathbf{q} & = \begin{bmatrix} q_x \\ q_y \\ q_z \end{bmatrix} \\
		+	& = \frac{2 \pi}{\lambda} \begin{bmatrix} \sin \theta_f \cos \alpha_f \\ \cos \theta_f \cos \alpha_f - 1 \\ \sin \alpha_f \end{bmatrix}
		+	\end{alignat}
		+	</math>

	====Scratch/working (contains errors)====		====Scratch/working (contains errors)====

@@ Line 49: / Line 49: @@
 \begin{alignat}{2}
 \left( \frac{q}{k} \right)^2
-     & = \left( \frac{x}{ \sqrt{x^2 + d^2 + z^2 d^2  }} \right)^2 + \left( \frac{d - \sqrt{x^2 + d^2 + z^2 d^2  } }{\sqrt{x^2 + d^2 + z^2 d^2  }} \right)^2 + \left( \frac{z d }{\sqrt{x^2 + d^2 + z^2 d^2  }} \right)^2 \\
+     & = \left( \frac{x}{ \sqrt{x^2 + d^2 + z^2  }} \right)^2 + \left( \frac{d - \sqrt{x^2 + d^2 + z^2  } }{\sqrt{x^2 + d^2 + z^2  }} \right)^2 + \left( \frac{z }{\sqrt{x^2 + d^2 + z^2  }} \right)^2 \\
-     & = \frac{x^2 + \left( d - \sqrt{x^2 + d^2 + z^2 d^2  }\right)^2 + z^2d^2 }{x^2 + d^2 + z^2d^2} \\
+     & = \frac{x^2 + \left( d - \sqrt{x^2 + d^2 + z^2   }\right)^2 + z^2 }{x^2 + d^2 + z^2} \\
-     & = \frac{x^2 + \left( d^2 - 2d \sqrt{x^2 + d^2 + z^2 d^2} + x^2 + d^2 + z^2 d^2  \right) + z^2d^2 }{x^2 + d^2 + z^2d^2} \\
+     & = \frac{x^2 + \left( d^2 - 2d \sqrt{x^2 + d^2 + z^2 } + x^2 + d^2 + z^2   \right) + z^2 }{x^2 + d^2 + z^2} \\
-     & = \frac{2 x^2 + 2 d^2 + 2 z^2d^2 - 2d \sqrt{x^2 + d^2 + z^2 d^2} }{x^2 + d^2 + z^2d^2} \\
+     & = \frac{2 x^2 + 2 d^2 + 2 z^2 - 2d \sqrt{x^2 + d^2 + z^2 } }{x^2 + d^2 + z^2} \\
-     & = 2 \frac{( x^2 + d^2 + z^2d^2 ) - d \sqrt{x^2 + d^2 + z^2 d^2} }{x^2 + d^2 + z^2d^2} \\
+     & = 2 \frac{( x^2 + d^2 + z^2 ) - d \sqrt{x^2 + d^2 + z^2 } }{x^2 + d^2 + z^2} \\
-     & = 2 \left( 1  - \frac{d}{\sqrt{x^2 + d^2 + z^2d^2}} \right)
+     & = 2 \left( 1  - \frac{d}{\sqrt{x^2 + d^2 + z^2}} \right)
 \end{alignat}
 </math>

@@ Line 17: / Line 17: @@
 \end{alignat}
 </math>
-Note that <math>\cos \theta_f = d^2/\sqrt{d^2+x^2}</math>, and <math>\cos^2 \theta_f = d^4/(d^2+x^2)</math> so:
+Note that <math>\cos \theta_f = d/\sqrt{d^2+x^2}</math>, and <math>\cos^2 \theta_f = d^2/(d^2+x^2)</math> so:
 :<math>
 \begin{alignat}{2}
 \frac{1}{\sqrt{d^2+z^2 \cos^2 \theta_f }}
-     & = \frac{1}{\sqrt{d^2+z^2 \left( d^4/(d^2+x^2) \right) }} \\
+     & = \frac{1}{\sqrt{d^2+z^2 \left( d^2/(d^2+x^2) \right) }} \\
-     & = \frac{1}{\sqrt{d^2} \sqrt{((d^2+x^2)+z^2 d^2)/(d^2+x^2)  }} \\
+     & = \frac{1}{\sqrt{d^2} \sqrt{((d^2+x^2)+z^2)/(d^2+x^2)  }} \\
-     & = \frac{\sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2 d^2  }} \\
+     & = \frac{\sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2  }} \\
 \end{alignat}
@@ Line 32: / Line 32: @@
 \mathbf{q}
 & = \frac{2 \pi}{\lambda} \begin{bmatrix}
-\frac{x d}{\sqrt{d^2+x^2 }} \frac{\sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2 d^2  }}  \\
+\frac{x d}{\sqrt{d^2+x^2 }} \frac{\sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2  }}  \\
-\frac{d}{\sqrt{d^2+x^2}} \frac{d \sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2 d^2  }} - 1 \\
+\frac{d}{\sqrt{d^2+x^2}} \frac{d \sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2  }} - 1 \\
-\frac{z \left( d^2/\sqrt{d^2+x^2} \right) \sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2 d^2  }} \end{bmatrix} \\
+\frac{z \left( d/\sqrt{d^2+x^2} \right) \sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2  }} \end{bmatrix} \\
 & = \frac{2 \pi}{\lambda} \begin{bmatrix}
-\frac{x}{ \sqrt{x^2 + d^2 + z^2 d^2  }}  \\
+\frac{x}{ \sqrt{x^2 + d^2 + z^2   }}  \\
-\frac{d }{\sqrt{x^2 + d^2 + z^2 d^2  }} - 1 \\
+\frac{d }{\sqrt{x^2 + d^2 + z^2   }} - 1 \\
-\frac{z d }{\sqrt{x^2 + d^2 + z^2 d^2  }} \end{bmatrix} \\
+\frac{z }{\sqrt{x^2 + d^2 + z^2   }} \end{bmatrix} \\