**4. Further Developments**

In Appendix A (Equations (A1) to (A12) and Figures A1–A3), a more complete collection of curved shapes to explain the different energy exchanges can be found. Following the discussion, we introduce an equation that correlates the radiative transfers of circular sectors with connected rectangles (Figure 7). This clearly applies to the geometry of the tunnel, a form that has been never been calculated previously due to is difficulty in integration [7].

For any vertical circular sector with a center situated in the middle of the edge *x* of a horizontal rectangle of dimensions *x*, *y*; by virtue of the Cabeza-Lainez seventh law, the form factor from the said rectangle to the sector of radius *r* will be (Figure 8):

$$\begin{array}{c} \mathbf{t} = \mathbf{r}^2 + \mathbf{y}^2 + \mathbf{x}^2; \; \mathbf{m} = \sqrt{\mathbf{x}^2 \times \sin^2 \theta\_1 + \mathbf{y}^2}; \; \mathbf{n} = \sqrt{\mathbf{x}^2 \times \sin^2 \theta\_2 + \mathbf{y}^2} \\\ F\_{21} = \frac{\mathbf{y}}{2\pi} \times \left( \frac{\cos \theta\_1}{m} \times \arctan \frac{r}{\frac{\cos \theta\_1 x}{m} \times (\cos \theta\_1 \mathbf{x} - r)} - \frac{\cos \theta\_2}{n} \times \arctan \frac{r}{n + \frac{\cos \theta\_2 x}{n} \times (\cos \theta\_2 \mathbf{x} - r)} \right) \\\ \qquad \qquad \qquad \qquad + \frac{\mathbf{y}}{4\pi \times \pi} \ln \left[ \frac{(t - 2 \cos \theta\_1 \times \mathbf{r} \times \mathbf{r})}{(t - 2 \cos \theta\_2 \times \mathbf{r} \times \mathbf{r})} \right] \end{array} \tag{15}$$

The sector is comprised between the angles θ2 and θ1, being its radius *r*.

**Figure 7.** The surfaces and quantities that constitute a tunnel figure.

**Figure 8.** Energy distribution on the floor of a tunnel with a radius of 3 and a length of 10 (non-dimensional).

These important results lead to more possibilities of a simulation where different fragments of figures like conoids, spheres, cones, or circular sections are involved (Figure 8). All of these are very important to the appropriate study of lighting in heritage architecture, given that, as mentioned previously, the cuboid rarely appears in Asian or baroque temples [8].

Careful numerical integration methods give us the Cabeza-Lainez sixth law (Figure 9), which might be considered as a further generalization of the fourth law. We present them together in the graphs below (Figure 10).

**Figure 9.** A quarter of a circle with a radius of 4 non-dimensional, and the energy spread over a semicircle with common edge.

The sixth law is described as:

$$F\_{23} = 1 - 2 \times F\_{21} - \frac{\alpha}{\pi} + \frac{1}{2} \left(\frac{\alpha}{\pi}\right)^2 \tag{16}$$

where α has the same angular meaning as before, but this time the law compares a quarter of the circles instead of semicircles.

**Figure 10.** Fourth law (in orange) compared to the new sixth law (Equation (16), red), which directly connects to the seventh law (blue) f.

At this point of the discussion, we believe that a vast repertoire of unsuspected curved shapes in the literature has been covered in a robust way. Some other forms that have also been defined by the authors can be found in [9,10]. The forms solved can be applied with perfect ease to any kind of radiant energy, for example, artificial lighting, emitted heat, or daylight. In each case, we need to find the climatic data and boundary conditions that will appear in the particular problem considered. Appendix B (Equations (A13) and (A14)) gives some hints as to the above issue.
