Talk:Geometry:TSAXS 3D

Working results 1

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{q} & = \frac{2 \pi}{\lambda} \begin{bmatrix} \sin \theta_f \cos \alpha_f \\ \cos \theta_f \cos \alpha_f - 1 \\ \sin \alpha_f \end{bmatrix} \\ & = \frac{2 \pi}{\lambda} \begin{bmatrix} \sin \left( \arctan\left[ \frac{x}{d} \right] \right) \cos \left( \arctan \left[ \frac{z }{d / \cos \theta_f} \right] \right) \\ \cos \left( \arctan\left[ \frac{x}{d} \right] \right) \cos \left( \arctan \left[ \frac{z }{d / \cos \theta_f} \right] \right) - 1 \\ \sin \left( \arctan \left[ \frac{z }{d / \cos \theta_f} \right] \right) \end{bmatrix} \\ & = \frac{2 \pi}{\lambda} \begin{bmatrix} \frac{x/d}{\sqrt{1+\left(x/d \right)^2}} \frac{d}{\sqrt{d^2+z^2\cos^2 \theta_f}} \\ \frac{1}{\sqrt{1+\left(x/d \right)^2}} \frac{d}{\sqrt{d^2+z^2\cos^2 \theta_f}} - 1 \\ \frac{z \cos \theta_f}{\sqrt{d^2+z^2 \cos^2 \theta_f }} \end{bmatrix} \\ & = \frac{2 \pi}{\lambda} \begin{bmatrix} \frac{x d}{\sqrt{d^2+x^2 }} \frac{1}{\sqrt{d^2+z^2\cos^2 \theta_f}} \\ \frac{d}{\sqrt{d^2+x^2}} \frac{d}{\sqrt{d^2+z^2\cos^2 \theta_f}} - 1 \\ \frac{z \cos \theta_f}{\sqrt{d^2+z^2 \cos^2 \theta_f }} \end{bmatrix} \\ \end{alignat} }

Note that 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 \cos \theta_f = d^2/\sqrt{d^2+x^2}} , 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 \cos^2 \theta_f = d^4/(d^2+x^2)} so:

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} \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} \sqrt{((d^2+x^2)+z^2 d^2)/(d^2+x^2) }} \\ & = \frac{\sqrt{d^2+x^2}}{d \sqrt{d^2 + x^2 + z^2 d^2 }} \\ \end{alignat} }

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 \begin{alignat}{2} \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{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{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{2 \pi}{\lambda} \begin{bmatrix} \frac{x}{ \sqrt{x^2 + d^2 + z^2 d^2 }} \\ \frac{d }{\sqrt{x^2 + d^2 + z^2 d^2 }} - 1 \\ \frac{z d }{\sqrt{x^2 + d^2 + z^2 d^2 }} \end{bmatrix} \\ \end{alignat} }

As a check:

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} \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 \\ & = \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^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{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} \\ & = 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 \left( 1 - \frac{d}{\sqrt{x^2 + d^2 + z^2d^2}} \right) \end{alignat} }

Working results 2 (contains errors)

As a check of these results, consider:

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} q & = \sqrt{ q_x^2 + q_y^2 + q_z^2 } \\ & = \frac{2 \pi}{\lambda} \sqrt{ \sin^2 \theta_f \cos^2 \alpha_f + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + \sin^2 \alpha_f } \\ \left( \frac{q}{k} \right)^2 & = (\sin \theta_f)^2 (\cos \alpha_f)^2 + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + (\sin \alpha_f)^2 \\ & = \left(\frac{x/d}{\sqrt{1+(x/d)^2}} \right)^2 \left(\cos \alpha_f \right)^2 + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + \left( \frac{z \cos \theta_f /d }{\sqrt{1+(z \cos \theta_f /d)^2}} \right)^2 \\ & = \left(\frac{x}{\sqrt{d^2+x^2}} \right)^2 \left(\cos \alpha_f \right)^2 + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + \left( \frac{z \cos \theta_f }{\sqrt{d^2+z^2 \cos^2 \theta_f }} \right)^2 \\ & = \frac{x^2}{d^2+x^2} \left(\cos \alpha_f \right)^2 + \left ( \cos \theta_f \cos \alpha_f - 1 \right )^2 + \frac{z^2 \cos^2 \theta_f }{d^2+z^2 \cos^2 \theta_f } \\ & = \frac{x^2}{d^2+x^2} \frac{d^4}{d^2+z^2 \cos^2 \theta_f} + \left ( \cos \theta_f \frac{d^2}{\sqrt{d^2+z^2 \cos^2 \theta_f}} - 1 \right )^2 + \frac{z^2 \cos^2 \theta_f }{d^2+z^2 \cos^2 \theta_f } \end{alignat} }

Where we used:

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} \sin( \arctan[u]) & = \frac{u}{\sqrt{1+u^2}} \\ \sin \theta_f & = \sin( \arctan [x/d] ) \\ & = \frac{x/d}{\sqrt{1 + (x/d)^2}} \\ & = \frac{x}{\sqrt{d^2+x^2}} \end{alignat} }

And, we further note that:

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} \cos( \arctan[u]) & = \frac{1}{\sqrt{1+u^2}} \\ \cos \theta_f & = \frac{1}{\sqrt{1 + (x/d)^2}} \\ & = \frac{d^2}{\sqrt{d^2+x^2}} \end{alignat} }

Continuing:

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} \left( \frac{q}{k} \right)^2 & = \frac{x^2}{d^2+x^2} \frac{d^4}{d^2+z^2 \cos^2 \theta_f} + \left ( \frac{d^2}{\sqrt{d^2+x^2}} \frac{d^2}{\sqrt{d^2+z^2 \cos^2 \theta_f}} - 1 \right )^2 + \frac{z^2 }{d^2+z^2 \cos^2 \theta_f } \frac{d^4}{d^2+x^2} \\ & = d^4\frac{x^2+z^2}{(d^2+x^2)(d^2+z^2 \cos^2 \theta_f)} + \left ( \frac{d^4}{\sqrt{(d^2+x^2)(d^2+z^2 \cos^2 \theta_f)}} - 1 \right )^2 \\ & = \frac{d^4x^2+d^4z^2}{d^4+d^2x^2+d^4z^2} + \left ( \frac{d^4}{\sqrt{d^4+d^2x^2+d^4z^2}} - 1 \right )^2 \\ & = \frac{d^2x^2+d^2z^2}{d^2+x^2+d^2z^2} + \left ( \frac{d^8}{d^4+d^2x^2+d^4z^2} -2 \frac{d^4}{\sqrt{d^4+d^2x^2+d^4z^2}} + 1 \right ) \\ & = \frac{d^2x^2+d^2z^2}{d^2+x^2+d^2z^2} + \frac{d^6}{d^2+x^2+d^2z^2} -2 \frac{d^3}{\sqrt{d^2+x^2+d^2z^2}} + 1 \\ & = \frac{d^2x^2+d^2z^2 + d^6 -2d^3\sqrt{d^2+x^2+d^2z^2} + d^2+x^2+d^2z^2}{d^2+x^2+d^2z^2} \\ & = \frac{d^6 + d^2 + d^2x^2 + x^2 + 2d^2z^2 -2d^3\sqrt{d^2+x^2+d^2z^2}}{d^2+x^2+d^2z^2} \\ & = ? \\ & = ? \\ & = \frac{\sqrt{d^2+x^2+z^2} - d}{2 \sqrt{d^2+x^2+z^2}} \\ & = \frac{1}{2}\left(1 - \frac{d}{\sqrt{d^2+x^2+z^2}} \right) \\ q & = \frac{4 \pi}{\lambda} \sin \left( \theta_s \right) \end{alignat} }