The approximation writes the reflection coefficient as$$R(\theta)=R_0+(1-R_0)(1-\cos\theta)^5,\,R_0=\left(\frac{n_1-n_2}{n_1+n_2}\right)^2.$$Why is the exponent 5? Schlick 1994 introduces this exponent in Eq. (24), claiming it's the right Fresnel approximation, but with no explanation.

  • $\begingroup$ Before equation 15, they have several constraints. I guess they extended those a bit further in order to get a higher continuity at $F_{\lambda}^{(k)}(1) = 0$. $\endgroup$
    – lightxbulb
    Jan 26, 2020 at 13:42
  • 2
    $\begingroup$ @lightxbulb Thanks. An exponent of $3$ or $4$ would meet the stated conditions, but I suspect $5$ was obtained either with an approximation of (13) or empirical trial & error. $\endgroup$
    – J.G.
    Jan 26, 2020 at 14:16
  • $\begingroup$ Just plug in a larger $k$ in what I wrote, and you'll get the higher degree. $\endgroup$
    – lightxbulb
    Jan 26, 2020 at 14:48
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    $\begingroup$ @lightxbulb I understand the calculus, but such conditions have no motive apparent in the paper. $\endgroup$
    – J.G.
    Jan 26, 2020 at 14:50


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