Finding the exact radiative field of triangular sources: application for more effective shading devices and windows

Citation: Cabeza-Lainez, J. Finding the Exact Radiative Field of Triangular Sources: Application for More Effective Shading Devices and Windows. Appl. Sci. 2023,13, 11318. https://doi.org/10.3390/ app132011318 Academic Editors: John D. Clayton and Weihua Cai Received: 23 September 2023 Revised: 11 October 2023 Accepted: 13 October 2023 Published: 15 October 2023 Copyright: © 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article Finding the Exact Radiative Field of Triangular Sources: Application for More Effective Shading Devices and Windows Joseph Cabeza-Lainez Department of Architectural Composition, University of Seville, 41012 Sevilla, Spain; cr[email protected] Featured Application: Design and construction of innovative shading devices and fins. Abstract: Triangles are an ever-present feature in nature, which the building construction industry duly echoes. However, an exact expression intended to supply the radiant field of any triangle in an upright or inclined position has not been identified by previous research. In this case, the author has been able to solve, via direct integration, the canonical expression of radiative transfer. This result alone confers a myriad of possibilities, that had been inconceivable before, for studying in detail the three-dimensional heat-transfer behavior of volumes and figures in which triangles manifest, such as fins, windows, roof-gables and louvers of various kinds. Typically, shading devices, when tilted, give rise in their extremes to rhomboidal shapes which were difficult to take into account or had to be subject to discretization and subsequent Monte Carlo methods in order to perform an approximate calculation of their emissions. This implied a lengthy and inexact procedure that induced many errors and consumed computing time. We can now avoid all these former downsides due to the advances hereby presented. As this novel expression can be converted into an algorithm, it will be advantageously employed for simulation. This significant finding dovetails into the intricate puzzle of radiated heat and we believe that its consequences will greatly affect the conception and design of HVAC devices, aircraft manufacturing and specifically the building or lighting industries, among others. Keywords: triangular geometry; radiative heat transfer; form-factor calculation tools; radiant interchanges; design of radiators; assessment and construction of LED emitters 1. Introduction It is known that form factors are a substantial entity in all the phenomena that concern radiation. As this powerful and subtle kind of heat transfer does not imply any sort of proximity between the involved elements, sheer form and a geometric approach become paramount for the distribution of energies [ 1 ]. However, the diffusion of radiative fields is heavily dependent on the spatial configuration of surface sources [ 2 ]. At the level of unit areas of no defined shape, the issue may seem nugatory [ 3 , 4 ]; but when we arrive at definite widespread forms like the circle or the triangle, it is frequently problematic to extend these well-known differential equations to concrete and finite forms arbitrarily displayed, as we need to manage two sets of different and often complex double integrals [ 4 ]. It is evident that, to achieve a manageable solution from quadruple integrands even of the most basic emitters implies a feat of calculus unsuitable for conventional engineering [ 5 ], but it is substantially needed to ascertain the energy balance achieved in the said elements. Even recent studies have failed to overcome this severe hindrance in the completion of important problems in radiative transfer [ 6 , 7 ]. Experts have too long resorted to numerical or statistical approaches like probabilistic methods [ 8 ], albeit that they are inconsequential in most cases, not merely due to the many errors that arise in the lengthy process but also due to the unfeasibility of proving their results, as they remained the sole procedure available to attempt to overcome such problems. Appl. Sci. 2023,13, 11318. https://doi.org/10.3390/app132011318 https://www.mdpi.com/journal/applsci Appl. Sci. 2023,13, 11318 2 of 30 In this case, the author has followed a special procedure of dummy variables to perform the inner part of the integral, that is referred to the receiving source in an exact manner. Formerly, only rectangles and parallel disks could have been submitted to total integration [ 7 ]. It is paradoxical that the relatively simple form of the triangle could not have been included in the catalog of solved form factors but we have to admit that some mathematical drawbacks impeded it for an undesirably long lapse of time [9]. In this manner, the problem can be prepared for the final phase of integration as the dummy variables now take an active value in the third and fourth round of operation that is usually performed by numerical methods [ 10 ]. We have to bear in mind that, in the early part, the more feasible equation due only to the so-called configuration factor (an equation that reduces the interchange to arbitrary points on the emitting surface) is treated and solved [ 8 ]. A considerable fraction of the difficulty of the problem lies in the three-dimensional nature of the elements under consideration [11]. Different approximations so far employed for the problem have shown increasing flaws [ 12 ], as, being an elementary unit in itself, the triangle is not apt for division into (let us say) square or round tiles, for example. The accurate finding of such expressions can be considered a postulate in itself, that can be added rightfully to the six principles of radiation exchanges that we have developed previously [ 13 ]. Such algebraic expressions can be automatically computed with the software that we have created and a detailed explanation of the procedure segues out. 2. Materials and Methods: General Flux Expression As has been enunciated several times since the last quarter of the 18th century [ 14 ], the most general expression that defines the probability of an arbitrary interchange of radiated flux for any kind of three-dimensional form positioned in the manner illustrated in Figure 1, follows the structure described as follows (Equation (1)): dφ12 =(E1−E2)cos θ1cos θ2 dA1dA2 πr2 12 (1) Appl. Sci. 2023, 13, x FOR PEER REVIEW 2 of 33 sequential in most cases, not merely due to the many errors that arise in the lengthy process but also due to the unfeasibility of proving their results, as they remained the sole procedure available to attempt to overcome such problems. In this case, the author has followed a special procedure of dummy variables to perform the inner part of the integral, that is referred to the receiving source in an exact manner. Formerly, only rectangles and parallel disks could have been submitted to total integration [7]. It is paradoxical that the relatively simple form of the triangle could not have been included in the catalog of solved form factors but we have to admit that some mathematical drawbacks impeded it for an undesirably long lapse of time [9]. In this manner, the problem can be prepared for the final phase of integration as the dummy variables now take an active value in the third and fourth round of operation that is usually performed by numerical methods [10]. We have to bear in mind that, in the early part, the more feasible equation due only to the so-called configuration factor (an equation that reduces the interchange to arbitrary points on the emitting surface) is treated and solved [8]. A considerable fraction of the difficulty of the problem lies in the three-dimen- sional nature of the elements under consideration [11]. Different approximations so far employed for the problem have shown increasing flaws [12], as, being an elementary unit in itself, the triangle is not apt for division into (let us say) square or round tiles, for example. The accurate finding of such expressions can be considered a postulate in itself, that can be added rightfully to the six principles of radiation exchanges that we have developed previously [13]. Such algebraic expressions can be automatically computed with the software that we have created and a detailed explanation of the procedure segues out. 2. Materials and Methods: General Flux Expression As has been enunciated several times since the last quarter of the 18th century [14], the most general expression that defines the probability of an arbitrary interchange of radiated flux for any kind of three-dimensional form positioned in the manner illustrated in Figure 1, follows the structure described as follows (Equation (1)): 𝑑𝜙=(𝐸−𝐸)cos𝜃cos𝜃𝑑 𝐴 𝑑 𝐴  𝜋𝑟  (1) Figure 1. Radiant interchanges for a couple of undefined surfaces, named A1 and A2. Equation (1) initiates in the reciprocal principle, originally enunciated by J. H. Lambert [15–17] and it yields the probable amount of radiation emitted by unit area that is transferred (W/m2), from each surface defined as scalar quantities E1 and E2. The respective angles θ1 and θ2 that appear in the figure represent the deviation from the perpendicular Figure 1. Radiant interchanges for a couple of undefined surfaces, named A1and A2. Equation (1) initiates in the reciprocal principle, originally enunciated by J. H. Lambert [ 15 – 17 ] and it yields the probable amount of radiation emitted by unit area that is transferred (W/m 2 ), from each surface defined as scalar quantities E 1 and E 2 . The respective angles θ1 and θ2 that appear in the figure represent the deviation from the perpendicular that the vector r ij encounters in its rectilinear way from a random point in surface 1 to its correspondent in surface 2 [18]. Appl. Sci. 2023,13, 11318 3 of 30 The solution of Equation (1), which renders the value of the form factor between the two surfaces, would normally involve four rounds of integration, following Equation (2) [15]. Fij =1 AiZAiZAj cos θicos θj dAidAj πr2 ij (2) However, we have concentrated in this article on the first two stages (Equation (3)), which are also simpler, leaving a sort of harbinger for the next step, which is a variable (x 0 ) that can be considered a constant (dummy) in these early phases and will only become active for the third and fourth rounds of integrals. fdAi−Aj=ZAj cos θicos θj πr2 ij dAj(3) The latter operation is usually performed by means of deterministic numerical methods through an extension of the results to the whole emitting surface. With it, we can obtain the value of the form factor for the two complete surfaces in a more comfortable and accurate manner [ 18 ]. Ensuingly, we can use these form factors in various three-dimensional volumes that appear in most shading devices, some kinds of buildings affected by triangles, like warehouses and gabled-ceiling spaces, and different industrial and aeronautical fixtures [19,20]. For other different positions of the surface sources, an exact solution has not been achieved yet [ 5 ], nor for irregular fragments of triangles, rectangles and circles or even for the familiar form of the sphere [20,21]. Intermediate steps to proceed only with the first and the second round of integration have been proposed [ 22 ]. They usually end at a particular point or elementary area that pertains to the receiving surface. Some of these are recorded under the inexact and cumbersome manner of copied nomograms for catalogs of radiation view factors [23,24] . Such results are somewhat indicative but they are by no means general and can be misleading in many situations. In order to proceed with the said method, we shall start with a general right triangle of sides a (vertical) and b (horizontal) located at the origin of coordinates (Figure 2). Appl. Sci. 2023, 13, x FOR PEER REVIEW 3 of 33 that the vector rij encounters in its rectilinear way from a random point in surface 1 to its correspondent in surface 2 [18]. The solution of Equation (1), which renders the value of the form factor between the two surfaces, would normally involve four rounds of integration, following Equation (2) [15]. 𝐹=1 𝐴   cos𝜃cos𝜃𝑑 𝐴 𝑑 𝐴  𝜋𝑟   (2) However, we have concentrated in this article on the first two stages (Equation (3)), which are also simpler, leaving a sort of harbinger for the next step, which is a variable (x0) that can be considered a constant (dummy) in these early phases and will only become active for the third and fourth rounds of integrals. 𝑓 =cos𝜃cos𝜃 𝜋𝑟 𝑑 𝐴  (3) The latter operation is usually performed by means of deterministic numerical methods through an extension of the results to the whole emitting surface. With it, we can obtain the value of the form factor for the two complete surfaces in a more comfortable and accurate manner [18]. Ensuingly, we can use these form factors in various three-dimen- sional volumes that appear in most shading devices, some kinds of buildings affected by triangles, like warehouses and gabled-ceiling spaces, and different industrial and aeronautical fixtures [19,20]. For other different positions of the surface sources, an exact solution has not been achieved yet [5], nor for irregular fragments of triangles, rectangles and circles or even for the familiar form of the sphere [20,21]. Intermediate steps to proceed only with the first and the second round of integration have been proposed [22]. They usually end at a particular point or elementary area that pertains to the receiving surface. Some of these are recorded under the inexact and cumbersome manner of copied nomograms for catalogs of radiation view factors [23,24]. Such results are somewhat indicative but they are by no means general and can be misleading in many situations. In order to proceed with the said method, we shall start with a general right triangle of sides a (vertical) and b (horizontal) located at the origin of coordinates (Figure 2). Figure 2. Coordinates for the right triangle on the plane ZX of base b and height a, a form-factor determination over a horizontal plane XY. Figure 2. Coordinates for the right triangle on the plane ZX of base b and height a, a form-factor determination over a horizontal plane XY. Appl. Sci. 2023,13, 11318 4 of 30 Under this disposition, the coordinate Z axis is vertical and X is horizontal and perpendicular, the sides of the triangle are, in Z, equal to a, and, in X, to b, as seen in Figure 2. The tangent of the angle formed from the upper edge of the triangle to the horizontal equates tanα= a/b, therefore, the equation of the superior line of the triangle is, z = (a/b)x The axis Y in the graph represents the free coordinate perpendicular to X. Introducing the “dummy” variable x 0 , the squared distance (Equation (5)) from one receiving point-source of the triangle to another emitting point moving freely over the plane XY, presented as r12 in Equation (2) above, would be: r2 12 =(x−x0)2+y2+z2(4) The respective cosines β and γ , to the normal from Equations (1)–(3), would be accordingly y/r12 and z/r12. The integral to solve deriving from Equation (3), would then turn to (Equation (6)): fdAi−Aj=I=1 πZb 0Za bx 0 yzdzdx h(x−x0)2+y2+z2i2(5) We need to solve the root equation for a generality of types of triangles in diverse positions [ 25 ]. To avoid special difficulties, we will provide the solution at the intermediate stage of configuration, that is, the double integral firstly, obtained point by point over a rectangular emitting field. Once we have achieved this, to advance towards the complete form factor of Equation (2), we need to apply another property previously enunciated by the author [ 26 ]; namely the form factor is the average over the emitting surface of the configuration factor previously obtained on a point basis. It is now feasible by numerical calculus alone to extract such a mean value for any kind of planar figures contained in the said field. With a similar approach, the total form factors are found for the usual sets of precincts, such as prisms or oblique cuboids. Inter-reflection within the so-generated space is also an issue that we have to foresee; in the case of triangular prisms, we would require a set of five equations with five unknowns, which is detailed in the Appendix A[27]. For example, when designing and solving inclined slats, the triangles that appear are not right, but usually they appear as obtuse or acute; in these situations, we cannot employ the already solved form factor for perpendicular rectangles; instead we need to resort to the extended formulas developed by the author [ 28 ] to take into account the exchanges between inclined rectangles. 3. Results: Solutions for Various Forms of Triangles and Trapezes 3.1. Right Triangle with Minor Vertex at the Origin of Coordinates Consequently, let us advance towards the solution set out above for the more representative triangles and rhomboids, beginning with the case depicted in Figure 3. In Equation (6), we mentioned the integral to be solved as, f12 =1 πZb 0Za bx 0 yzdzdx h(x−x0)2+y2+z2i2(6) Appl. Sci. 2023,13, 11318 5 of 30 Appl. Sci. 2023, 13, x FOR PEER REVIEW 5 of 33 Figure 3. Coordinates for the triangle form factor over a horizontal plane. As in Figure 2 but presented again to facilitate following the calculations. In Equation (6), we mentioned the integral to be solved as, 𝑓 =1𝜋 𝑦𝑧𝑑𝑧𝑑𝑥 [(𝑥−𝑥)+𝑦+𝑧]     (6) In this case, the primitive is the quotient of the denominator and integration with respect to z gives, 𝑦𝑧𝑑𝑧 [(𝑥−𝑥)+𝑦+𝑧]  =−𝑦21 (𝑥−𝑥)+𝑦+𝑧 (4) Substituting, it yields, =−𝑦21 (𝑥−𝑥)+𝑦+𝑧=𝑦2󰇭1 (𝑥−𝑥)+𝑦−1 (𝑥−𝑥)+𝑦+𝑎 𝑏𝑥󰇮 (5) And regrouping, the result is 𝑦2󰇧1 (𝑥−𝑥)+𝑦−𝑏 𝑎𝑥+𝑏(𝑥−𝑥)+𝑏𝑦󰇨 (6) Now, we only need to integrate with respect to x. The first part presents not much difficulty to identify, 𝑑𝑥 (𝑥−𝑥)+𝑦  =1𝑦 𝑎𝑟𝑐𝑡𝑎𝑛𝑥 𝑦+𝑎𝑟𝑐𝑡𝑎𝑛𝑏−𝑥 𝑦 (7) The second part is slightly lengthier in calculation, 𝑏𝑑𝑥 𝑎𝑥+𝑏(𝑥−𝑥)+𝑏𝑦  =𝑏 𝑎𝑥+𝑦(𝑎+𝑏)󰇩𝑎𝑟𝑐𝑡𝑎𝑛 𝑥(𝑎+𝑏)−𝑥𝑏 𝑏𝑎𝑥+𝑦(𝑎+𝑏)󰇪 =𝑏 𝑎𝑥+𝑦(𝑎+𝑏)󰇩𝑎𝑟𝑐𝑡𝑎𝑛 𝑎+𝑏−𝑥𝑏 𝑎𝑥+𝑦(𝑎+𝑏) + 𝑎𝑟𝑐𝑡𝑎𝑛 𝑥𝑏  𝑎𝑥+𝑦(𝑎+𝑏)󰇪 (8) Figure 3. Coordinates for the triangle form factor over a horizontal plane. As in Figure 2but presented again to facilitate following the calculations. In this case, the primitive is the quotient of the denominator and integration with respect to z gives, Za bx 0 yzdz h(x−x0)2+y2+z2i2=−y 2"1 (x−x0)2+y2+z2#a bx 0 (7) Substituting, it yields, =−y 2"1 (x−x0)2+y2+z2#a bx 0 =y 2 1 (x−x0)2+y2−1 (x−x0)2+y2+a2 b2x2!(8) And regrouping, the result is y 2  1 (x−x0)2+y2−b2 a2x2+b2(x−x0)2+b2y2 (9) Now, we only need to integrate with respect to x. The first part presents not much difficulty to identify, Zb 0 dx (x−x0)2+y2=1 yarctan x0 y+arctan b−x0 y(10) The second part is slightly lengthier in calculation, Rb 0b2dx a2x2+b2(x−x0)2+b2y2 =b qa2x2 0+y2(a2+b2)"arctan x(a2+b2)−x0b2 bqa2x2 0+y2(a2+b2)#b 0 =b qa2x2 0+y2(a2+b2)"arctan a2+b2−x0b qa2x2 0+y2(a2+b2) +arctan x0b qa2x2 0+y2(a2+b2)# (11) Appl. Sci. 2023,13, 11318 6 of 30 Arranging the two parts and multiplying by y/2, which was outside of the integral in Equation (8), we obtain: 1 2harctan x0 y+arctan b−x0 yi −by qa2x2 0+y2(a2+b2)harctan a2+b2−x0b qa2x2 0+y2(a2+b2) +arctan x0b √a2x2 0+y2(a2+b2)i (12) Finally, we need to divide everything by π to keep the factor dimensionless in terms of radiant power, f21 =1 2π arctan b−x0 y+arctan x0 y) −by qa2x2 0+y2(a2+b2) (arctan a2+b2−x0b qa2x2 0+y2(a2+b2) +arctan x0b qa2x2 0+y2(a2+b2)! (13) If x0= 0, which means that we are calculating at the vertex point of the triangle, f21 =1 2π arctan b y−by √y2(a2+b2)arctan a2+b2 √y2(a2+b2)!=1 2π arctan b y− b √a2+b2arctan a2+b2 y√a2+b2!(14) This was obtained previously by the author for the triangle at its minor vertex [5], f12 =1 2π"arctan b y−b √a2+b2arctan a2+b2 y√a2+b2#(15) If x0= b, the end of the base, then, f21 =1 2π arctan b y −by qa2b2+y2(a2+b2) arctan a2 qa2b2+y2(a2+b2) +arctan b2 √a2b2+y2(a2+b2)!! (16) This was the formula previously obtained for the horizontal extreme corresponding to the right angle, which coincides exactly with the former, (Equation (17)) [29]. f12 =1 2πharctan b yi −by √a2b2+y2(a2+b2)harctan b2 √a2b2+y2(a2+b2) +arctan a2 √a2b2+y2(a2+b2)i (17) Finally, if we make x 0 = x, igniting the dummy variable, the above expression compares well with the one used for the rectangle of vertical side aand horizontal side bwhen the study point moves freely on the horizontal plane (Equation (18)) [7]. Appl. Sci. 2023,13, 11318 7 of 30 f21 =1 2π arctan b−x y+arctan x y−y pa2+y2 arctan b−x pa2+y2+arctan x pa2+y2!! (18) Let us now calculate, for the sake of generality, the same triangle in a more general position outside the origin of coordinates, see Figure 4. Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 of 33 𝑓 =1 2𝜋󰇧𝑎𝑟𝑐𝑡𝑎𝑛𝑏𝑦 −𝑏𝑦 𝑎𝑏+𝑦(𝑎+𝑏)󰇩𝑎𝑟𝑐𝑡𝑎𝑛 𝑏 𝑎𝑏+𝑦(𝑎+𝑏) + 𝑎𝑟𝑐𝑡𝑎𝑛 𝑎  𝑎𝑏+𝑦(𝑎+𝑏)󰇪󰇨 (17) Finally, if we make x0 = x, igniting the dummy variable, the above expression compares well with the one used for the rectangle of vertical side a and horizontal side b when the study point moves freely on the horizontal plane (Equation (18)) [7]. 𝑓 21=1 2𝜋󰇧𝑎𝑟𝑐𝑡𝑎𝑛𝑏−𝑥 𝑦+𝑎𝑟𝑐𝑡𝑎𝑛𝑥𝑦−𝑦 √𝑎2+𝑦2󰇡𝑎𝑟𝑐𝑡𝑎𝑛 𝑏−𝑥 √𝑎2+𝑦2+𝑎𝑟𝑐𝑡𝑎𝑛 𝑥 √𝑎2+𝑦2󰇢󰇨 (18) Let us now calculate, for the sake of generality, the same triangle in a more general position outside the origin of coordinates, see Figure 4. Figure 4. Coordinates for the triangle form factor over a horizontal plane. In this case, integration limits in z, would change between, 𝑧=𝑥−  and z = 0 𝑦𝑧𝑑𝑧 [(𝑥−𝑥)+𝑦+𝑧]   =−𝑦21 (𝑥−𝑥)+𝑦+𝑧  =−󰇣 ()󰇤  =󰇧 ()− ()󰇡  󰇢󰇨 (19) But 󰇡𝑎𝑏𝑥−𝑎𝑚 𝑏󰇢=𝑎 𝑏𝑥−2𝑎 𝑏𝑚𝑥 −𝑎𝑚 𝑏=𝑎 𝑏(𝑥−2𝑚𝑥+𝑚) (20) And regrouping, the partial result is Figure 4. Coordinates for the triangle form factor over a horizontal plane. In this case, integration limits in z, would change between, z=a bx−am band z = 0 Ra bx−am b 0 yzdz [(x−x0)2+y2+z2]2=−y 21 (x−x0)2+y2+z2a bx−am b 0 =−y 21 (x−x0)2+y2+z2a bx−am b 0 =y 21 (x−x0)2+y2−1 (x−x0)2+y2+(a bx−am b)2 (19) But a bx−am b2=a2 b2x2−2a2 b2mx −a2m2 b2=a2 b2x2−2mx +m2(20) And regrouping, the partial result is y 2  1 (x−x0)2+y2−1 (x−x0)2+y2+a2(x2−2mx+m2) b2 (21) Conveniently developed, this gives, y 2 1 (x−x0)2+y2−b2 b2(x−x0)2+b2y2+a2x2−2a2mx +a2m2!(22) That is, y 2 1 (x−x0)2+y2−b2 b2(x−x0)2+b2y2+a2(x2−2mx +m2)!(23) Appl. Sci. 2023,13, 11318 8 of 30 The first term of the integral is already solved in Equation (10), Zb+m m dx (x−x0)2+y2=1 yarctan b+m−x0 y+arctan x0−m y(24) For the second part, we need to integrate, Zb+m m b2dx b2(x−x0)2+b2y2+a2(x2−2mx +m2)(25) Developing all the terms in the denominator, we receive, b2(x−x0)2+b2y2+a2x2−2mx +m2 =b2x2+x02−2xx0+b2y2+a2x2−2a2mx +a2m2(26) And grouping in terms of x, (a2+b2x2−2b2x0+a2mx+b2y2+x02+a2m2(27) The final integral to be solved is, Rb+m mb2dx (a2+b2)x2−2(b2x0+a2m)x+b2(y2+x02)+a2m2= =2b2 bqa2(y2+(x0−m)2)+b2y2harctan 2x(a2+b2)−2(a2m+b2x0) 2bqa2(y2+(x0−m)2)+b2y2ib+m m =2b qa2(x0−m)2+y2(a2+b2)harctan b(a2+b(b+m−x0)) bqa2(x0−m)2+y2(b2+a2) −arctan b2(m−x0) bqa2(x0−m)2+y2(b2+a2)i (28) =2b qa2(x0−m)2+y2(a2+b2)harctan a2+b(b+m−x0) qa2(x0−m)2+y2(a2+b2) −arctan b(m−x0) qa2(x0−m)2+y2(a2+b2)i(29) Arranging all the necessary terms and multiplying by y/2, we obtain, f21 =1 2πharctan b+m−x0 y+arctan x0−m yi −by qa2(x0−m)2+y2(a2+b2)harctan a2+b(b+m−x0) qa2(x0−m)2+y2(a2+b2) −arctan b(m−x0) qa2(x0−m)2+y2(a2+b2)i (30) By making m = 0, we would find that, f21 =1 2πharctan b−x0 y+arctan x0 yi −by √a2x2 0+y2(a2+b2)harctan a2+b2−bx0 √a2x2 0+y2(a2+b2) +arctan bx0 √a2x2 0+y2(a2+b2)i (31) Appl. Sci. 2023,13, 11318 9 of 30 This is exactly the same as previously found in Equation (13), which we now define as the Cabeza-Lainez seventh postulate, that is, f21 =1 2πarctan b−x0 y+arctan x0 y −by qa2x2 0+y2(a2+b2)arctan a2+b2−bx0 qa2x2 0+y2(a2+b2) +arctan bx0 qa2x2 0+y2(a2+b2) 3.2. Right Triangle with Minor Vertex Removed from the Origin of Coordinates To continue with the former mathematical reasoning, we could apply the same kind of integration procedure to the triangle in the reverse position, referring to the X-axis (Figure 5). Appl. Sci. 2023, 13, x FOR PEER REVIEW 9 of 33 𝑓  =1 2𝜋𝑎𝑟𝑐𝑡𝑎𝑛𝑏+𝑚−𝑥 𝑦+𝑎𝑟𝑐𝑡𝑎𝑛𝑥−𝑚 𝑦 −𝑏𝑦 𝑎(𝑥−𝑚)+𝑦(𝑎+𝑏)⎣ ⎢ ⎢ ⎡ 𝑎𝑟𝑐𝑡𝑎𝑛 𝑎+𝑏(𝑏+𝑚−𝑥) 𝑎(𝑥−𝑚)+𝑦(𝑎+𝑏) − 𝑎𝑟𝑐𝑡𝑎𝑛 𝑏(𝑚−𝑥)  𝑎(𝑥−𝑚)+𝑦(𝑎+𝑏)⎦ ⎥ ⎥ ⎤ (30) By making m = 0, we would find that, 𝑓 =1 2𝜋𝑎𝑟𝑐𝑡𝑎𝑛𝑏−𝑥 𝑦+𝑎𝑟𝑐𝑡𝑎𝑛𝑥 𝑦 −𝑏𝑦 𝑎𝑥+𝑦(𝑎+𝑏)󰇩𝑎𝑟𝑐𝑡𝑎𝑛 𝑎+𝑏−𝑏𝑥 𝑎𝑥+𝑦(𝑎+𝑏) + 𝑎𝑟𝑐𝑡𝑎𝑛 𝑏𝑥  𝑎𝑥+𝑦(𝑎+𝑏)󰇪 (31) This is exactly the same as previously found in Equation (13), which we now define as the Cabeza-Lainez seventh postulate, that is, 𝑓 =1 2𝜋󰇭𝑎𝑟𝑐𝑡𝑎𝑛𝑏−𝑥 𝑦+𝑎𝑟𝑐𝑡𝑎𝑛𝑥 𝑦 −𝑏𝑦 𝑎𝑥+𝑦(𝑎+𝑏)󰇧𝑎𝑟𝑐𝑡𝑎𝑛 𝑎+𝑏−𝑏𝑥 𝑎𝑥+𝑦(𝑎+𝑏) + 𝑎𝑟𝑐𝑡𝑎𝑛 𝑏𝑥  𝑎𝑥+𝑦(𝑎+𝑏)󰇨󰇮 3.2. Right Triangle with Minor Vertex Removed from the Origin of Coordinates To continue with the former mathematical reasoning, we could apply the same kind of integration procedure to the triangle in the reverse position, referring to the X-axis (Figure 5). Figure 5. The triangle in a reversed position with the rectangular angle at the origin of coordinates. Figure 5. The triangle in a reversed position with the rectangular angle at the origin of coordinates. In this case, integration limits in z would change to, z=a−a bx0and z = 0 Ra−a bx 0 yzdz [(x−x0)2+y2+z2]2=−y 2h1 (x−x0)2+y2+z2ia−a bx 0 =−y 2h1 (x−x0)2+y2+z2ia−a bx 0=y 21 (x−x0)2+y2−1 (x−x0)2+y2+(a−a bx)2 (32) This binomial yields, a−a bx2=a2 b2x2+a2−2a2 bx=a2x2+a2b2−2a2bx b2(33) And regrouping, the result of the first part amounts to, y 2  1 (x−x0)2+y2−1 (x−x0)2+y2+a2x2+a2b2−2a2bx b2 (34) The expression, expanded, gives, y 2 1 (x−x0)2+y2−b2 b2(x−x0)2+b2y2+a2x2+a2b2−2a2bx !(35) Appl. Sci. 2023,13, 11318 16 of 30 Applying the same results obtained previously, we will find that, f21 =1 2πmy qa2x2 0+y2(a2+m2)arctan a2+m2−x0m qa2x2 0+y2(a2+m2) +arctan x0m qa2x2 0+y2(a2+m2) −y √y2+a2harctan x0 √y2+a2+arctan m−x0 √y2+a2i (55) This gives the eleventh postulate of Cabeza-Lainez. If now we subtract the latter expression from the general factor of the rectangle, which was given by Equation (56) (formerly (18)), and also the one of the displaced triangle, Equation (57), f21 =1 2πarctan (m+b)−x0 y+arctan x0 y −y √a2+y2arctan (m+b)−x0 √a2+y2+arctan x0 √a2+y2 (56) f21 =1 2πharctan b+m−x0 y+arctan x0−m yi −by qa2(x0−m)2+y2(a2+b2)harctan a2+b(b+m−x0) qa2(x0−m)2+y2(a2+b2) −arctan b(m−x0). qa2(x0−m)2+y2(a2+b2)i (57) Equation (56) minus Equations (55) and (57) offers the result for the rhomboid, f21 =1 2πarctan (m+b)−x0 y+arctan x0 y −y √a2+y2arctan (m+b)−x0 √a2+y2+arctan x0 √a2+y2 −1 2πharctan b+m−x0 y+arctan x0−m yi −by qa2(x0−m)2+y2(a2+b2)harctan a2+b(b+m−x0) qa2(x0−m)2+y2(a2+b2) −arctan b(m−x0) qa2(x0−m)2+y2(a2+b2)i −1 2πmy qa2x2 0+y2(a2+m2)arctan a2+m2−x0m qa2x2 0+y2(a2+m2) +arctan x0m qa2x2 0+y2(a2+m2) −y √y2+a2harctan x0 √y2+a2+arctan m−x0 √y2+a2i) (58) Appl. Sci. 2023,13, 11318 17 of 30 The final result, which becomes the twelfth postulate, would be (Equation (59)), f21 =1 2πarctan x0 y−arctan x0−m y −y √a2+y2arctan (m+b)−x0 √a2+y2−arctan m−x0 √y2+a2a +by qa2(x0−m)2+y2(a2+b2)harctan a2+b(b+m−x0) qa2(x0−m)2+y2(a2+b2) −arctan b(m−x0) qa2(x0−m)2+y2(a2+b2)i −my qa2x2 0+y2(a2+m2)arctan a2+m2−x0m qa2x2 0+y2(a2+m2) +arctan x0m qa2x2 0+y2(a2+m2) (59) These algorithms might seem complicated but they can be easily programmed for simulation once the process is duly automated [31]. 3.4. Vertical Semicircle at the Origin of Coordinates Although it is not related to triangles (and due to its singularity, it has been the subject for study of other articles by the author [ 32 ]), it is interesting to remark that, applying the same rationale of integration, we could find the configuration factors for several curves and among them the circumference (Figure 13). Appl. Sci. 2023, 13, x FOR PEER REVIEW 19 of 33 𝑓 21 =1 2𝜋󰇭𝑎𝑟𝑐𝑡𝑎𝑛𝑥0 𝑦−𝑎𝑟𝑐𝑡𝑎𝑛𝑥0−𝑚 𝑦 −𝑦 √𝑎2+𝑦2󰇧𝑎𝑟𝑐𝑡𝑎𝑛(𝑚+𝑏)−𝑥0 √𝑎2+𝑦2−𝑎𝑟𝑐𝑡𝑎𝑛 𝑚−𝑥0 𝑦2+𝑎2𝑎󰇨󰇮 +𝑏𝑦 𝑎2(𝑥0−𝑚)2+𝑦2(𝑎2+𝑏2)󰇩𝑎𝑟𝑐𝑡𝑎𝑛 𝑎2+𝑏(𝑏+𝑚−𝑥0) 𝑎2(𝑥0−𝑚)2+𝑦2(𝑎2+𝑏2) − 𝑎𝑟𝑐𝑡𝑎𝑛 𝑏(𝑚−𝑥0) 𝑎2(𝑥0−𝑚)2+𝑦2(𝑎2+𝑏2)󰇪 −󰇭𝑚𝑦 𝑎2𝑥02+𝑦2(𝑎2+𝑚2)󰇧𝑎𝑟𝑐𝑡𝑎𝑛 𝑎2+𝑚2−𝑥0𝑚 𝑎2𝑥02+𝑦2(𝑎2+𝑚2) + 𝑎𝑟𝑐𝑡𝑎𝑛 𝑥0𝑚 𝑎2𝑥02+𝑦2(𝑎2+𝑚2)󰇨󰇮 (59) These algorithms might seem complicated but they can be easily programmed for simulation once the process is duly automated [31]. 3.4. Vertical Semicircle at the Origin of Coordinates Although it is not related to triangles (and due to its singularity, it has been the subject for study of other articles by the author [32]), it is interesting to remark that, applying the same rationale of integration, we could find the configuration factors for several curves and among them the circumference (Figure 13). Figure 13. Coordinates for the quarter-circle form factor over a horizontal plane. In this case for the quarter of a circle, in Figure 13 the limits of integration for z would have been accordingly, √𝑎−𝑥 and 0. And the first integral would have produced: Figure 13. Coordinates for the quarter-circle form factor over a horizontal plane. In this case for the quarter of a circle, in Figure 13 the limits of integration for z would have been accordingly, √a2−x2and 0. And the first integral would have produced: =−y 21 (x−x0)2+y2+z2√a2−x2 0 =y 21 (x−x0)2+y2−1 (x−x0)2+y2+a2−x2= y 21 (x−x0)2+y2−1 x2 0+y2+a2−2x0x (60) Since the second term in x is linear, the integration to perform is easier than before. Appl. Sci. 2023,13, 11318 18 of 30 In fact, the result of the integral would be, Rb 0dx x2 0+y2+a2−2xx0=1 2x0h−logx2 0+y2+a2−2xx0ib 0 =1 2x0hlogx2 0+y2+a2−logx2 0+y2+a2−2bx0i(61) Grouping all the results again as the latter integral bears the minus sign in the preceding expression, we arrive at: f21 =1 2πarctan b−x0 y+arctan x0 y +y 2x0hlog(x2 0+y2+a2−2bx0)−log(x2 0+y2+a2)i (62) But as a = b, and represents the radius r of the quarter of circle, f21 =1 2πarctan a−x0 y+arctan x0 y +y 2x0hlog(x2 0+y2+a2−2ax0)−log(x2 0+y2+a2)i (63) This coincides exactly with the factor found via other means by the author for the quarter of a circle, (angle = π /2), [ 32 , 33 ]. There it was named the Cabeza-Lainez eighth postulate, but now it must be accordingly renumbered as the thirteenth postulate. We must bear in mind that a = b = r and as previously x0= x, f21 =1 2πarctanr−x y+arctan x y +y 4πx[logx2+y2+r2−2rx−logx2+y2+r2](64) It is obvious that, for the whole semicircle (angle = π ), with the limits of the integral from b to –b instead of 0, we would receive, (fourteenth postulate), f21 =1 2π(arctanr+x y+arctanr−x y) + y 4π∗x[log(r2+y2+x2−2r∗x) −log(r2+y2+x2+2r∗x)] (65) For the quarter of a circle (angle = π /2), as we saw in Equation (63), the same form factor is, f21 =1 2π(arctanr−x y+arctan x y) + y 4π∗x[log(r2+y2+x2−2r∗x) −log(r2+y2+x2)] (66) If the semicircle were now displaced from the origin of coordinates, we could just have the integral of the logarithm in the second term. =−y 21 (x−x0)2+y2+z2√a2−(x−d)2 0 =y 21 (x−x0)2+y2−1 (x−x0)2+y2+a2−(x−d)2= =y 21 (x−x0)2+y2−1 x2 0+y2+a2+d2−2x(d−x0)(67) The first integral was solved previously with limits d + r and d −r, respectively, as, 1 2arctan d+r−x0 y−arctan d−r−x0 y(68) The result of the second integral for the same limits is, y 2 1 2(d−x0)log2x(d−x0)−x2 0+y2+a2+d2d+r d−r (69) Appl. Sci. 2023,13, 11318 19 of 30 And the latter, developed, gives, y 2(1 2(d−x0))(log|2(d+r)(d−x0)−(x2 0+y2+a2+d2)|−log|2(d−r)(d−x0) −(x2 0+y2+a2+d2)|)(70) The total result would be, f21 =1 2π[arctan d+r−x0 y−arctan d−r−x0 y] + 1 4π(y (d−x0))(log|2(d+r)(d −x0)−(x2 0+y2+a2+d2)|−log|2(d−r)(d−x0)−(x2 0+y2+a2+d2)|)(71) The latter expression becomes the fifteenth postulate of Cabeza-Lainez. A wide set of curves like ellipses and parabolas could be resolved with the same procedure. For instance, in the case of a semi-ellipse, we only need to change the upper integral limit for b/a√a2−x2. In such a situation, the first integral would have given, =−y 2[1 (x−x0)2+y2+z2]b/a√a2−x2 0 =y 2(1 (x−x0)2+y2−1 (x−x0)2+y2+b2/a2(a2−x2)) = =y 2(1 (x−x0)2+y2−1 x2(1−b2/a2)+x2 0+y2+a2+b2−2x0x) (72) And, f12 =1 2π([arctan x0 y+arctan b−x0 y] −a √a2(a2+y2)−b2(b2+y2)−b2x2 0 [arctan b(a2−b2)−a2x0 a√a2(a2+y2)−b2(b2+y2)−b2x2 0 +arctan a2x0 a√a2(a2+y2)−b2(b2+y2)−b2x2 0 ]) (73) If ∆=a2a2+y2−b2b2+y2−b2x2 0>0 (74) Alternately, f12 =1 2πharctan x0 y+arctan b−x0 yi −a √a2(a2+y2)−b2(b2+y2)−b2x2 0harctan a(b−x0)−b3/a √a2(a2+y2)−b2(b2+y2)−b2x2 0 +arctan ax0 √a2(a2+y2)−b2(b2+y2)−b2x2 0i (75) Equation (75) becomes the sixteenth postulate of Cabeza-Lainez. That the former expression is greater than zero, seems likely once that, in an ellipse, a > b as the major semiaxis is larger than the minor one, but this must be checked against the x and y dimensions of the horizontal plane beforehand. For example, in an ellipse of major semiaxis = 4 m and minor = 1 m for an x, y grid of 10 by 10 m, all values of ∆ are positive and the result of Equation (75) is valid. This is a crucial development, since many researchers have been so far unable to give a consistent expression for the form factor of elliptical shapes [33]. 4. Discussion: Finding of the Global Form Factor As previously explained, we need to find the average over the chosen emitting surface (normally rectangular) of the previously found configuration factors for the triangles and other shapes identified in Section 3. Appl. Sci. 2023,13, 11318 20 of 30 4.1. Determination Procedures It has been proved by the author [ 34 , 35 ] that the form factor amounts to the average over the domain of the emitting surface of the configuration factor previously obtained on a point-to-point basis, as it precisely represents the integral extended to this second area. It is possible by any numerical automated procedure to obtain the mean value of all the point factors for any sort of figure contained in the said transferring field. For instance, in the nephograph in Figure 14 with the results of Equation (13) by virtue of the seventh postulate, we have automatically calculated the form factor F 21 from a rectangle of 8 m by 5 m to a triangle of 5 m height by 5 m width; it amounts to 0.090. In Figure 15, we can see the radiative field in plan of the triangle over the rectangle; the curves of received radiation are more pronounced in the left lower part of the nephograph due to the higher altitude of the triangle in that area as it nears its upper vertex. Appl. Sci. 2023, 13, x FOR PEER REVIEW 22 of 33 𝑓  =1 2𝜋 󰇧𝑎𝑟𝑐𝑡𝑎𝑛𝑥 𝑦+𝑎𝑟𝑐𝑡𝑎𝑛𝑏−𝑥 𝑦 −𝑎 𝑎(𝑎+𝑦)−𝑏(𝑏+𝑦)−𝑏𝑥󰇩𝑎𝑟𝑐𝑡𝑎𝑛 𝑎(𝑏−𝑥)−𝑏/𝑎 𝑎(𝑎+𝑦)−𝑏(𝑏+𝑦)−𝑏𝑥 + 𝑎𝑟𝑐𝑡𝑎𝑛 𝑎𝑥  𝑎(𝑎+𝑦)−𝑏(𝑏+𝑦)−𝑏𝑥󰇪󰇨 (75) Equation (75) becomes the sixteenth postulate of Cabeza-Lainez. That the former expression is greater than zero, seems likely once that, in an ellipse, a > b as the major semiaxis is larger than the minor one, but this must be checked against the x and y dimensions of the horizontal plane beforehand. For example, in an ellipse of major semiaxis = 4 m and minor = 1 m for an x, y grid of 10 by 10 m, all values of Δ are positive and the result of Equation (75) is valid. This is a crucial development, since many researchers have been so far unable to give a consistent expression for the form factor of elliptical shapes [33]. 4. Discussion: Finding of the Global Form Factor As previously explained, we need to find the average over the chosen emitting surface (normally rectangular) of the previously found configuration factors for the triangles and other shapes identified in Section 3. 4.1. Determination Procedures It has been proved by the author [34,35] that the form factor amounts to the average over the domain of the emitting surface of the configuration factor previously obtained on a point-to-point basis, as it precisely represents the integral extended to this second area. It is possible by any numerical automated procedure to obtain the mean value of all the point factors for any sort of figure contained in the said transferring field. For instance, in the nephograph in Figure 14 with the results of Equation (13) by virtue of the seventh postulate, we have automatically calculated the form factor F21 from a rectangle of 8 m by 5 m to a triangle of 5 m height by 5 m width; it amounts to 0.090. Figure 14. Average for a triangle of 5 m by 5 m over a horizontal rectangle of length 8 m. The form factor F21 from the rectangle (2) to the triangle (1) is equal in this case to 0.090. Figure 14. Average for a triangle of 5 m by 5 m over a horizontal rectangle of length 8 m. The form factor F21 from the rectangle (2) to the triangle (1) is equal in this case to 0.090. Appl. Sci. 2023, 13, x FOR PEER REVIEW 23 of 33 In Figure 15, we can see the radiative field in plan of the triangle over the rectangle; the curves of received radiation are more pronounced in the left lower part of the nephograph due to the higher altitude of the triangle in that area as it nears its upper vertex. Figure 15. Nephograph in plan of the radiative field from the triangle to the rectangle. In Figure 16, we have represented the same field in a three-dimensional view to understand better how the exchanges are manifested, for example, in luminous emissions such as windows or LED devices. Figure 16. Nephograph of radiative distribution for the same case in 3D. With a similar approach, the total form factors can be found for the usual sets of precincts, such as prisms or oblique cuboids. Inter-reflection within the so-generated volumes is also an issue that we have to foresee; in the case of triangular prisms, we would require a set of five equations with five unknowns, which is detailed in the Appendix A [36]. For example, when designing and solving inclined slats, the triangles that appear are not right, but usually they appeat as obtuse or acute angles; in these situations, we cannot employ the already solved form factor for perpendicular rectangles [5]; instead, we need to resort to different extended formulas developed by the author [37] to take into account the exchanges between inclined rectangles. Let us proceed to explain how the twenty form factors Fij with i and j from 1 to 5 but excluding the dummy sub-indexes like F11, F22, as all surfaces are planar. Figure 15. Nephograph in plan of the radiative field from the triangle to the rectangle. In Figure 16, we have represented the same field in a three-dimensional view to understand better how the exchanges are manifested, for example, in luminous emissions such as windows or LED devices. With a similar approach, the total form factors can be found for the usual sets of precincts, such as prisms or oblique cuboids. Inter-reflection within the so-generated Appl. Sci. 2023,13, 11318 21 of 30 volumes is also an issue that we have to foresee; in the case of triangular prisms, we would require a set of five equations with five unknowns, which is detailed in the Appendix A[ 36 ]. For example, when designing and solving inclined slats, the triangles that appear are not right, but usually they appeat as obtuse or acute angles; in these situations, we cannot employ the already solved form factor for perpendicular rectangles [ 5 ]; instead, we need to resort to different extended formulas developed by the author [ 37 ] to take into account the exchanges between inclined rectangles. Appl. Sci. 2023, 13, x FOR PEER REVIEW 23 of 33 In Figure 15, we can see the radiative field in plan of the triangle over the rectangle; the curves of received radiation are more pronounced in the left lower part of the nephograph due to the higher altitude of the triangle in that area as it nears its upper vertex. Figure 15. Nephograph in plan of the radiative field from the triangle to the rectangle. In Figure 16, we have represented the same field in a three-dimensional view to understand better how the exchanges are manifested, for example, in luminous emissions such as windows or LED devices. Figure 16. Nephograph of radiative distribution for the same case in 3D. With a similar approach, the total form factors can be found for the usual sets of precincts, such as prisms or oblique cuboids. Inter-reflection within the so-generated volumes is also an issue that we have to foresee; in the case of triangular prisms, we would require a set of five equations with five unknowns, which is detailed in the Appendix A [36]. For example, when designing and solving inclined slats, the triangles that appear are not right, but usually they appeat as obtuse or acute angles; in these situations, we cannot employ the already solved form factor for perpendicular rectangles [5]; instead, we need to resort to different extended formulas developed by the author [37] to take into account the exchanges between inclined rectangles. Let us proceed to explain how the twenty form factors Fij with i and j from 1 to 5 but excluding the dummy sub-indexes like F11, F22, as all surfaces are planar. Figure 16. Nephograph of radiative distribution for the same case in 3D. Let us proceed to explain how the twenty form factors F ij with i and j from 1 to 5 but excluding the dummy sub-indexes like F11, F22, as all surfaces are planar. With the first direct integration using Equation (13) and detailed in Figures 14–16, we obtain the exchange between the horizontal surface (4) and the side triangles (surfaces 1 and 2 in Figure 17), by symmetry, F41 = F42 (76) Appl. Sci. 2023, 13, x FOR PEER REVIEW 24 of 33 With the first direct integration using Equation (13) and detailed in Figures 14–16, we obtain the exchange between the horizontal surface (4) and the side triangles (surfaces 1 and 2 in Figure 17), by symmetry, F41 = F42 (76) But by reciprocity (Equation (1)) A4F41 = A1F14 (77) F14 = F24 = A4F41/A1 (78) F43 and F34 are known, since we have deducted the expression for perpendicular rectangles with a common edge [38]. Figure 17. The five surfaces of the volume of a triangular prism ending in right triangles. The form factor algebra demonstrated elsewhere [1,5,34,35] states that, F12 + F13 + … F1J = 1 (79) Consequently, F41 + F42+ F43+ F45 = 1 (80) F45 = 1 − 2F41– F43 (81) As F41 and F43 are known, we can extract F45, A4F45 = A5F54 (82) Once we know F54, we can write, F51 + F52+ F53+ F54 = 1 (83) F53= 1 − 2F51 − F54 (84) F51 is unknown, but similarly to Equation (76), rotating the triangles to the vertical side of the prism (Surf. 3), we would find, F31 = F32 (85) And also, we can deduct F13 and F23 as, A3F31 = A1F13 (86) F34 is already known as it was the perpendicular rectangle factor, Figure 17. The five surfaces of the volume of a triangular prism ending in right triangles. But by reciprocity (Equation (1)) A4F41 = A1F14 (77) F14 = F24 = A4F41/A1(78) Appl. Sci. 2023,13, 11318 22 of 30 F 43 and F 34 are known, since we have deducted the expression for perpendicular rectangles with a common edge [38]. The form factor algebra demonstrated elsewhere [1,5,34,35] states that, F12 + F13 + . . . F1J = 1 (79) Consequently, F41 + F42+ F43+ F45 = 1 (80) F45 = 1 −2F41−F43 (81) As F41 and F43 are known, we can extract F45, A4F45 = A5F54 (82) Once we know F54, we can write, F51 + F52+ F53+ F54 = 1 (83) F53= 1 −2F51 −F54 (84) F 51 is unknown, but similarly to Equation (76), rotating the triangles to the vertical side of the prism (Surf. 3), we would find, F31 = F32 (85) And also, we can deduct F13 and F23 as, A3F31 = A1F13 (86) F34 is already known as it was the perpendicular rectangle factor, F35 = 1 −2F31 −F34 (87) From here, we receive, A3F35 = A5F53 (88) Knowing F53 implies that, 1−F53 −F54 = 2F51 (89) With which, once we have F51 = F52 the problem is completely solved and the matrix of four by five elements is known, for, A5F51 = A1F15 (90) While, F12 + F13 + F14 + F15 = 1 (91) And thus, F12 = 1 −F13 −F14 −F15 (92) Knowing, for example, F 12 = F 21 carries implicitly the solution of a long-unresolved problem, namely, the rate of exchange between two parallel triangles, which also opens the way for a wide set of forms that we are not in a position to detail in this article. Appl. Sci. 2023,13, 11318 23 of 30 4.2. The Case of Inclined Rectangles With the former, we might consider now that all kinds of polygonal prisms are completely solved. However, this is not the case, since there are still some configurations in which the triangle is not right but obtuse or acute. Then, the form factors for perpendicular or parallel rectangles cannot be applied. Therefore, we have to employ other formulas apt for inclined rectangles, also developed by the author. In order to tackle such a complex issue, let us consider a rectangle positioned in the form described in Figure 18. As we already know, the general formula to apply for a particular point located at an inclined distance D, according to the figure, is: Appl. Sci. 2023, 13, x FOR PEER REVIEW 25 of 33 F35 = 1 − 2F31 − F34 (87) From here, we receive, A3F35 = A5F53 (88) Knowing F53 implies that, 1 − F53 − F54 = 2F51 (89) With which, once we have F51 = F52 the problem is completely solved and the matrix of four by five elements is known, for, A5F51 = A1F15 (90) While, F12 + F13 + F14 + F15 = 1 (91) And thus, F12 = 1 − F13 − F14 − F15 (92) Knowing, for example, F12 = F21 carries implicitly the solution of a long-unresolved problem, namely, the rate of exchange between two parallel triangles, which also opens the way for a wide set of forms that we are not in a position to detail in this article. 4.2. The Case of Inclined Rectangles With the former, we might consider now that all kinds of polygonal prisms are completely solved. However, this is not the case, since there are still some configurations in which the triangle is not right but obtuse or acute. Then, the form factors for perpendicular or parallel rectangles cannot be applied. Therefore, we have to employ other formulas apt for inclined rectangles, also developed by the author. In order to tackle such a complex issue, let us consider a rectangle positioned in the form described in Figure 18. As we already know, the general formula to apply for a particular point located at an inclined distance D, according to the figure, is: Figure 18. Configuration factor between a rectangle and a point at an inclined plane forming an angle φ with respect to the lower vertex of the said rectangle and distanced D from it on the inclined cyan line. Figure 18. Configuration factor between a rectangle and a point at an inclined plane forming an angle ϕ with respect to the lower vertex of the said rectangle and distanced D from it on the inclined cyan line. This represents a logical evolution from the previously described formulas [39] f21 =ZA1 cosθ1cosθ2dA1 r2(93) With, r2=x2+D2sin2ϕ+z2 1(94) The configuration factor f, on the given point with angle ϕ is the sum of two integrals. f21 =DRb 0Ra−Dcosϕ −Dcosϕ Dsin2ϕcosϕdz1dx (x2+D2sin2ϕ+z2 1)2+Rb 0Ra−Dcosϕ −Dcosϕ z1sin2ϕdz1dx (x2+D2sin2ϕ+z2 1)2(95) After considerable operations and adding all necessary terms, we conclude that the equation governing the whole process is the following. f21 =1 2π((arctan b D)+ acosϕ−D √a2+D2+2aDcosϕ(arctan b √a2+D2+2aDcosϕ)+ bcosϕ √b2+D2sin2ϕ(arctan a−Dcosϕ √b2+D2sin2ϕ+arctan Dcosϕ √b2+D2sin2ϕ)) (96) Appl. Sci. 2023,13, 11318 24 of 30 For a general position of the point on the XY plane, we find, f21 =1 2πarctan b−x y+arctan x y +acosϕ−y √a2+y2+2aycosϕarctan x √a2+y2+2aycosϕ +arctan b−x √a2+y2+2aycosϕ +xcosϕ √x2+y2sin2ϕarctan a−ycosϕ √x2+y2sin2ϕ +arctan ycosϕ √x2+y2sin2ϕ +(b−x)cosϕ √(b−x)2+y2sin2ϕarctan a−ycosϕ √(b−x)2+y2sin2ϕ +arctan ycosϕ √(b−x)2+y2sin2ϕ (97) In order, this will be the seventeenth postulate of Cabeza-Lainez. Thanks to this new property, it is feasible to calculate the interchanges in the rhomboidal prism whose end section is presented in Figures 19 and 20. Many other polygonal prisms can be solved in the same manner. We have to notice that, beforehand, the parallel faces of the prism could be found using combinations of the equations for parallel rectangles but there was no exact way to find the exchanges between the inclined rectangles, only rough approximations or blurred tables [ 40 ]. A singular advance has been produced in this fashion. Appl. Sci. 2023, 13, x FOR PEER REVIEW 27 of 33 Figure 19. Coordinates for the rhomboidal form factor over a horizontal plane. Figure 20. Rhomboidal prism formed by two inclined slats that appear in many shading devices. 4.3. Examples and Applications for Shading Devices A great deal of shading devices present this rhomboid configuration and their radiative performance could not be assessed in an accurate manner. In Figures 21 and 22, we can find details of the brise-soleil for the Sri Aurobindo Ashram designed and built by the architect Antonin Raymond in Pondicherry in 1937 [41,42]. Figure 19. Coordinates for the rhomboidal form factor over a horizontal plane. Appl. Sci. 2023, 13, x FOR PEER REVIEW 27 of 33 Figure 19. Coordinates for the rhomboidal form factor over a horizontal plane. Figure 20. Rhomboidal prism formed by two inclined slats that appear in many shading devices. 4.3. Examples and Applications for Shading Devices A great deal of shading devices present this rhomboid configuration and their radiative performance could not be assessed in an accurate manner. In Figures 21 and 22, we can find details of the brise-soleil for the Sri Aurobindo Ashram designed and built by the architect Antonin Raymond in Pondicherry in 1937 [41,42]. Figure 20. Rhomboidal prism formed by two inclined slats that appear in many shading devices. Appl. Sci. 2023,13, 11318 25 of 30 4.3. Examples and Applications for Shading Devices A great deal of shading devices present this rhomboid configuration and their radiative performance could not be assessed in an accurate manner. In Figures 21 and 22, we can find details of the brise-soleil for the Sri Aurobindo Ashram designed and built by the architect Antonin Raymond in Pondicherry in 1937 [41,42]. Appl. Sci. 2023, 13, x FOR PEER REVIEW 28 of 33 Figure 21. Section of the movable shading system where we can read “Window Arrangement for tropical countries”, notice that there is no glazing in such a window arrangement. Figure 22. View from the outside of the same shading system in Pondicherry (India). Another famous shading device in architectural terms is the one employed around the same time in 1938 by architects Oscar Niemeyer and Lucio Costa in Rio de Janeiro Figure 21. Section of the movable shading system where we can read “Window Arrangement for tropical countries”, notice that there is no glazing in such a window arrangement. Appl. Sci. 2023, 13, x FOR PEER REVIEW 28 of 33 Figure 21. Section of the movable shading system where we can read “Window Arrangement for tropical countries”, notice that there is no glazing in such a window arrangement. Figure 22. View from the outside of the same shading system in Pondicherry (India). Another famous shading device in architectural terms is the one employed around the same time in 1938 by architects Oscar Niemeyer and Lucio Costa in Rio de Janeiro Figure 22. View from the outside of the same shading system in Pondicherry (India).