**1. Introduction**

The fundamental question of light in heritage architecture remains unsettled due to a lack of scientific advances in the field. The main objective of our research in this article was to implement the adroit correlations and expressions that we found, in order to deal with the variegated phenomena of light as it distributes over complex architectural surfaces from the past. It is known that daylight stems from surface sources. However, many designers do not properly understand the radiative performance of such surfaces, especially when typical cuboid shapes are absent, which is the main reason behind our efforts.

Disregard of surface sources of a three-dimensional shape in radiative transfer analysis is altogether usual. This can be explained by the complexities of the integral equations involved and is partly due to the want of a sufficiently geometric approach to the problem, if scholars come from fields of expertise outside design or architecture. When dealing with complex radiant surfaces, assimilation to a cluster of point sources is the base for the application of the iterative method common to finite element approaches in software, and sometimes direct integration is unable to yield adequate expressions, so only insufficient results exist for fewer basic forms [1,2]. By virtue of this, an ample list of meaningful

geometries that are often present in heritage architecture has remained neglected and energy waste or inaccuracies are evident. In order to increase the sustainability of the whole process from the restoration of ancient vestiges to public exhibition, we considered it necessary to start from a different approach.

In the ensuing research, we begin by focusing on the curved surfaces and derived shapes. Following a similar approach to radiation exchanges through detailed reflection on the particular details of different volumes, we hereby offer innovative and simpler expressions obtained from geometric adjustments that we have found in many sites around Asia.

As an initial discussion, we depart from the well-known spherical trigonometry. By means of basic calculation, we devised a new set of promising laws. These, in turn, result in previously unknown configuration factors for several fragments that involve the sphere. Afterward, the method can be extended to other organic shapes often contained in the architecture of the past.

In the second part, we accomplish the research by adding the equations concerned with finding inter-reflections in the said curved elements.

At all times, we conveyed the findings to an innovative software that enhances luminous radiative transfer simulation for a significant quantity of products and designs, especially of the paradigmatic cases that we give in our conclusions.

### **2. Physical Outline of the Problem**

The reciprocity principle appears in Lambert's treatise *Photometria* [3], written in Latin in 1761 and is explicit in Equation (1). It produces the ensuing differential Equation (2), which still constitutes the scientific fundament of form factor exchanges for all kind of surfaces.

$$d\mathcal{Q}\_{1-2} = (E\_{b1} - E\_{b2})\cos\theta\_1 \times \cos\theta\_2 \times \frac{dA\_1 \times dA\_2}{\pi \times r^2} \tag{1}$$

$$\mathcal{Z}\_{1-2} = (E\_{b1} - E\_{b2}) \int\_{A\_1} \int\_{A\_2} \cos \theta\_1 \times \cos \theta\_2 \times \frac{dA\_1 \mathbf{x} dA\_2}{\pi \mathbf{x} r^2} \tag{2}$$

where *Ebi* is the radiative power exiting from surface 1 or 2; *Ai* is the extension of each surface; *dAi* is the elementary area; r is the vector that measures distance; and θ*i* is the angle between distance vector and the normal to the surface for each elementary area considered (Figure 1).

**Figure 1.** The reciprocity principle and equation for arbitrary surfaces A1 and A2.

The above equation indicates that radiative exchanges in any shape depend partly on the size, but also on its relative situation within the space under consideration (Figure 1). From the influential book of Lambert to the 21st century, engineers and scholars of either optics or radiation heat transfer have tried hard to find ways of solving the canonical Equation (2) for sundry elements [4]. This implies a

considerable e ffort as the said equation takes us on many occasion to a four-fold integration and the doubled primitive of simple functions implies many complex operations.

The variable, or internal part of the integral, is a dimensionless ratio called the form factor [1]. From the quantum electrodynamic point of view, the form factor represents the probability that the photons uniformly emitted from a certain surface can hit another to which it exposes in some manner [5]. Curvilinear forms may acquire distinct features that make a new system of calculation based on stable di ffuse radiation, but without treating the integral directly, feasible [6].

### **3. Derivations from Spherical Caps**

Beginning with elementary forms, it is possible to obtain a number of useful factors with perfect ease. It is only necessary to know the geometric properties of the intervening shapes. In a volume encircled by one surface, the sphere, let us consider di ffuse irradiation to the inside of the form; in this case, the energy going to the interior of the volume has no way out and must be received on the same surface; if we call the spherical surface 1, the configuration factor to be found must be:

$$F\_{11} = 1\tag{3}$$

Considering the former for one-half of the sphere, the form factor could give *F*11 = 1 2 . In concordance with this familiar result, we decided to delve into the analysis of basic volumes composed of a limited number of surfaces. If we have two elements and one is flat (the base) and the other covers such a base in a sphere-like fashion, we would have a spherical cap.

Employing the previous finding to a spherical cap (called element 1), and the said disk of radius a (element 2); the radiative transfers in Figure 2 are deducted from the algebraic properties:

**Figure 2.** A spherical segmen<sup>t</sup> of uniform brightness featuring *h* as the height and radius *a*.

*A*1<sup>x</sup>*F*12 = *A*2<sup>x</sup>*F*21. As F21 = 1, the expression produces *F*12 = *A*2 *A*1 , and in this particular case:

$$F\_{12} = \frac{a^2}{a^2 + h^2} \tag{4}$$

$$F\_{11} = \frac{h^2}{a^2 + h^2} = \frac{h}{2 \times R} \tag{5}$$

Note that in planar surfaces F22 equals null.

A couple of basic principles are inferred from here. These are defined as the laws of Cabeza-Lainez. The first principle says that:

For a volume composed of perfectly fitting elements presenting no internal obstruction and one of them being a disk, the view factor from the non-planar surface to the disk equates a fraction whose numerator is the area of the disk and the denominator is the area of the curved geometry.

The second principle of Cabeza-Lainez states that:

Under any spherical element, the so-called reflexive factor (or self-given energy) can be determined by the fraction of the area of the element divided by the total area of the spherical surface.

Bearing in mind that any segmen<sup>t</sup> of the sphere is the Nth fraction of the total surface (of radius R), and using Pythagoras's theorem,

$$(h^2 + a^2) = 2 \times R \text{x}h \tag{6}$$

Therefore,

$$N \times (h^2 + a^2) = 4 \times R^2 \; ; \; N = 2 \times \frac{R}{h} \; ; \; h = 2 \times \frac{R}{N} \tag{7}$$

This implies that

$$F\_{11} = \frac{h}{2 \times R} = \frac{h^2 \text{xN}}{4 \times R^2} = \frac{1}{N} \tag{8}$$

The second principle informs us that the self-given energy for an Nth part of the sphere is a fraction of value 1/N. In this way, the previous supposition for one-half of the sphere is true; ensuing that divisions for one third or one quarter would give 1/3, 1/4, and so forth.

Such a principle will be fulfilled for many imaginable elements of the sphere that do not fall on typical categories as segments. Consequently, the properties expressed in Equations (5) and (8) are unique to the spherical surface and critical to settle the issue.

The former two principles can be extended to other adjustments that involve the sphere. Take for instance, a volume limited by two half-disks under any angle *x*, which varies between zero and π degrees (Figure 3):

**Figure 3.** Depiction of elements in a spherical wedge, areas 1 and 2 show half-disks, and surface 3 is a part of the sphere.

Recalling the findings from above, the Nth fraction in the wedge gives 1/*N* = *<sup>x</sup>*/2π and so,

$$F\_{33} = \frac{\varkappa}{2\pi} \tag{9}$$

Consequently,

$$F\_{31} = F\_{32} = \frac{1}{2} \times \left(1 - \frac{x}{2\pi}\right) \tag{10}$$

Incorporating the surfaces of the half-disks, π*<sup>R</sup>*2/2

$$F\_{13} = F\_{23} = \frac{2\mathbf{x}}{\pi} \times \left(1 - \frac{\mathbf{x}}{2\pi}\right) \tag{11}$$

Continuing with the deduction at Equation (11), the pair of half-disks can cover any angle x ranging from 0 to π degrees (Figure 4). In this manner, the expression found below, and termed as the Cabeza-Lainez third law, is not described in any reference that we many know of [1]. Thus, we claim it to define the configuration factors that involve two semicircles with a common diameter, if x covers the magnitude of the angle comprised between them (Figure 5).

**Figure 4.** Two half-disks with equal radius R, sharing their diameters for any angle of X◦ degrees.

The above result simplified produces:

$$F\_{12} = 1 - \frac{2\alpha}{\pi} + \left(\frac{\alpha}{\pi}\right)^2\tag{12}$$

Transforming the equation into degrees,

$$F\_{12} = 1 - \frac{\mathbf{x}}{90} + \frac{\mathbf{x}^2}{32400} \tag{13}$$

**Figure 5.** Luminous energy factors for two half-disks sharing the diameter and covering an angle of *x<sup>º</sup>* sexagesimal degrees.

Cabeza-Lainez invented a fourth law based on the former (unpublished) and we can express it in the following way. For any couple of planar sections of a sphere (1–2) with a common edge and irrespective of the angle they form α, or if they pass through the center of the sphere or not, the form factor from surface 1 to surface 2 is:

$$F\_{12} = \frac{A\_1 + A\_2 + A\_3 \times (F\_{33} - 1)}{2 \times A\_1} = \frac{1}{2} \left( 1 + \frac{A\_2 - A\_3}{A\_1} + \frac{A\_3^2}{A\_5 \times A\_1} \right) \tag{14}$$

F33 is the factor of a sphere's fragment over itself, which is defined in the Cabeza-Lainez second law as the ratio between the area of the said fragment and the whole sphere (Equation (8)), in other words, 12for a hemisphere, 14for a quarter of sphere, and successively, it equals A3/Asphere.

Ai is the respective areas of surfaces 1 and 2 (segments of circle or ellipse), and 3 is the comprised fragment of the sphere (Figure 6).

**Figure 6.** A spherical underground chamber susceptible to solve with our laws of radiation.

It is important to stress that Equations (12) to (14) have been demonstrated by numerical computation methods.
