A new principle for building simulation of radiative heat transfer in the presence of spherical surfaces

Citation: Cabeza-Lainez, J. A New Principle for Building Simulation of Radiative Heat Transfer in the Presence of Spherical Surfaces. Buildings 2023,13, 1447. https:// doi.org/10.3390/buildings13061447 Academic Editor: Christopher Yu-Hang Chao Received: 24 April 2023 Revised: 16 May 2023 Accepted: 30 May 2023 Published: 1 June 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/). buildings Article A New Principle for Building Simulation of Radiative Heat Transfer in the Presence of Spherical Surfaces Joseph Cabeza-Lainez Department of Architectural Composition, University of Seville, Av. Reina Mercedes 2, 41012 Sevilla, Spain; [email protected] Abstract: Radiant heat interchanges are pivotal to assessing the energy use of buildings and facilities that channel some sort of solar radiation. Form factor integrals are needed for an accurate simulation of the main features of the envelope of such buildings. However, the expressions required when the space under analysis is curved, for instance, in domes and vaults, are not feasible. The calculation process of algorithms is usually addressed by cumbersome analytical deductions or else by rough statistical approximations included in the simulations, such as ray-tracing methods. Neither of which works properly under curved geometries. The following article deals with an innovative methodology for employing an exact property that solves any spherical configuration of the radiant surfaces. The newly found relationship is validated by comparison with other solutions previously deducted by the author and by numerical simulations when available. Since there is no other exact method of calculating radiation exchanges within spherical fragments, we consider that this finding represents an advance which contributes to overcoming a variety of unexplained and practical problems of radiative heat transfer applicable to architectural developments, lighting elements and aircraft components. Keywords: radiative heat transfer; form factor calculation; buildings thermal simulation; algorithms for engineering; radiative properties of curved geometries; retrofit of heritage architecture 1. Introduction One of the main objectives of environmental sciences applied to building engineering has been to determine how the built environment is transformed due to the physiognomy of the constructions and how the design should be adjusted to obtain an improved climate performance [ 1 , 2 ]. In other words, how to optimize the design to attain a satisfactory and coherent distribution of available solar energy. A similar problem appears in lighting and aerospace technologies for different reasons [3]. The exact determination of radiative exchanges between solids and surfaces has been a long sought-for question [ 4 ] in heat transfer for building simulation [ 5 ]. Since the canonical equation rules such phenomena, a fourfold integral, it is extremely difficult to obtain an accurate solution such as a formula or abacus [ 6 ]. Several researchers have devoted their life to cataloging the relatively few explicit expressions that solve such complex problems with contributions to such developments from the present author [7–9]. The general problem of finding the form factor inside a spherical fragment has not been addressed in the literature, and until recently, only partial solutions for hemispheres and spherical caps have been proposed [ 10 – 12 ]. As the spherical surface is evidently nonplanar, the integration of the form factor over it becomes more and more complicated. Based on previous findings [ 13 , 14 ], the authors have identified a new property capable of regulating the matter in the domain of spherical geometries often employed in architectural and industrial buildings. However, the issue seems to have further unexpected derivates hereby presented in a detailed way. Buildings 2023,13, 1447. https://doi.org/10.3390/buildings13061447 https://www.mdpi.com/journal/buildings Buildings 2023,13, 1447 2 of 21 Such relevant matter appears initially as a by-product of the form factor between complete sections of the sphere, such as circles or disks. The configuration factors between a complete disk and a point, or as they are usually called, the point-to-point view factors from a circular plate, have been treated to a certain extent in the literature [ 15 – 17 ] but not the fragments of the disk, for instance, semicircles or circular segments and sectors. In the same way, the form factor between disks only appears in the references when the disks are complete (not fragmentary) and coaxial, but the case of intersecting disks remained undiscussed until introduced by the present authors [ 18 ]. The rationale of our finding is that intersecting circles usually enclose a portion of a sphere and, in this manner, constitute a volume composed of three surfaces, one of which is spherically curved. The algebra of the form factors could display and establish, in this way, the necessary equations to solve the problem, but one link was missing, namely, the form factor of the fraction of the sphere over itself. This form factor was solved by the authors by means of Cabeza-Lainez’s first principle [13,14]. The consistent application of such property offers a more complete panorama on the manner in which surface sources related or contained in spheres can be interpreted and accounted for with the straightforward advantage that integration is not required. An advance that will greatly facilitate the exact solution of variegated geometric problems in heat transfer of buildings. 2. Materials and Methods Spherical sources are extremely common in human designs or even natural forms [ 19 ]. Architecture has often been configurated by domes and vaults from monuments of the past to modern-day realizations [ 1 ]. Their radiative performance in terms of heat transfer from surface to surface was not well-known and often discarded or neglected and so the use of such adroit forms was discontinued. In this sense, we would start from the first principle of Cabeza-Lainez [ 14 ]: it states that for a spherical surface numbered for convenience A 3 , the so-called form factor F 33 , in other words, the fraction of energy emitted by such sphere’s fragment over itself, is defined as the ratio between the area of the said fragment and the whole sphere (Equation (1)). Hence, it yields 1 2for a hemisphere, 1 4for a quarter of sphere and successively. Due to its simplicity, it constituted a paramount finding [13,14,19,20]. It can be expressed algebraically as [13]: F33 =A3 As =A3 4πR2(1) Therefore, this extremely complex factor, if we had to calculate it with conventional methods, equals the area of surface A 3 divided by the total area of the sphere, A3/Asphere [13], as in Figure 1. Buildings 2023, 13, x FOR PEER REVIEW 2 of 22 Such relevant matter appears initially as a by-product of the form factor between complete sections of the sphere, such as circles or disks. The configuration factors between a complete disk and a point, or as they are usually called, the point-to-point view factors from a circular plate, have been treated to a certain extent in the literature [15–17] but not the fragments of the disk, for instance, semicircles or circular segments and sectors. In the same way, the form factor between disks only appears in the references when the disks are complete (not fragmentary) and coaxial, but the case of intersecting disks remained undiscussed until introduced by the present authors [18]. The rationale of our finding is that intersecting circles usually enclose a portion of a sphere and, in this manner, constitute a volume composed of three surfaces, one of which is spherically curved. The algebra of the form factors could display and establish, in this way, the necessary equations to solve the problem, but one link was missing, namely, the form factor of the fraction of the sphere over itself. This form factor was solved by the authors by means of Cabeza-Lainez’s first principle [13,14]. The consistent application of such property offers a more complete panorama on the manner in which surface sources related or contained in spheres can be interpreted and accounted for with the straightforward advantage that integration is not required. An advance that will greatly facilitate the exact solution of variegated geometric problems in heat transfer of buildings. 2. Materials and Methods Spherical sources are extremely common in human designs or even natural forms [19]. Architecture has often been configurated by domes and vaults from monuments of the past to modern-day realizations [1]. Their radiative performance in terms of heat transfer from surface to surface was not well-known and often discarded or neglected and so the use of such adroit forms was discontinued. In this sense, we would start from the first principle of Cabeza-Lainez [14]: it states that for a spherical surface numbered for convenience A3, the so-called form factor F33, in other words, the fraction of energy emitted by such sphere’s fragment over itself, is defined as the ratio between the area of the said fragment and the whole sphere (Equation (1)). Hence, it yields ½ for a hemisphere, ¼ for a quarter of sphere and successively. Due to its simplicity, it constituted a paramount finding [13,14,19,20]. It can be expressed algebraically as [13]: 𝐹 = 𝐴  𝐴 = 𝐴  4𝜋𝑅 (1) Therefore, this extremely complex factor, if we had to calculate it with conventional methods, equals the area of surface A3 divided by the total area of the sphere, A3/Asphere [13], as in Figure 1. Figure 1. Hemisphere with fragment of sphere A3 in red tones, which is used to determine F33, or the factor of the fraction over itself. Figure 1. Hemisphere with fragment of sphere A 3 in red tones, which is used to determine F 33 , or the factor of the fraction over itself. The first Cabeza-Lainez principle is compact and deft in the sense that it avoids the resolution of the difficult canonical expression described in Equation (2) in differential form, Buildings 2023,13, 1447 3 of 21 regardless of the shape of the spherical fragment [21], while to find the surface area of the said fragment is almost immediate (Figure 2). dφ12 =(E1−E2)cos θ1cos θ2 dA1dA2 πr2 12 (2) Buildings 2023, 13, x FOR PEER REVIEW 3 of 22 The first Cabeza-Lainez principle is compact and deft in the sense that it avoids the resolution of the difficult canonical expression described in Equation (2) in differential form, regardless of the shape of the spherical fragment [21], while to find the surface area of the said fragment is almost immediate (Figure 2). 𝑑𝜙 =(𝐸−𝐸)cos𝜃cos𝜃𝑑 𝐴 𝑑 𝐴  𝜋𝑟  (2) Figure 2. Radiative exchanges due to manifold surface sources A1 and A2, separated by an arbitrary distance r. A complete nomenclature appears at the end of the article. In the meanwhile, for Equation (2), E1 and E2 represent the amount of energy (in W/m2) emitted in Lambertian mode by each surface 1 and 2 [21]. θ1 and θ2, stand for the inclination angles formed (in radians) by the perpendiculars to the unit area elements of areas dA1 and dA2, r12 being the random radius vector (in meters) that connects both elementary surfaces dA1 and dA2 [21]. The exact and complete solution of Equation (2) is restricted, after lengthy calculations, to a few spare geometries, such as rectangles and parallel concentric circles [6]. The author has contributed actively to the list of solved surfaces, as can be seen in references [9,22]. Ensuing we will describe how and why the property under study was discovered [13]. It is known that, in most cases, the fourfold integration of the canonical equation over a volume composed of two curve-edged but planar surfaces, usually circular segments, towards an enclosing fragment of sphere is not attainable. So far, several authors have had to desist after the third round of integration [6,23]. Recently, some limited results have been obtained by virtue of numerical calculus and computational graphics [24,25]. However, a sphere cannot be discretized in equal elements, and this induces many inconsistencies and flaws in the approximate methods [24]. On the contrary, as we will demonstrate, Cabeza-Lainez’ derived property is sufficient to solve the problem for all the volumes specified, regardless of the position of the two planar surfaces, i.e., if they cut each other, they are tangent or they have no contact points. We will express such a new property in the following terms: For any couple of planar sections of a sphere (A1 − A2) that enclose within them a fragment of the same sphere (A3), the form factor from surface 1 to surface 2 can be defined as: 𝐹 = 𝐴 + 𝐴 + 𝐴 (𝐹−1) 2 𝐴  =1 2 󰇧1+ 𝐴 − 𝐴  𝐴  + 𝐴   𝐴  𝐴 󰇨 (3) F33 is the factor of a sphere’s fragment over itself that was defined previously in Cabeza-Lainez’s first principle as the ratio between the area of the said fragment and the Figure 2. Radiative exchanges due to manifold surface sources A 1 and A 2 , separated by an arbitrary distance r. A complete nomenclature appears at the end of the article. In the meanwhile, for Equation (2), E 1 and E 2 represent the amount of energy (in W/m 2 ) emitted in Lambertian mode by each surface 1 and 2 [21]. θ1 and θ2 , stand for the inclination angles formed (in radians) by the perpendiculars to the unit area elements of areas dA1 and dA2 , r12 being the random radius vector (in meters) that connects both elementary surfaces dA1and dA2[21]. The exact and complete solution of Equation (2) is restricted, after lengthy calculations, to a few spare geometries, such as rectangles and parallel concentric circles [ 6 ]. The author has contributed actively to the list of solved surfaces, as can be seen in references [9,22]. Ensuing we will describe how and why the property under study was discovered [ 13 ]. It is known that, in most cases, the fourfold integration of the canonical equation over a volume composed of two curve-edged but planar surfaces, usually circular segments, towards an enclosing fragment of sphere is not attainable. So far, several authors have had to desist after the third round of integration [ 6 , 23 ]. Recently, some limited results have been obtained by virtue of numerical calculus and computational graphics [ 24 , 25 ]. However, a sphere cannot be discretized in equal elements, and this induces many inconsistencies and flaws in the approximate methods [ 24 ]. On the contrary, as we will demonstrate, Cabeza-Lainez’ derived property is sufficient to solve the problem for all the volumes specified, regardless of the position of the two planar surfaces, i.e., if they cut each other, they are tangent or they have no contact points. We will express such a new property in the following terms: For any couple of planar sections of a sphere (A 1− A 2 ) that enclose within them a fragment of the same sphere (A 3 ), the form factor from surface 1 to surface 2 can be defined as: F12 =A1+A2+A3(F33 −1) 2A1 =1 2 1+A2−A3 A1 +A2 3 AsA1!(3) F 33 is the factor of a sphere’s fragment over itself that was defined previously in Cabeza-Lainez’s first principle as the ratio between the area of the said fragment and the whole sphere (Equation (1)), which resulted in a value of 1 2 for a hemisphere, 1 4 for a quarter of sphere and successively. It is written A 3 /A s . Where A s stands for the total area of the sphere or the constant 4πR2, and where R is the radius of the said sphere [13]. Buildings 2023,13, 1447 4 of 21 A 1 and A 2 are, as described, the respective areas of surfaces 1, 2 (segments of a circle), and A 3 is the comprised fragment of a sphere (Figures 3and 4). Determination of areas 1 and 2 offers no particular problem, but finding the surface of a fragment of a sphere might at times require the employ of spherical trigonometry and the internal angles involved, for instance, through Girard’s theorem, to be able to substitute in the formula of the third property. Buildings 2023, 13, x FOR PEER REVIEW 4 of 22 whole sphere (Equation (1)), which resulted in a value of ½ for a hemisphere, ¼ for a quarter of sphere and successively. It is written A 3 /A s . Where A s stands for the total area of the sphere or the constant 4πR 2 , and where R is the radius of the said sphere [13]. A 1 and A 2 are, as described, the respective areas of surfaces 1, 2 (segments of a circle), and A 3 is the comprised fragment of a sphere (Figures 3 and 4). Determination of areas 1 and 2 offers no particular problem, but finding the surface of a fragment of a sphere might at times require the employ of spherical trigonometry and the internal angles involved, for instance, through Girard’s theorem, to be able to substitute in the formula of the third property. Figure 3. Profile view of surfaces A 1 , A 2 (in orange) and A 3 (in cyan) forming part of a complete sphere of radius R. Figure 4. Imaginary perspective view of A 1 a circular segment with radius r 1 and another A 2 of radius r 2 , in concordance with A 3 (spherical fragment). The property verifies, irrespective of the relative position of the planar areas A 1 and A 2 , if they pass or not through the center of the sphere or even if they possess any tangent point. That is, both planar surfaces can even be independent circular sections of the sphere. The form factors between the sphere and any of the planar surfaces cannot be obtained at present in any other way or procedure, hence the novelty presented. Projective Figure 3. Profile view of surfaces A 1 , A 2 (in orange) and A 3 (in cyan) forming part of a complete sphere of radius R. Buildings 2023, 13, x FOR PEER REVIEW 4 of 22 whole sphere (Equation (1)), which resulted in a value of ½ for a hemisphere, ¼ for a quarter of sphere and successively. It is written A 3 /A s . Where A s stands for the total area of the sphere or the constant 4πR 2 , and where R is the radius of the said sphere [13]. A 1 and A 2 are, as described, the respective areas of surfaces 1, 2 (segments of a circle), and A 3 is the comprised fragment of a sphere (Figures 3 and 4). Determination of areas 1 and 2 offers no particular problem, but finding the surface of a fragment of a sphere might at times require the employ of spherical trigonometry and the internal angles involved, for instance, through Girard’s theorem, to be able to substitute in the formula of the third property. Figure 3. Profile view of surfaces A 1 , A 2 (in orange) and A 3 (in cyan) forming part of a complete sphere of radius R. Figure 4. Imaginary perspective view of A 1 a circular segment with radius r 1 and another A 2 of radius r 2 , in concordance with A 3 (spherical fragment). The property verifies, irrespective of the relative position of the planar areas A 1 and A 2 , if they pass or not through the center of the sphere or even if they possess any tangent point. That is, both planar surfaces can even be independent circular sections of the sphere. The form factors between the sphere and any of the planar surfaces cannot be obtained at present in any other way or procedure, hence the novelty presented. Projective Figure 4. Imaginary perspective view of A 1 a circular segment with radius r 1 and another A 2 of radius r2, in concordance with A3(spherical fragment). The property verifies, irrespective of the relative position of the planar areas A 1 and A 2 , if they pass or not through the center of the sphere or even if they possess any tangent point. That is, both planar surfaces can even be independent circular sections of the sphere. The form factors between the sphere and any of the planar surfaces cannot be obtained at present in any other way or procedure, hence the novelty presented. Projective or geometric methods were detained at the inaccessibility of finding the projection of a conical section generated by a sphere on the view cone [ 25 ]; finite element and other numerical procedures encountered the obstacle of non-discretizable surfaces when curves were involved [26]. Based on this new and innovative principle, obtaining the form factor is only a matter of knowing the areas of surfaces 1, 2 and 3, which is usually trivial, and additionally, it Buildings 2023,13, 1447 5 of 21 is exact and does not require integration. We have to note that this principle was hinted at by the author in early 2019 [ 20 ], but for several reasons, it was not published until fully developed. 3. Methodology of Proof and Results In order to demonstrate the validity of the newlyfound property, we will proceed to check it under different instances. Additionally, we will compare it with another principle discovered by the author, the number four principle of radiative exchange, that can be partially applied in some special cases [13,14]. The form factor algebra demonstrated elsewhere [1,5,12,27] states that, F11 + F12 + F13 + . . . F1N = 1 (4) Let us consider a volume composed of threes surfaces in the manner previously described (Figure 5), a fragment of sphere (surface 3) and two limiting planar surfaces (areas 1 and 2). Buildings 2023, 13, x FOR PEER REVIEW 5 of 22 or geometric methods were detained at the inaccessibility of finding the projection of a conical section generated by a sphere on the view cone [25]; finite element and other numerical procedures encountered the obstacle of non-discretizable surfaces when curves were involved [26]. Based on this new and innovative principle, obtaining the form factor is only a matter of knowing the areas of surfaces 1, 2 and 3, which is usually trivial, and additionally, it is exact and does not require integration. We have to note that this principle was hinted at by the author in early 2019 [20], but for several reasons, it was not published until fully developed. 3. Methodology of Proof and Results In order to demonstrate the validity of the newlyfound property, we will proceed to check it under different instances. Additionally, we will compare it with another principle discovered by the author, the number four principle of radiative exchange, that can be partially applied in some special cases [13,14]. The form factor algebra demonstrated elsewhere [1,5,12,27] states that, F11 + F12 + F13 + … F1N = 1 (4) Let us consider a volume composed of threes surfaces in the manner previously described (Figure 5), a fragment of sphere (surface 3) and two limiting planar surfaces (areas 1 and 2). Figure 5. The three surfaces involved in radiative exchange for the third property. Two planar sections (usually circular) and a spherical fragment that works as an enclosure of both sections. Then, the sole reflexive or form factor over the same surface that appears in the calculations is logically F33, which has been previously defined in Cabeza-Lainez’s first principle of form factors (Equation (1)). It is the ratio between the surface area of the spherical fragment and the total area of the sphere in which surface 3 is contained. F33 is, therefore, a constant for a given subsurface pertaining to the total sphere [22]. In this situation, if we apply the above algebra to find the unknown factors, we will find that, F31 + F32 + F33 = 1; (5) F12 + F13 = 1; (6) F21 + F23 = 1; (7) Figure 5. The three surfaces involved in radiative exchange for the third property. Two planar sections (usually circular) and a spherical fragment that works as an enclosure of both sections. Then, the sole reflexive or form factor over the same surface that appears in the calculations is logically F 33 , which has been previously defined in Cabeza-Lainez’s first principle of form factors (Equation (1)). It is the ratio between the surface area of the spherical fragment and the total area of the sphere in which surface 3 is contained. F 33 is, therefore, a constant for a given sub-surface pertaining to the total sphere [22]. In this situation, if we apply the above algebra to find the unknown factors, we will find that, F31 + F32 + F33 = 1; (5) F12 + F13 = 1; (6) F21 + F23 = 1; (7) Consequently, F12 = 1 −F13; F23 = 1 −F21; F13 = 1 −F12; Buildings 2023,13, 1447 6 of 21 However, the form factor algebra applied to finite surfaces in a closed volume [ 2 ] implies that: A1F12 = A2F21 (8) Thus, F32 =A2 A3 (1−F21); (9) F31 =A1 A3 (1−F12); (10) From Equation (5), we know that F31 = 1 −F32 −F33; (11) Substituting (9) and (10) in (11) A1 A3 (1−F12)=1−A2 A3 (1−F21)−F33; (12) Additionally, substituting (8) in (12): A1 A3 (1−F12)=1−A2 A3 +A1F12 A3−F33; (13) Multiplying both terms by A3, we obtain A1−A1F12 =A3−A2+A1F12 −A3F33; (14) Additionally, simplifying, 2A1F12 =A1+A2−A3+A3F33; (15) Thus, we arrive to F12 =1 21+A2−A3 A1 +A3 A1 F33; (16) As Equation (16) is equal to Equation (3), the expression for the new property of the form factors has been demonstrated; Quod Erat Demostrandum or QED (Latin for “which was what we intended to demonstrate”). Knowing F12, we can immediately find F21 by using reciprocity theorem (8) Consequently, we have arrived at the values of F 23 , F 13 , F 32 (9) and F 31 (10 and 11), and the exact exchanges between the sphere and the planar limiting surfaces are found. All the unknown factors of the problem are solved with perfect ease and exactitude for a great variety of shapes previously unexplored. To ensue, in a manner of comprobation, let us check the form factor due to several typical configurations. 3.1. Perpendicular Circles within the Sphere We shall begin, as is customary, with perpendicular circles by virtue of Cabeza-Lainez’ 3rd property (Figures 6and 7). Figure 6is an extreme case of Figure 7, in which the perpendicular circles have been enlarged to become tangent in a point. Buildings 2023,13, 1447 7 of 21 Buildings 2023, 13, x FOR PEER REVIEW 7 of 22 Figure 6. Perpendicular circles which possess a tangent point in common. Figure 7. Depiction of perpendicular circles of the same area A1 and A2 in a sphere. Figure 6 is an extreme case of Figure 7, in which the perpendicular circles have been enlarged to become tangent in a point. The third property, that we have seen in Equation (3) states that: For any couple of planar sections of a sphere (1–2) with a common edge or point, the form factor from surface 1 to surface 2 is in Equation (3) Additionally, assuming that the area of the circles has to be πR2/2 and A1 = A2, such equation turns out to be: 𝐹 =1+ 𝐴 (𝐹−1) 2 𝐴  =1+ 𝐴   2 𝐴  𝐴 − 𝐴  2 𝐴  (17) In this case, A3 is found by subtraction of the area of the two identical caps of the sphere; the area of one segment is π(a2 + h2), h is the height and a the radius of the base circumference of the cap. Theretofore, a = R√2/2 and h = R(1 − √2/2), where R is the radius of the sphere. 𝑎=𝑅 2 ; ℎ=𝑅3 2 − √ 2=𝑅 2 3 − 2 ∗ √ 2 (18) 𝐴 =4𝜋𝑅−2𝜋𝑅 2 1+3− 2 √ 2=4𝜋𝑅−2𝜋𝑅2 + √ 2=𝜋𝑅2 √ 2 (19) and the former yields 𝐴  =8𝜋𝑅 (20) Figure 6. Perpendicular circles which possess a tangent point in common. Buildings 2023, 13, x FOR PEER REVIEW 7 of 22 Figure 6. Perpendicular circles which possess a tangent point in common. Figure 7. Depiction of perpendicular circles of the same area A1 and A2 in a sphere. Figure 6 is an extreme case of Figure 7, in which the perpendicular circles have been enlarged to become tangent in a point. The third property, that we have seen in Equation (3) states that: For any couple of planar sections of a sphere (1–2) with a common edge or point, the form factor from surface 1 to surface 2 is in Equation (3) Additionally, assuming that the area of the circles has to be πR2/2 and A1 = A2, such equation turns out to be: 𝐹 =1+ 𝐴 (𝐹−1) 2 𝐴  =1+ 𝐴   2 𝐴  𝐴 − 𝐴  2 𝐴  (17) In this case, A3 is found by subtraction of the area of the two identical caps of the sphere; the area of one segment is π(a2 + h2), h is the height and a the radius of the base circumference of the cap. Theretofore, a = R√2/2 and h = R(1 − √2/2), where R is the radius of the sphere. 𝑎=𝑅 2 ; ℎ=𝑅3 2 − √ 2=𝑅 2 3 − 2 ∗ √ 2 (18) 𝐴 =4𝜋𝑅−2𝜋𝑅 2 1+3− 2 √ 2=4𝜋𝑅−2𝜋𝑅2 + √ 2=𝜋𝑅2 √ 2 (19) and the former yields 𝐴  =8𝜋𝑅 (20) Figure 7. Depiction of perpendicular circles of the same area A1and A2in a sphere. The third property, that we have seen in Equation (3) states that: For any couple of planar sections of a sphere (1–2) with a common edge or point, the form factor from surface 1 to surface 2 is in Equation (3). Additionally, assuming that the area of the circles has to be πR2/2 and A1= A2, such equation turns out to be: F12 =1+A3(F33 −1) 2A1 =1+A2 3 2AsA1−A3 2A1 (17) In this case, A 3 is found by subtraction of the area of the two identical caps of the sphere; the area of one segment is π (a 2 + h 2 ), h is the height and a the radius of the base circumference of the cap. Theretofore, a = R √ 2/2 and h = R(1 −√ 2/2), where R is the radius of the sphere. a2=R2 2;h2=R23 2−√2=R2 23−2∗√2(18) A3=4πR2−2πR2 21+3−2√2=4πR2−2πR22+√2=πR22√2(19) and the former yields A2 3=8π2R4(20) Buildings 2023,13, 1447 8 of 21 F12 =1+8π2R4 8πR2πR2/2 − πR22√2 πR2(21) Thus, the sought factor F12 is obtained as F12 =1+2−2√2=3−2√2=0.1715 (22) This value appears as the radiative exchange in a sort of cube composed of circles, which in turn are enclosed in a fragment of a sphere. The same value would be obtained for the parallel circles of the same figure or a sort of spherical barrel. In the cuboid made of six squares [ 2 , 5 ], the value of F 12 would be, as it is known, 1/5 = 0.2, a very similar quantity which attests to the validity of our calculations. Knowing the F 12 and the F 33 form factors of the problem, the radiant exchanges between the circles and the enclosing sphere, namely, F 23 , F 13 , F 32 and F 31, are obtained from Equations (5)–(7). This implies a complete novelty even for such simple and usual arrangement of surfaces. In an additional finding called the fourth principle [ 16 ], by logical deduction from the shape of the sphere, the author was able to obtain the form factor for semicircles of equal radius R possessing a common edge, depending on the internal angle formed by the half disks α . These, together with the consequent factors between the half disks and the sphere [9], are considered significant contributions so as to be reproduced in technical notes of journals and university catalogues [13,22] (Figure 8). Buildings 2023, 13, x FOR PEER REVIEW 8 of 22 𝐹 =1+ 8𝜋𝑅 8𝜋𝑅𝜋𝑅/2−𝜋𝑅2 √ 2 𝜋𝑅 (21) Thus, the sought factor F12 is obtained as 𝐹 =1+2−2 √ 2=3−2 √ 2=0.1715 (22) This value appears as the radiative exchange in a sort of cube composed of circles, which in turn are enclosed in a fragment of a sphere. The same value would be obtained for the parallel circles of the same figure or a sort of spherical barrel. In the cuboid made of six squares [2,5], the value of F12 would be, as it is known, 1/5 = 0.2, a very similar quantity which attests to the validity of our calculations. Knowing the F12 and the F33 form factors of the problem, the radiant exchanges between the circles and the enclosing sphere, namely, F23, F13, F32 and F31, are obtained from Equations (5)–(7). This implies a complete novelty even for such simple and usual arrangement of surfaces. In an additional finding called the fourth principle [16], by logical deduction from the shape of the sphere, the author was able to obtain the form factor for semicircles of equal radius R possessing a common edge, depending on the internal angle formed by the half disks α. These, together with the consequent factors between the half disks and the sphere [9], are considered significant contributions so as to be reproduced in technical notes of journals and university catalogues [13,22] (Figure 8). Figure 8. Two semicircles of same radius R with a common edge and an internal angle α. In this case, the defining equation is 𝐹 =1−2𝛼 𝜋+󰇡𝛼 𝜋󰇢 (23) Introducing the parameter p, 𝑝=𝛼 𝜋 The equation, represented in Figure 9 can be expressed as: 𝐹 =1−2𝑝+𝑝 (24) Figure 8. Two semicircles of same radius R with a common edge and an internal angle α. In this case, the defining equation is F12 =1−2α π+α π2(23) Introducing the parameter p, p=α π The equation, represented in Figure 9can be expressed as: F12 =1−2p+p2(24) The expression of this factor may be seen as somewhat simple, but we need to note that it is impossible to obtain by any other known procedures, for if we were to look at it in more detail (Figure 10): Buildings 2023,13, 1447 9 of 21 Buildings 2023, 13, x FOR PEER REVIEW 9 of 22 Figure 9. Graphic representation of the Cabeza-Lainez fourth principle (Equation (24)) of form factors between semicircles, marking its value for angle = Pi/2 =1.57 that is 0.25 or 1/4. The expression of this factor may be seen as somewhat simple, but we need to note that it is impossible to obtain by any other known procedures, for if we were to look at it in more detail (Figure 10): Figure 10. Radiative exchange between two semicircles forming a common angle α. The integral equation that we would be forced to solve would be for the semicircles (Equation (25)), 𝐹 =(𝑟 ∗𝑟∗𝑠𝑖𝑛𝜃∗𝑟∗𝑟∗𝑠𝑖𝑛𝜃)𝑑𝑟𝑑𝑟𝑑𝜃𝑑𝜃 𝜋∗(𝑟 +𝑟+2∗𝑟∗𝑟∗𝑐𝑜𝑠𝜃∗𝑐𝑜𝑠𝜃)        =1/4 (25) Equation (25), which is only valid for α = π/2, describes the unusual complexity of the factor F 12 to evaluate the radiative exchanges that occur in Figures 8 and 10, such exchanges, are unfeasible to calculate in an exact manner even for a perpendicular angle not to mention any other inclinations of the semicircles [28,29]. Therefore, the fourth principle can be considered as a pivotal advance, which is now subsumed in the third property, as we will show below. As it happens, the fourth property was discovered before the third property, but being less general, the author has chosen to alter their correlative order. Let us now proceed to check more examples based on the former finding. 3.2. Calculations for an Eighth of Sphere The area of a fraction of a sphere is 𝐴=2𝛼𝑅, where α is the angle between two semicircles with a common edge. For the whole sphere α = 2π. Figure 9. Graphic representation of the Cabeza-Lainez fourth principle (Equation (24)) of form factors between semicircles, marking its value for angle = Pi/2 =1.57 that is 0.25 or 1/4. Buildings 2023, 13, x FOR PEER REVIEW 9 of 22 Figure 9. Graphic representation of the Cabeza-Lainez fourth principle (Equation (24)) of form factors between semicircles, marking its value for angle = Pi/2 =1.57 that is 0.25 or 1/4. The expression of this factor may be seen as somewhat simple, but we need to note that it is impossible to obtain by any other known procedures, for if we were to look at it in more detail (Figure 10): Figure 10. Radiative exchange between two semicircles forming a common angle α. The integral equation that we would be forced to solve would be for the semicircles (Equation (25)), 𝐹 =(𝑟 ∗𝑟∗𝑠𝑖𝑛𝜃∗𝑟∗𝑟∗𝑠𝑖𝑛𝜃)𝑑𝑟𝑑𝑟𝑑𝜃𝑑𝜃 𝜋∗(𝑟 +𝑟+2∗𝑟∗𝑟∗𝑐𝑜𝑠𝜃∗𝑐𝑜𝑠𝜃)        =1/4 (25) Equation (25), which is only valid for α = π/2, describes the unusual complexity of the factor F 12 to evaluate the radiative exchanges that occur in Figures 8 and 10, such exchanges, are unfeasible to calculate in an exact manner even for a perpendicular angle not to mention any other inclinations of the semicircles [28,29]. Therefore, the fourth principle can be considered as a pivotal advance, which is now subsumed in the third property, as we will show below. As it happens, the fourth property was discovered before the third property, but being less general, the author has chosen to alter their correlative order. Let us now proceed to check more examples based on the former finding. 3.2. Calculations for an Eighth of Sphere The area of a fraction of a sphere is 𝐴=2𝛼𝑅, where α is the angle between two semicircles with a common edge. For the whole sphere α = 2π. Figure 10. Radiative exchange between two semicircles forming a common angle α. The integral equation that we would be forced to solve would be for the semicircles (Equation (25)), F12 =Zπ 0Zπ 0ZR 0ZR 0 (r2 1∗r2∗sinθ1∗r2 2∗r1∗sinθ2dr1dr2dθ1dθ2 π∗h(r2 1+r2 2+2∗r1∗r2∗cosθ1∗cosθ2i2=1/4 (25) Equation (25), which is only valid for α = π /2, describes the unusual complexity of the factor F 12 to evaluate the radiative exchanges that occur in Figures 8and 10, such exchanges, are unfeasible to calculate in an exact manner even for a perpendicular angle not to mention any other inclinations of the semicircles [ 28 , 29 ]. Therefore, the fourth principle can be considered as a pivotal advance, which is now subsumed in the third property, as we will show below. As it happens, the fourth property was discovered before the third property, but being less general, the author has chosen to alter their correlative order. Let us now proceed to check more examples based on the former finding. 3.2. Calculations for an Eighth of Sphere The area of a fraction of a sphere is A3= 2 αR2 , where α is the angle between two semicircles with a common edge. For the whole sphere α= 2π. In this case, an octave, α=π/4 and then A3=πR2/2. However, we know by virtue of Cabeza’s first principle [3] that F33 =1 8. That fraction means the area of the octave of sphere divided by the total area. Buildings 2023,13, 1447 16 of 21 F2, 3+4 = 0.303317 Buildings 2023, 13, x FOR PEER REVIEW 16 of 22 Figure 17. The quarter of sphere that appears at the Scenic Triclinium in Tivoli. Figure 18. Archaeologically reconstructed section of the Scenic Triclinium in Tivoli. Following the graphic in Figure 19, and considering as reflective surface the pool in front of the vault, we use the known properties of form factor algebra to obtain that F1+2, 3+4 = 0.085805 F2 3 = 0.228031 F1+2, 3 = 0.04815 F2, 3+4 = 0.303317 Figure 19. Simplified graph of the layout of the Scenic Triclinium in Tivoli (Rome), excerpted to study the radiation exchange between the reflecting pool and the adjoining domed structure. In this initial moment, we have equated the arched aperture to a rectangle for the sole purpose of finding the total amount of energy by solar radiation that impinges on this particular surface. Accordingly as, A1F14 = 36 ×14 ×0.085805 + 14 ×16 ×0.228031 −36 ×14 ×0.04815 −14 ×6×0.303317 = 14 ×30 ×F14 = 12.31908 Then, F14 = 0.029331 F41 = 0.125705 is the desired value to find the exchange of energy. For example, on the 21 June, if in the afternoon we consider the solar altitude to be of 50 ◦ , according to the weather data of Rome, we find that the value of horizontal illuminance is 74,252 lux. With a set reflectance of the pool of 0.4, we would then receive 29,700.8 lux. As F41 = 0.125705, the value of illuminance on surface 4 would be 3733.53 lux, the lower surface of the quarter of sphere as we have just discussed in the fourth principle (Section 3.4), receives exactly 1 4 of the former value, that is: 933.38 lux. Accordingly, the floor plan, by simple operations not presented here, would receive an average of 350 lux (See Figure 20). This proves the extraordinary reflective power of the pool, as well as the efficacy of the vault in channelling radiation. Although paradoxically, it is situated to the Northwest of the Triclinium, it has an exposure to the sun appropriate for cooling. We have just shown with “conservative” figures that the upper clerestory of the building alone produces more than 300 lux on the floor plan. To be on the safe side, we did not take into account the illuminance from the lower portico because today, we are unsure about the obstructions that this element would have presented to the light. It is said that the dome in its heyday was covered with bright enameled tiles, and that could have added a great amount to the former value of illuminance. With the findings described in previous sections of the article, we could discuss the probable amount of internal reflections involved. On the other hand, the building is roughly oriented to the North and protected by an artificial hill, so direct radiation rarely gets into the halls. However, by virtue of the pool and dome described, the spaces have adequate daylight avoiding the heat at the same time. There is no doubt that the Roman builders had intuitively noticed this effect and used it for special buildings such as the Triclinium. If we add the patent effect of earth-coupling and that of water running around the floor and walls Buildings 2023,13, 1447 17 of 21 of the structure in nymphaea, it is hard to imagine an architectural icon more exquisitely integrated with its surroundings. This ancient vault is undoubtedly an outstanding example of sustainable design. Buildings 2023, 13, x FOR PEER REVIEW 17 of 22 Figure 19. Simplified graph of the layout of the Scenic Triclinium in Tivoli (Rome), excerpted to study the radiation exchange between the reflecting pool and the adjoining domed structure. In this initial moment, we have equated the arched aperture to a rectangle for the sole purpose of finding the total amount of energy by solar radiation that impinges on this particular surface. Accordingly as, A 1 F 14 = 36 × 14 × 0.085805 + 14 × 16 × 0.228031 − 36 × 14 × 0.04815 − 14 × 6 × 0.303317 = 14 × 30 × F 14 = 12.31908 Then, F 14 = 0.029331 F 41 = 0.125705 is the desired value to find the exchange of energy. For example, on the 21 June, if in the afternoon we consider the solar altitude to be of 50°, according to the weather data of Rome, we find that the value of horizontal illuminance is 74,252 lux. With a set reflectance of the pool of 0.4, we would then receive 29,700.8 lux. As F 41 = 0.125705, the value of illuminance on surface 4 would be 3733.53 lux, the lower surface of the quarter of sphere as we have just discussed in the fourth principle (Section 3.4), receives exactly ¼ of the former value, that is: 933.38 lux. Accordingly, the floor plan, by simple operations not presented here, would receive an average of 350 lux (See Figure 20). Figure 20. Illuminance distribution under the dome of the Triclinium due only to the thermal (arched) aperture in the summer’s solstice. This proves the extraordinary reflective power of the pool, as well as the efficacy of the vault in channelling radiation. Although paradoxically, it is situated to the Northwest of the Triclinium, it has an exposure to the sun appropriate for cooling. We have just shown with “conservative” figures that the upper clerestory of the building alone produces more than 300 lux on the floor plan. To be on the safe side, we Figure 20. Illuminance distribution under the dome of the Triclinium due only to the thermal (arched) aperture in the summer’s solstice. 5.2. Recollection on the Findings Thus far, we have established and verified a new unbeknownst property to determine: 1. The total exchanges of radiant energy between two planar surfaces that cut a sphere in any way and the comprising fragment of the same sphere. The said planar surfaces can intersect, be tangent or stay completely independent of one another. The exchange case for arbitrary intersections and of these with the fragment of the sphere is an absolute novelty which cannot be obtained by any other procedure. The remaining non-intersecting layout is perfectly encompassed under a modern and inclusive formulation, and the exchanges between the disks and the inner sphere are completely innovative. 2. The possible inter-reflections of the compound despite the existence of a curved surface. This was not previously solved in the references since reflections were always limited to cuboidal spaces. 3. The introduction of the commutative property and its subsequent algebra in form factors on a purely surface-to-surface basis for volumes that integrate the circle in their composition. Thus, the property enunciated is original and quite efficient, a few cases covered by it, for instance, radiating parallel circles, were partially addressed earlier in the literature by other clumsier means, usually involving lengthy integrations [ 36 , 37 ]. These limited cases had absolutely no regard for the radiative transfer between the circles and the surrounding fragment of a sphere that we have now settled in an exact manner [ 38 ]. Particularly, if the two planar surfaces intersect themselves in any fashion and cease to be entirely circular to become circular segments, then such relevant stance was ignored by the literature [ 11 ] or was termed unsolvable [ 6 ], and we have proved that it can only be determined through the third property of Cabeza-Lainez, hereby presented. Such findings will have paramount repercussions in various fields of engineering and heat transfer simulations that are not completely foreseeable at the moment, but we can outline just a few in building construction or heritage retrofit and surely in the lighting and Buildings 2023,13, 1447 18 of 21 aero-spatial fields. Sundry architectural configurations, as we have seen in the example of Section 5.1, feature the sphere and the circular fragment as an important part of their fenestration or ceiling system (Figures 21–23), especially as we mentioned for monumental or singular buildings. The analysis of the thermal, lighting or even acoustic performance of these features was severely restricted, and the possibility of increasing energy savings and comfort for the users was almost impeded, which subsequently led to its abandonment. With the findings developed, we expect that they will become better preserved and future enhancements will turn mandatory for this field of knowledge [30]. Buildings 2023, 13, x FOR PEER REVIEW 19 of 22 Figure 21. Ilja Doganoff. Railway Sheds of conoidal shape and semicircular clerestories in Bulgaria (1958). Source: Author. Figure 22. Interior spherical dome of the church of St. Louis (1730). Seville. Spain. Source: Author. Figure 21. Ilja Doganoff. Railway Sheds of conoidal shape and semicircular clerestories in Bulgaria (1958). Source: Author. Buildings 2023, 13, x FOR PEER REVIEW 19 of 22 Figure 21. Ilja Doganoff. Railway Sheds of conoidal shape and semicircular clerestories in Bulgaria (1958). Source: Author. Figure 22. Interior spherical dome of the church of St. Louis (1730). Seville. Spain. Source: Author. Figure 22. Interior spherical dome of the church of St. Louis (1730). Seville. Spain. Source: Author. Buildings 2023,13, 1447 19 of 21 Buildings 2023, 13, x FOR PEER REVIEW 19 of 22 Figure 21. Ilja Doganoff. Railway Sheds of conoidal shape and semicircular clerestories in Bulgaria (1958). Source: Author. Figure 22. Interior spherical dome of the church of St. Louis (1730). Seville. Spain. Source: Author. Figure 23. The Latina Market (1960) with spherical vaults and openings. Madrid. Spain. Source: Author. In aerospace techniques, parts of the vessels or mechanisms that feature spheres and/or circles will experience a more accurate prediction of their performance and capabilities thanks to this property introduced in computational geometry. Lighting industries will surely benefit from the display of an ampler palette of algorithms and simulation tools in order to be able to offer more conscious products [ 39 ]. This is our expectation for beyond the manuscript. 6. Conclusions In architecture, curved spherical surfaces have been used since antiquity, especially in domes and vaults and their respective openings. However, a consistent procedure to treat them from the point of view of radiative transfer was not readily available or feasible. Radiance and other ray-tracing methods of simulation stayed at this precise point as there is no exact or even customized discretization in equal elements for spherical or other curved fragments. In this manuscript, we solved the issue with total exactitude by thoroughly introducing the property found by the author, which has demonstrated a robust capacity to overcome the most important caveats appearing in the integration procedure. Such principle is based on Cabeza-Lainez’s first expression for spheres, and it does not require integration; it only demands finding the respective surface areas of the emitters involved. It provides us with a new synoptic vision about the way in which surface emitters are enclosed in a sphere, and considering its effect can be calculated with the direct advantage that cumbersome integral procedures or inexact approximations that consume computer time and resources are ruled out. The pivotal advance of this property resides in the fact that it can be easily incorporated into novel algorithms for computation and thus applicable to a variety of previously unconsidered problems of radiated energy conveyed to the building sector, lighting sciences and aerospace technologies. In this way, capital issues of solar radiation channelling, ever present in architecture and buildings, can be sorted out as exactly as unequivocally. In this sense, the authors consider that they have been able to provide a crucial development in their research field by imagining it by virtue of scientific intelligibility and its formidable potential for renewal. Funding: This research received no external funding. Data Availability Statement: Not applicable. Acknowledgments: This article is dedicated to the Odajima family. I wish to honor the memory of Manuel Bendala Garcia. My appreciation to Juan Manuel Bonilla Martinez for his continuous support. Thanks to Restituto Bravo and Desiree Gonzalez for their help. Conflicts of Interest: The author declares no conflict of interest. Buildings 2023,13, 1447 20 of 21 Nomenclature Nomenclature of canonical equation, E1 and E2 are the amounts of energy (in W/m 2 ) emitted in Lambertian fashion by surfaces 1 and 2. θ1 and θ2 , stand for the inclination angles (in radians) formed by the perpendiculars to the unit area elements of area dA1and dA2. r12 is the arbitrary vector (in m) that connects the surfaces dA1and dA2. References 1. Cabeza-Lainez, J. Fundamentals of Luminous Radiative Transfer: An Application to the History and Theory of Architectural Design; Crowley Editions: Seville, Spain, 2006. 2. Hensen, J.L.M.; Lamberts, R. Building Performance Simulation for Design and Operation, 2nd ed.; Routledge: London, UK, 2019; ISBN 9781138392199. 3. Moon, P.H. The Scientific Basis of Illuminating Engineering; McGraw-Hill Book Co. Dover Publications: New York, NY, USA, 1963. 4. Lambert, J.H. Photometria. Sive de Mensura et Gradibus Luminis, Colorum et Umbrae; DiLaura, D., Ed.; IESNA: New York, NY, USA, 1764. 5. Howell, J.R.; Siegel, R.; Mengüç, M.P. Thermal Radiation Heat Transfer, 5th ed.; Taylor and Francis/CRC: New York, NY, USA, 2010. 6. Hamilton, D.C.; Morgan, W. Radiant-Interchange Configuration Factors. NASA. 1952; Online Version. Available online: https://ntrs.nasa.gov/citations/19930083529 (accessed on 9 May 2023). 7. Howell, J.R.; Mengüç, M.P. Radiative transfer configuration factor catalogue: A listing of relations for common geometries. J. Quant. Spectrosc. Radiat. Transf. 2011,112, 910–912. [CrossRef] 8. John, R. A Catalogue of Radiation Heat Transfer Configuration Factors. Howell. University of Texas at Austin. Available online: http://www.thermalradiation.net/indexCat.html (accessed on 29 March 2023). 9. Howell, J.R. A Catalogue of Radiation Heat Transfer Configuration Factors. Factor C-43b. Available online: http://www. thermalradiation.net/sectionc/C-43b.html (accessed on 12 April 2023). 10. Howell, J.R. A Catalogue, Factor C-140b. Available online: http://www.thermalradiation.net/sectionc/C-140b.html (accessed on 12 April 2023). 11. Moon, P.H.; Spencer, D.E. The Photic Field; The MIT Press: Cambridge, MA, USA, 1981. 12. Holman, J.P. Heat Transfer, 7th ed.; Mac Graw Hill: New York, NY, USA, 1995. 13. Cabeza-Lainez, J. New configuration factors for curved surfaces. J. Quant. Spectrosc. Radiat. Transf. 2013,111, 71–80. [CrossRef] 14. Cabeza Lainez, J. Solar Radiation in Buildings. Transfer and Simulation Procedures. In Solar Radiation; Babatunde, E.B., Ed.; InTech: London, UK, 2012; Chapter 16; ISBN 978-953-51-0384-4. 15. Gershun, A. The Light Field (translated from Russian by P. Moon and G. Timoshenko). J. Math. Phys. 1939 ,18, 51–151. [CrossRef] 16. Vladimir, F. Zur Berechnung der Beleuchtungsstärke; Optisches Institut: St. Petersburg, Russia, 1924. 17. Feingold, A. A new look at radiation configuration factors between disks. J. Heat Transf. 1978,100, 742–744. [CrossRef] 18. Cabeza-Lainez, J.M.; Pulido-Arcas, J.A. New Configuration Factor between a Circle, a Sphere and a Differential Area al Random Positions. J. Quant. Spectrosc. Radiat. Transf. 2013,129, 272–276. [CrossRef] 19. Shukuya, M. Hikari to Netsu no KenchikuKankyogaku. In Light Temperature Environment Science; Maruzen: Tokyo, Japan, 1993. (In Japanese) 20. Cabeza-Lainez, J.M.; Rodriguez-Cunill, I. The Problem of Lighting in Underground Domes, Vaults, and Tunnel-Like Structures of Antiquity; An Application to the Sustainability of Prominent Asian Heritage (India, Korea, China). Sustainability 2019 ,11, 5865. [CrossRef] 21. Cabeza-Lainez, J.M.; Rodríguez-Cunill, I. Prevention of Hazards Induced by a Radiation Fireball through Computational Geometry and Parametric Design. Mathematics 2022,10, 387. [CrossRef] 22. Howell, J.R. A Catalogue of Radiation Heat Transfer Configuration Factors. Factor C-43a. Available online: http://www. thermalradiation.net/sectionc/C-43a.html (accessed on 10 May 2023). 23. Salguero-Andujar, F.; Cabeza-Lainez, J.-M. New Computational Geometry Methods Applied to Solve Complex Problems of Radiative Transfer. Mathematics 2020,8, 2176. [CrossRef] 24. Subramaniam, S.; Hoffmann, S.; Thyageswaran, S.; Ward, G. Calculation of View Factors for Building Simulations with an Open-Source Raytracing Tool. Appl. Sci. 2022,12, 2768. [CrossRef] 25. Nußelt, W. Graphische Bestimmung des Winkelverhältnisses bei der Wärmestrahlung. Z. Ver. Dtsch. Ing. 1928,72, 673. 26. Schröder, P.; Hanrahan, P. On the Form Factor between Two Polygons. In Proceedings of the SIGGRAPH 93: 20th Annual Conference on Computer Graphics and Interactive Techniques, Anaheim CA, USA, 2–6 August 1993; pp. 163–164. [CrossRef] 27. Modest, M.F. View Factors. In Radiative Heat Transfer, 3rd ed.; Academic Press: Cambridge, MA, USA, 2013; pp. 129–159. 28. Naraghi, M.H.N. Radiation View Factors from Differential plane sources to disks—A general formulation. Tech. Notes Am. Inst. Aeronaut. Astronaut. J. 1988,2, 3. [CrossRef] 29. MacAllister, A.S. Graphical solutions of problems involving plane surface lighting sources. Light. World 1911,56, 135. Buildings 2023,13, 1447 21 of 21 30. Cabeza-Lainez, J. Architectural Characteristics of Different Configurations Based on New Geometric Determinations for the Conoid. Buildings 2022,12, 10. [CrossRef] 31. Cabeza Lainez, J.M.; Pulido Arcas, J.A.; Bellido, C.R.; Bonilla Martínez, J.M. Radiative Heat Transfer for Curvilinear Surfaces. In Solar Radiation Applications; InTech: London, UK, 2015; Chapter 1. 32. Howell, J.R. A Catalogue, Factor C-140a. Available online: http://www.thermalradiation.net/sectionc/C-140a.html (accessed on 18 May 2023). 33. Howell, J.R. A Catalogue, Factor C-140c. Available online: http://www.thermalradiation.net/sectionc/C-140c.html (accessed on 18 May 2023). 34. Howell, J.R. A Catalogue, Factor C-140d. Available online: http://www.thermalradiation.net/sectionc/C-140d.html (accessed on 21 May 2023). 35. Howell, J.R. A Catalogue, References. Available online: http://www.thermalradiation.net/references.html (accessed on 21 May 2023). 36. Naraghi, M.H.N. Radiative View Factors from Spherical Segments to Planar Surfaces. J. Thermophys. Heat Transf. 1988 ,2, 373–375. [CrossRef] 37. Chung, B.T.F.; Naraghi, M.H.N. Some Exact Solutions for Radiation View Factors from Spheres. AIAA J. 1981 ,19, 1077–1108. [CrossRef] 38. Sasaki, K.; Sznajder, M. Analytical view factor solutions of a spherical cap from an infinitesimal surface. Int. J. Heat Mass Transf. 2020,163, 120477. [CrossRef] 39. Cabeza Lainez, J. Scientific designs of sky-lights. In Proceedings of the Conference on Passive and Low Energy Architecture (PLEA), Brisbane, Australia, 22–24 September 1999. 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