*3.4. Opaque Flakes of Fixed Size as a Limiting Case*

Obviously, BRDF scaling for opaque flakes of fixed size that was initially investigated in [7] can be calculated now as the limiting case when the distribution of area is a deltafunction and *t* = 0.

In case this distribution is sharp, *γm*(*Z*) quickly change at *Z* = *R* and

$$\gamma\_{\mathfrak{m}}(Z) = \overline{D}\pi^{\frac{3-\mathfrak{m}}{2}}\overline{\mathcal{R}}^{3-\mathfrak{m}}\begin{cases} 0, & Z > \overline{\mathcal{R}} \\ 1, & Z < \overline{\mathcal{R}} \end{cases}$$

Thus,

$$\alpha(Z) = \frac{2}{\pi} \overline{\text{DSR}} \Big( 1.6361 \min(\zeta, 1) - 1.2722 \left( \min(\zeta, 1) \right)^2 + 0.30278 \left( \min(\zeta, 1) \right)^3 \Big)$$

$$\zeta \equiv \frac{Z}{R}$$

As a result, the integral of attenuation is

$$\int\_0^H a\left(v', u', z\right) dz \approx \frac{2b}{\overline{DS}} \int\_0^{\frac{\sqrt{2}b}{2R}} e^{-4\left(1-t\right)b\left(x - \left(1-t\right)\frac{1}{\pi}\left(1.6361x - 1.2722x^2 + 0.30278x^3\right)\right)} dx$$

where *<sup>b</sup>* <sup>≡</sup> *DSR <sup>c</sup>* <sup>=</sup> *<sup>F</sup> c R <sup>H</sup>* in case *cH* <sup>2</sup> < *R*, and

$$\begin{split} \int\_{0}^{H} a(\upsilon', \mu', z) dz &\quad \approx \frac{2h}{\mathrm{DS}} \int\_{0}^{1} e^{-4(1-t)b(x-(1-t)\frac{1}{H}(1.6361x - 1.2722x^{2} + 0.30278x^{3}))} dx \\ &\quad + e^{4b(1-t)} \, ^2\_{\mathrm{NT}} \frac{e^{-4b(1-t)} - e^{-2F(1-t)}}{2\overline{D}\overline{S}(1-t)} \end{split}$$

in case *cH* <sup>2</sup> > *R*.

For opaque flakes this gives

$$\int\_{0}^{H} a(v', u', z) \, dz = \begin{cases} \frac{2b}{\widetilde{DS}} \int\_{0}^{\frac{r}{\widetilde{\Delta}}} e^{-4b(x - \frac{1}{\pi}(1.6361x - 1.2722x^2 + 0.30278x^3))} \, d\chi, & \frac{cH}{2} < \overline{R} \\\frac{2b}{\widetilde{DS}} \int\_{0}^{1} e^{-4b(x - \frac{1}{\pi}(1.6361x - 1.2722x^2 + 0.30278x^3))} \, d\chi + e^{\frac{4b}{\widetilde{DS}}} \frac{e^{-4b} - e^{-2F}}{2\widetilde{DS}}, & \frac{cH}{2} > \overline{R} \end{cases}$$

The difference from [7] is very small because 2 *<sup>x</sup>* <sup>−</sup> <sup>1</sup> *π* 1.6361*<sup>x</sup>* − 1.2722*x*<sup>2</sup> + 0.30278*x*<sup>3</sup> ≈ *x* + 0.58*x*2.
