**5. Inter-Reflections**

One last point needs to be explained before presenting some relevant case studies, which is the question of inter-reflection. To this aim, we developed the following procedure.

The total balance of energy depends on the equation

$$E\_{tot} = E\_{dir} + E\_{ref} \tag{17}$$

where *Edir* is the fraction of direct energy and *Eref* is the reflected fraction. If these quantities are summed, we obtain the global amount of radiation *Etot*. Usually, several surfaces intervene in the process of reflection, and thus a set of expressions can be built. It is possible to introduce a pair of instrumental arrays that we will call *Fd* and *Fr*. For a volume enclosed by three surfaces (see Figure 11), such matrices would have the ensuing form:

$$F\_d = \begin{pmatrix} F\_{11} \times \rho\_1 & F\_{12} \times \rho\_2 & F\_{13} \times \rho\_3 \\ F\_{21} \times \rho\_1 & F\_{22} \times \rho\_2 & F\_{23} \times \rho\_3 \\ F\_{31} \times \rho\_1 & F\_{32} \times \rho\_2 & F\_{33} \times \rho\_3 \end{pmatrix} \tag{18}$$

$$F\_{\mathbf{r}} = \begin{pmatrix} 1 & -F\_{12} \times \rho\_2 & -F\_{13} \times \rho\_3 \\ -F\_{21} \times \rho\_1 & 1 & -F\_{23} \times \rho\_3 \\ -F\_{31} \times \rho\_1 & -F\_{32} \times \rho\_2 & 1 \end{pmatrix} \tag{19}$$

with factors *Fij* as previously found, yielding the energy transfer between the respective surfaces involved. Here, ρi represents the ratio of reflection (direct or otherwise) assigned to a particular element *i* [1].

Notice that in the matrix called *Fd* (Equation (18)) the diagonals are not null as the *Fii* (with two ii) elements have definite values for curved surfaces, unlike the exchange in a cuboid.

If after careful integration by the procedures described previously, we are able to fix all the elements in Equations (18) and (19), we can establish new expressions (Equations (20)–(22)) in order to correlate the primary direct energy with the one extracted by reflections.

$$F\_r \times E\_{nvf} = F\_d \times E\_{dir} \tag{20}$$

$$F\_{rd} = F\_r^{-1} \times F\_d \tag{21}$$

$$E\_{ref} = F\_{rd} \times E\_{dir} \tag{22}$$

As the amount of internally diffused energy is now a function of direct transfer, the problem is settled. A novel operation defined by Cabeza-Lainez is termed as the *product of form factors*, and being a sort of convolution operator it is still unrevealed how it applies in the calculation of these shapes.

**Figure 11.** Volume composed only of curved figures, which is usual in heritage architecture.
