Talk:Geometry:WAXS 3D - Revision history

KevinYager at 21:22, 13 January 2016

2016-01-13T21:22:32Z

KevinYager: Blanked the page

2016-01-13T21:21:25Z

Blanked the page

KevinYager: /* Check of Total Magnitude #2: Doesn't work */

2016-01-13T21:21:11Z

‎Check of Total Magnitude #2: Doesn't work

KevinYager: /* Check of Total Magnitude #2: Doesn't work */

2016-01-13T21:20:46Z

‎Check of Total Magnitude #2: Doesn't work

KevinYager: /* Check of Total Magnitude #2: Doesn't work */

2016-01-13T21:16:42Z

‎Check of Total Magnitude #2: Doesn't work

KevinYager: /* Check of Total Magnitude #2: Doesn't work */

2016-01-13T21:06:01Z

‎Check of Total Magnitude #2: Doesn't work

KevinYager at 20:25, 13 January 2016

2016-01-13T20:25:58Z

KevinYager: /* Check */

2016-01-13T20:25:20Z

‎Check

KevinYager: /* Check */

2016-01-13T20:14:47Z

‎Check

KevinYager: /* Check */

2016-01-13T20:11:46Z

‎Check

← Older revision	Revision as of 21:22, 13 January 2016
Line 1:	Line 1:
	+	<source lang="python" line>
	+	#!/usr/bin/python
	+	# Quick/rough Python code to check (numerically) the derivations...
	+	import numpy as np

	+	for i in range(10):
	+
	+	x = np.random.uniform(-10,10)
	+	z = np.random.uniform(-10,10)
	+	d = np.random.uniform(10,1000)
	+
	+
	+	phi = np.random.uniform(-np.pi, +np.pi)
	+	theta = np.random.uniform(0, +np.pi)
	+
	+	print( 'x{:+05.1f} d{:+06.1f} z{:+05.1f} phi{:+06.1f} theta{:+06.1f}'.format(x, d, z, np.degrees(theta), np.degrees(phi)) )
	+
	+	#k = 1.0
	+
	+	v2y = dnp.cos(theta) - znp.sin(theta)
	+	dprime = np.sqrt( np.square(x) + np.square(d) + np.square(z) )
	+
	+	qx = xnp.cos(phi) - np.sin(phi)v2y
	+	qy = xnp.sin(phi) + np.cos(phi)v2y - dprime
	+	qz = dnp.sin(theta) + znp.cos(theta)
	+
	+	q = np.sqrt( np.square(qx) + np.square(qy) + np.square(qz) )
	+
	+
	+
	+
	+	print( ' pieces qx{:g} qy{:g} qz{:g} q{:g}'.format(qx, qy, qz, q) )
	+	#print( '{}{:g}'.format( ' '*50, q) )
	+
	+	if False:
	+	l1 = np.square(qx)
	+	l2 = np.square(qy)
	+	l3 = np.square(qz)
	+	qalt2 = l1 + l2 + l3
	+
	+	if False:
	+	l1 = np.square(xnp.cos(phi) - np.sin(phi)(dnp.cos(theta) - znp.sin(theta)))
	+	l2 = np.square(xnp.sin(phi) + np.cos(phi)(dnp.cos(theta) - znp.sin(theta)) - dprime)
	+	l3 = np.square(dnp.sin(theta) + znp.cos(theta))
	+	qalt2 = l1 + l2 + l3
	+
	+	if True:
	+	l1 = np.square(x)np.square(np.cos(phi)) - 2xnp.cos(phi)np.sin(phi)v2y + np.square(np.sin(phi))np.square(v2y)
	+	l2 = np.square(x)np.square(np.sin(phi)) + xnp.sin(phi)np.cos(phi)v2y - dprimexnp.sin(phi)
	+	l3 = xnp.sin(phi)np.cos(phi)v2y + np.square(np.cos(phi))np.square(v2y) - dprimenp.cos(phi)v2y
	+	l4 = -1dprimexnp.sin(phi) - dprimenp.cos(phi)*v2y + np.square(dprime)
	+	l5 = np.square(d)np.square(np.sin(theta)) + 2dznp.sin(theta)np.cos(theta) + np.square(z)np.square(np.cos(theta))
	+
	+	qalt2 = l1 + l2 + l3 + l4 + l5
	+
	+	if True:
	+
	+	qalt2 = 2dprime( dprime - xnp.sin(phi) - np.cos(phi)v2y )
	+
	+
	+	qalt = np.sqrt(qalt2)
	+
	+	print( ' qalt {:g}'.format(qalt) )
	+	print( '{}{:g}'.format( ' '*50, q-qalt) )
	+
	+	</source>

← Older revision		Revision as of 21:21, 13 January 2016
Line 1:		Line 1:
−	~~====Check of Total Magnitude #1: Doesn't work====~~
−	~~:<math>~~
−	~~\begin{alignat}{2}~~
−	~~\left ( \frac{q}{k} \right )^2 d^{\prime 2}~~
−	& = \begin{alignat}{2} [ & \left( x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right)^2 \\ & + \left( x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) - d^{\prime} \right)^2 \\ & + \left( d \sin \theta_g + z \cos \theta_g \right)^2 ] \end{alignat} \\

−	~~& = \begin{alignat}{2} [~~
−	~~& x^2 \cos^2 \phi_g - x \cos \phi_g \sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) + \sin^2 \phi_g ( d \cos \theta_g - z \sin \theta_g )^2 \\~~
−	~~& + x^2 \sin^2 \phi_g + x \sin \phi_g \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) - d^{\prime} x \sin \phi_g \\~~
−	~~& + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )x \sin \phi_g + \cos^2 \phi_g ( d \cos \theta_g - z \sin \theta_g )^2 - d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\~~
−	~~& - d^{\prime} x \sin \phi_g - d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) + d^{\prime 2} \\~~
−	~~& + d^2 \sin^2 \theta_g + 2 d \sin \theta_g z \cos \theta_g + z^2 \cos^2 \theta_g ] \end{alignat} \\~~
−
−	~~& = \begin{alignat}{2} [~~
−	~~& x^2 \cos^2 \phi_g - x \cos \phi_g \sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) + \sin^2 \phi_g ( d \cos \theta_g - z \sin \theta_g )^2 \\~~
−	~~& + x^2 \sin^2 \phi_g + 2 x \sin \phi_g \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) - 2 d^{\prime} x \sin \phi_g \\~~
−	~~& + \cos^2 \phi_g ( d \cos \theta_g - z \sin \theta_g )^2 - 2 d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) + d^{\prime 2} \\~~
−	~~& + d^2 \sin^2 \theta_g + 2 d \sin \theta_g z \cos \theta_g + z^2 \cos^2 \theta_g ] \end{alignat} \\~~
−
−	~~& = \begin{alignat}{2} [~~
−	~~& x^2 - x \sin \phi_g \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) + ( d \cos \theta_g - z \sin \theta_g )^2 \\~~
−	~~& + 2 x \sin \phi_g \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) - 2 d^{\prime} x \sin \phi_g \\~~
−	~~& - 2 d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) + d^{\prime 2} \\~~
−	~~& + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g ] \end{alignat} \\~~
−
−	~~& = \begin{alignat}{2} [~~
−	~~& x^2 + d^2 \cos^2 \theta_g - 2 dz \cos \theta_g \sin \theta_g + z^2 \sin^2 \theta_g \\~~
−	~~& + ( - x \sin \phi_g \cos \phi_g + 2 x \sin \phi_g \cos \phi_g - 2 d^{\prime} \cos \phi_g )( d \cos \theta_g - z \sin \theta_g ) \\~~
−	~~& - 2 d^{\prime} x \sin \phi_g \\~~
−	~~& + d^{\prime 2} \\~~
−	~~& + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g ] \end{alignat} \\~~
−
−	~~& = \begin{alignat}{2} [~~
−	~~& d^{\prime 2} + x^2 + d^2 + z^2 - 2 dz \cos \theta_g \sin \theta_g \\~~
−	~~& + ( x \sin \phi_g \cos \phi_g - 2 d^{\prime} \cos \phi_g )( d \cos \theta_g - z \sin \theta_g ) \\~~
−	~~& + 2 d z \sin \theta_g \cos \theta_g - 2 d^{\prime} x \sin \phi_g ] \end{alignat} \\~~
−
−	~~& = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g + ( x \sin \phi_g \cos \phi_g - 2 d^{\prime} \cos \phi_g )( d \cos \theta_g - z \sin \theta_g ) \\~~
−
−	~~& = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g + ( x \sin \phi_g - 2 d^{\prime} )\cos \phi_g( d \cos \theta_g - z \sin \theta_g ) \\~~
−
−
−	~~& = ? \\~~
−	~~& = ? \\~~
−	~~& = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g + 2 d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\~~
−	~~& = 2 d^{\prime} \left( d^{\prime} - x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right) \\~~
−	~~\left( \frac{q}{k} \right)^2~~
−	~~& = 2 \left( 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{d^{\prime} } \right)~~
−	~~\end{alignat}~~
−	~~</math>~~

@@ Line 45: / Line 45: @@
      & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g + 2 d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )  \\
      & = 2 d^{\prime} \left( d^{\prime} - x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right) \\
 \left( \frac{q}{k} \right)^2
      & = 2 \left( 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{d^{\prime} } \right)
 \end{alignat}
 </math>

@@ Line 103: / Line 103: @@
      & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} )\\
+     & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )  \\
-    & = ? \\
+     & = 2 d^{\prime} \left( d^{\prime} - x \sin \phi_g - \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right) \\
 \left( \frac{q}{k} \right)^2
      & = 2 \left( 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{d^{\prime} } \right)
 \end{alignat}
 </math>

@@ Line 57: / Line 57: @@
 ( v_{2y} ) & = ( d \cos \theta_g - z \sin \theta_g ) \\
 ( v_{2y} )^2 & = ( d \cos \theta_g - z \sin \theta_g )^2 \\
-     & = d^2 \cos^2 \theta_g -dz \cos \theta_g \sin\theta_g + z^2 \sin^2 \theta_g
+     & = d^2 \cos^2 \theta_g - 2dz \cos \theta_g \sin\theta_g + z^2 \sin^2 \theta_g
 \end{alignat}
 </math>
@@ Line 86: / Line 86: @@
 \left ( \frac{q}{k} \right )^2 d^{\prime 2}
      & = \begin{alignat}{2} [
-       & x^2  + ( v_{2y} )^2  \\
+       & x^2 + ( v_{2y} )^2  \\
        & - 2 d^{\prime} x \sin \phi_g \\
-       & + x \sin \phi_g \cos \phi_g ( v_{2y} ) - d^{\prime} \cos \phi_g ( v_{2y} ) \\
+       & - 2 d^{\prime} \cos \phi_g ( v_{2y} ) \\
-       & - d^{\prime} \cos \phi_g ( v_{2y} ) + d^{\prime 2} \\
+       & + d^{\prime 2} \\
-       & + d^2 \sin^2 \theta_g + d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g  ] \end{alignat}  \\
+       & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g  ] \end{alignat}  \\
      & = \begin{alignat}{2} [
-       & x^2  + ( d^2 \cos^2 \theta_g -dz \cos \theta_g \sin\theta_g + z^2 \sin^2 \theta_g )  \\
+       & d^{\prime 2} + x^2 + ( d^2 \cos^2 \theta_g - 2dz \cos \theta_g \sin\theta_g + z^2 \sin^2 \theta_g )  \\
-       & - 2 d^{\prime} x \sin \phi_g \\
+       & - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} ) \\
-      & + (x \sin \phi_g - d^{\prime} - d^{\prime}) \cos \phi_g ( v_{2y} )  \\
+       & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g  ] \end{alignat}  \\
      & = \begin{alignat}{2} [
-       & d^{\prime 2} + x^2  + d^2 + z^2    \\
+       & d^{\prime 2} + x^2 + d^2 + z^2   \\
-       & - 2 d^{\prime} x \sin \phi_g \\
+       & - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} ) ] \end{alignat}  \\
-      & + (x \sin \phi_g - 2 d^{\prime}) \cos \phi_g ( v_{2y} ) ] \end{alignat}  \\
      & = ? \\

@@ Line 74: / Line 74: @@
      & = \begin{alignat}{2} [
-       & x^2 \cos^2 \phi_g - x \cos \phi_g \sin \phi_g ( v_{2y} ) + \sin^2 \phi_g ( v_{2y} )^2  \\
+       & x^2 \cos^2 \phi_g - 2 x \cos \phi_g \sin \phi_g ( v_{2y} ) + \sin^2 \phi_g ( v_{2y} )^2  \\
        & + x^2 \sin^2 \phi_g + x \sin \phi_g \cos \phi_g ( v_{2y} ) - d^{\prime} x \sin \phi_g \\
        & + x \sin \phi_g \cos \phi_g ( v_{2y} ) + \cos^2 \phi_g ( v_{2y} )^2 - d^{\prime} \cos \phi_g ( v_{2y} ) \\
        & - d^{\prime} x \sin \phi_g - d^{\prime} \cos \phi_g ( v_{2y} ) + d^{\prime 2} \\
-       & + d^2 \sin^2 \theta_g + d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g  ] \end{alignat}  \\
+       & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g  ] \end{alignat}  \\
 \end{alignat}
 </math>

@@ Line 1: / Line 1: @@
-====Check====
+====Check of Total Magnitude #1: Doesn't work====
 We define:
 ::<math>

@@ Line 10: / Line 10: @@
 </math>
-And solve:
+And calculate:
 :<math>
 \begin{alignat}{2}
@@ Line 22: / Line 22: @@
        & + \left( d \sin \theta_g + z \cos \theta_g \right)^2  ] \end{alignat}  \\
      & = \begin{alignat}{2} [
        & x^2 \cos^2 \phi_g - x \cos \phi_g \sin \phi_g ( v_{2y} ) + \sin^2 \phi_g ( v_{2y} )^2  \\

@@ Line 16: / Line 16: @@
 \left ( \frac{q}{k} \right )^2 d^{\prime 2}
      & = \begin{alignat}{2} [ & \left( x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right)^2 \\ & + \left( x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) - d^{\prime} \right)^2 \\ & + \left( d \sin \theta_g + z \cos \theta_g \right)^2  ] \end{alignat}  \\
      & = \begin{alignat}{2} [ & \left( x \cos \phi_g -\sin \phi_g ( v_{2y} ) \right)^2 \\ & + \left( x \sin \phi_g + \cos \phi_g ( v_{2y} ) - d^{\prime} \right)^2 \\ & + \left( d \sin \theta_g + z \cos \theta_g \right)^2  ] \end{alignat}  \\