Cedrick at Lucas suggested some nice optimisations for the in-scattering equation I posted last time.
I had left off at:
\[L_{s} = \frac{\sigma_{s}I}{v}( \tan^{-1}\left(\frac{d+b}{v}\right) - \tan^{-1}\left(\frac{b}{v}\right) )\]
But we can remove one of the two inverse trigonometric functions by using the following identity:
\[\tan^{-1}x - \tan^{-1}y = \tan^{-1}\frac{x-y}{1+xy}\]
Which simplifies the expression for $L_{s}$ to:
\[L_{s} = \frac{\sigma_{s}I}{v}( \tan^{-1}\frac{x-y}{1+xy} )\]
With $x$ and $y$ being replaced by:
\[\begin{array}{lcl} x = \frac{d+b}{v} \\ y = \frac{b}{v}\end{array}\]
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This demo shows an analytic solution to the differential in-scattering equation for light in participating media. It's a similar but simplified version of equations found in[1], [2] and as I recently discovered [3]. However I thought showing the derivation might be interesting for some out there, plus it was a good excuse for me to brush up on my \(\LaTeX\).
You might notice I also updated the site's theme, unfortunately you need a white background to make wordpress.
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