# Why does Schlick's approximation contain a \$(1-\cos\theta)^5\$ term?

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.

• Before equation 15, they have several constraints. I guess they extended those a bit further in order to get a higher continuity at $F_{\lambda}^{(k)}(1) = 0$. – lightxbulb Jan 26 at 13:42
• @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. – J.G. Jan 26 at 14:16
• Just plug in a larger $k$ in what I wrote, and you'll get the higher degree. – lightxbulb Jan 26 at 14:48
• @lightxbulb I understand the calculus, but such conditions have no motive apparent in the paper. – J.G. Jan 26 at 14:50