New Computational Geometry Methods Applied to Solve Complex Problems of Radiative Transfer

Article New Computational Geometry Methods Applied to Solve Complex Problems of Radiative Transfer

Francisco Salguero-Andújar 1 and Joseph-Maria Cabeza-Lainez 2,*

1 School of Engineering, University of Huelva, Campus de El Carmen, 21007 Huelva, Spain; [email protected] 2 Higher Technical School of Architecture, University of Seville, 41012 Seville, Spain * Correspondence: [email protected]; Tel.: +34-696-749344

Received: 11 November 2020; Accepted: 4 December 2020; Published: 6 December 2020

Abstract: Diverse problems of radiative transfer remain as yet unsolved due to the difficulties of the calculations involved, especially if the intervening shapes are geometrically complex. The main goal of our investigation in this domain is to convert the equations that were previously derived into a graphical interface based on the projected solid-angle principle. Such a procedure is now feasible by virtue of several widely diffused programs for Algorithms Aided Design (AAD). Accuracy and reliability of the process is controlled in the basic examples by means of subroutines from the analytical software DianaX, developed at an earlier stage by the authors, though mainly oriented to closed cuboidal or curved volumes. With this innovative approach, the often cumbersome calculation procedure of lighting, thermal or even acoustic energy exchange can be simplified and made available for the neophyte, with the undeniable advantage of reduced computer time.

Keywords: mathematics applied to lighting and radiative transfer; conﬁguration factors; computational geometry; parametric design; new solutions for equations of geometric optics; numerical computation of quadruple integrals

1. Introduction

1.1. Form Factors To our knowledge, one of the main problems in Science as applied to aerospace, solar and industrial design technology has been to determine how the existing physical fields are transformed according to the physiognomy of fixtures or products and in which direction we should orient our developments to seek a more correct transmission of environmental and heat transfer phenomena, or, in other words, in which way can the design of three-dimensional forms be improved to obtain an optimal and coherent distribution of energies that effectively contributes to mitigate global warming and thereby helps resolve climatic issues. We must outline that radiation in a physical or spatial environment is made manifest through fields of a fundamentally vector nature. Therefore, our first objective would lie in the assessment of these types of fields in an unaltered state. On such issues, there exist the relevant contributions of Yamauchi [1] and Moon [2,3] amongst others in seeking radiative potential. However, successive modifications of the spatial features of the elements entailed possess the capacity to substantially alter the layout of the field. Such geometric alterations contribute to changes in energy distribution that manufacturers and users are heavily demanding, as it is already vital for them to acquire accurate and simple notions regarding the issue. The problem is not recent. In fact, Lambert’s statement of the famous theorem XVI in his treatise Photometry [4] speculated about the amount of rays (flux) that issues from any two equally luminous surfaces onto the adjacent, and established that if the two surfaces A1 and A2 were equally luminous

Mathematics 2020, 8, 2176; doi:10.3390/math8122176 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 2176 2 of 25 Mathematics 2020, 8, x FOR PEER REVIEW 2 of 29 andsurfaces faced onto each the other adjacent, in some and way, esta theblished amount that of if incidentthe two surfaces rays from A either1 and A of2 thewere two equally surfaces luminous on the secondand faced is identical. each other The in formersome way, is also the known amount as of reciprocity incident rays theorem from and either to develop of the two it to surfaces some extent on the is crucialsecond tois identical. assess the The mathematical former is also reach known of the as matter reciprocity that we theorem are discussing. and to develop it to some extent is crucialThe energyto assess that the leaves mathematical surface A re1 achand of reaches the matter surface that A we2 will are be: discussing. The energy that leaves surface A1 and reaches surface A2 will be: E1A1F12, (1) E1A1F12, (1) and, reciprocally, thethe energyenergy thatthat passespasses fromfrom surfacesurface AA22 toto AA11 is:

EE22AA22FF2121,, (2) (2)

2 where E1 and E2 are the equivalent amounts of energy (in W/m2 ) emitted by surfaces A1 and A2; and where E1 and E2 are the equivalent amounts of energy (in W/m ) emitted by surfaces A1 and A2; and F12 F12 or F21 are dimensionless entities called “form factors”. If we assume, in principle, that there are no or F are dimensionless entities called “form factors”. If we assume, in principle, that there are no inter-reflections21 or re-emissions from one surface to the reciprocal, and theoretically no other inter-reflections or re-emissions from one surface to the reciprocal, and theoretically no other significant significant sources of radiation have access, by any means, to the boundaries of the problem, all sources of radiation have access, by any means, to the boundaries of the problem, all incident flux will incident flux will be absorbed and the flow of energy (dΦ), according to Lambert’s theorem, should be absorbed and the flow of energy (dΦ), according to Lambert’s theorem, should be zero, that is: be zero, that is:

dΦ =dΦE 1=A E11FA121F12-EE22AA22F21 == 0 ⇔0 A1AF112F =12 A=2FA21.2 F21.(3) (3) − ⇔ To this end, we would consider the surface elements dA1 and dA2. The angles θ1 and θ2 are To this end, we would consider the surface elements dA1 and dA2. The angles θ1 and θ2 are measured between the line from dA1 to dA2 (direction of propagation) and the normal to each surface; measured between the line from dA1 to dA2 (direction of propagation) and the normal to each surface; r is the distance from thethe centercenter ofof oneone didifferentialﬀerential areaarea toto thethe otherother (Figure(Figure1 ):1):

Figure 1. General three-dimensional arrangement of surface source elements through which the radiationFigure 1. formGeneral factor three-dimensional is defined. arrangement of surface source elements through which the radiation form factor is defined. The cosine emission law, also attributed to J. H. Lambert, states that the flux per unit of solid angleThe in acosine given emission direction law,θ (or also radiant attributed intensity) to J. H. is Lambert, equal to states the flux that in the the flux normal per unit direction of solid to theangle surface in a given (maximum direction intensity) θ (or radiant multiplied intensity) by the iscosine equal ofto the angleflux in between the normal the normaldirection and to the directionsurface (maximum considered. intensity) multiplied by the cosine of the angle between the normal and the directionLuminance considered. or radiance is the radiant flux or power emitted per unit area and solid angle (steradian) Luminance or radiance is the radiant flux or power emitted per unit area and solid angle in a given direction. To obtain the flow emitted by the surface element dA1 it is necessary to multiply (steradian) in a given direction. To obtain the flow emitted by the surface element dA1 it is necessary the radiance or luminance by the projected surface in the fixed direction of the angle θ1, and thus: to multiply the radiance or luminance by the projected surface in the fixed direction of the angle θ1, and thus: dΦ I = = L1dA1 cos θ1, (4) dΩ I= =L dA cos θ , (4) where I is the radiant intensity, dΩ is the solid angle and L1 represents the radiance of surface A1. where I is the radiant intensity, dΩ is the solid angle and L1 represents the radiance of surface A1. It is easily deducted that L1 remains independent of the direction and therefore L1θ = L1.

Mathematics 2020, 8, 2176 3 of 25

It is easily deducted that L1 remains independent of the direction and therefore L1θ = L1. Since the previously found flux refers to the solid angle unit, we must now evaluate the amount of flux that reaches a differential surface element dAn whose distance to A1 is precisely r.

dAn dΦ = L1dA1 cos θ1 (5) r2 It is often useful to establish a relationship between luminance or radiance and lighting or irradiance; with this aim, we could simply extend the ﬂux to a hemisphere of radius r in which dA1, is inscribed. The surface dAn in spherical co-ordinates amounts to:

2 dAn = r sin θ1dθdϕ (6) then from (5) and (6):

Z Z π Z π dA 2 /2 Φ = L A cos θ n = L A cos θ sin θ dθdϕ = πL A = E A (7) 1 1 1 1 2 1 1 1 1 1 1 1 1 An r 0 0

2 Thus, E1 = πL1 (E is emitted radiation per area unit, expressed in W/m ). Returning to the problem of energy exchange, if we assimilate the diﬀerential surface element dAn with dA2, we ﬁnd that: dAn = cos θ2dA2. (8)

In this manner, the ﬂux or radiant power that goes from dA1 to dA2, substituting L1 in Equation (5), will be: dA1dA2 dΦ1 2 = E1 cos θ1 cos θ2 (9) − πr2 And respectively, dA2dA1 dΦ2 1 = E2 cos θ2 cos θ1 . (10) − πr2 The energy exchange is eventually set at: Z Z dA1dA2 Φ1 2 = (E1 E2) cos θ1 cos θ2 . (11) − π 2 − A2 A1 r

The above integral equation responds to the fundamental or canonical formula of radiation which in differential terms is: 2 dA1dA2 d Φ = Ei cos θ1 cos θ2 (12) πr2 In such a situation, the integral found by virtue of the fundamental expression adopts the value of A1F12 or A2F21, following the above definitions. In general, these are quadruple integrals and to solve them becomes so complex that extreme dexterity in geometry problems along with haphazard techniques of integration is required. Therefore, we want to avoid this difficulty as much as possible by virtue of the procedure presented in this article. The fundamental equation of the much-sought form factors is: "Z Z # 1 cos θ cos θ F = 1 2 dA dA (13) 12 π 2 1 2 A1 A2 A1 r and its asymmetric reciprocal in algebraic terms, "Z Z # 1 cos θ cos θ F = 1 2 dA dA (14) 21 π 2 1 2 A2 A2 A1 r Mathematics 2020, 8, 2176 4 of 25

If we assume, for easier visualization and since it is a frequent case in luminous radiative transfer (but not so in thermal problems involving, for instance. gases), that the surface A2 does not emit energy and only receives it from A1, the total ﬂow received in A2 will be: "Z Z # M cos θ cos θ A N = 1 1 2 dA dA (15) 2 2 π 2 1 2 A2 A1 r

2 where M1 = N2/F21 and N2 is the energy received by surface 2 in W/m .

1.2. Configuration Factors Let us behold the meaning of the form factor, defined as an integral equation. Actually, what it permits us to calculate is just the net radiant exchange from surface to surface. A surface that emits a 2 certain amount of power M1 (in W/m ) successively produces a certain power N2 ‘as an average or percentage’ on another surface, which tells us nothing about the point-by-point distribution of that same power. However, in many engineering, architectural and medical problems, a precise description of the flux function, that is, the radiant field, is often deemed necessary to establish the exact nature of the radiation “map” and to be able to control the possible deficit or surplus which could, in turn, become critical. Therefore, if we look for the point distribution of power that reaches surface A2 instead of the mere average, we could reduce the fourfold integrals in the expression of the form factor to a simpler surface integral as follows: Z Z M cos θ cos θ cos θ cos θ n = 1 1 2 dA = M 1 2 dA (16) 2 π 2 1 1 π 2 1 A1 r A1 r

This new expression is, like the previous one, strictly geometric and as such only depends on angles and areas, given that for a certain moment of time the value of M1 can be assumed as constant. The double integral equation, Z cos θ cos θ f = 1 2 dA (17) 12 π 2 1 A1 r will be called the conﬁguration factor and it presents several fundamental applications in the method that we will develop hereby. The ﬁrst elementary consequences are:

n2 = f12M1 (18)

and "Z # A2N2 = M1 f12dA2 = A1F12M1 = A2F21M1 (19) A2 Then, "Z # 1 F12 = f12dA2 (20) A1 A2 Accordingly, "Z # 1 F21 = f12dA2 (21) A2 A2

Equation (21) can be clearly identiﬁed with the “average” of f12“over surface” A2, especially since we had already obtained that N2 = F21M1. In quantum dynamics terms, the form factor expresses the probability of a surface being stricken with photonic energy emitted by an adjacent source. Mathematics 2020, 8, 2176 5 of 25

Similarly, we would deﬁne N2 as the average of n2 on the surface A2; "Z # 1 N = n dA (22) Mathematics 2020, 8, x FOR PEER REVIEW 2 2 2 5 of 29 A2 A2

ThisThis importantimportant finding finding represents represents the ignoredthe igno relationshipred relationship between formbetween factors form and configurationfactors and configurationfactors that will factors allow that us will to easily allow simplify us to easily and visualizesimplify and many visualize complex many calculations complex in calculations the following in thechapters following by restrictingchapters by them restricting onlyto them second only orderto second primitives, order primitives, and the ensuing and the ensuing “average” “average” is ready isaccessible ready accessible by numerical by numerical procedures. procedures. To our knowledge, To our knowledge, it has only it been has only defined been by defined Cabeza-Lainez by Cabeza- [5,6]. Lainez [5,6]. 2. Symbolic Calculus of Basic Elements 2. Symbolic Calculus of Basic Elements 2.1. Triangle 2.1. TriangleA set of basic forms can be derived from the triangle, but only if the calculation axis lies perpendicularA set of basic to the forms center can ofbe the derived figure from or the the vertex triangle, of thebut triangleonly if the or rectanglecalculation considered. axis lies perpendicularThe principle equationto the center is, of the figure or the vertex of the triangle or rectangle considered. The Z b Z z y2 principle equation is, n = L dzdx (23) 2 0 0 (x2 + y2 + z2) y n=L dzdx for an instant of time t in which the luminance or radiance L can be considered constant and keeping(23) (x + y +z ) in mind that Lπ = M. for an instant of time t in which the luminance or radiance L can be considered constant and keeping The solution for triangle A1 in Figure2, integrated by parts is: in mind that Lπ = M.   The solution for triangle A1 in Figure 2, integrated by parts is:    L b 1 a  n =  q tan− q  (24) 2L b 2 a 2  n=  y2 + b tan y2 + b  (24) 2 y +b y +b

Figure 2. Conﬁguration factor between a rectangle consisting of two united triangles (A1 and A2) and Figurea point 2. belonging Configuration to a parallel factor between plane. a rectangle consisting of two united triangles (A1 and A2) and a point belonging to a parallel plane. 2.2. Rectangle 2.2. Rectangle The full rectangle is just the sum of solutions for surfaces A1 (24) and A2 (see AppendixA): The full rectangle is just the sum of solutions for surfaces A1 (24) and A2 (see Appendix A):   LL aa 1 b bb 1 a a  n =  p tan− p + q tan− q  (25) n=2  2 2 tan 2 2 + tan  (25) 2 yy++aa y +a+ a yy2+b+ b2 y y+b2 + b2  However, we need to be reminded that we have just obtained the value of radiative exchange at a single point located precisely under the vertex of the rectangle and in a particular direction (perpendicular) to the surface source. Deft combinations of triangles would give us the equilateral triangle, hexagon and, as a limit case that we will discuss in Appendix A, the circle, which has been only obtained in such “polygonal” fashion by Cabeza-Lainez.

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However, we need to be reminded that we have just obtained the value of radiative exchange at a single point located precisely under the vertex of the rectangle and in a particular direction (perpendicular) to the surface source. Deft combinations of triangles would give us the equilateral triangle, hexagon and, as a limit case that we will discuss in AppendixA, the circle, which has been only obtained in such “polygonal” fashion by Cabeza-Lainez. Mathematics 2020, 8, x FOR PEER REVIEW 6 of 29 2.3. Calculations in a Plane Perpendicular to the Figure 2.3. Calculations in a Plane Perpendicular to the Figure In the above discussion, we have solely found the component of radiation that was perpendicular to theIn emitting the above surface discussion, at an isolated we point.have solely found the component of radiation that was perpendicularIt is easy toto imaginethe emitting the severe surface complexity at an isolated that point. arises when trying to find the solution for points movingIt is freelyeasy to in imagine a plane, the let alonesevere space. complexity that arises when trying to find the solution for points movingEven freely so, in radiance a plane, being let alone a vector, space. to complete the problem means finding values of other auxiliaryEven components. so, radiance being a vector, to complete the problem means finding values of other auxiliaryIn this components. situation, the principle integral turns out to be (if we multiply by the corresponding cosine): In this situation, the principle integral turns out to be (if we multiply by the corresponding ZZ cosine): zy n = L dxdz (26) (x2 +zyy2 + z2) n=L dxdz (26) (x + y +z) Applying this proposition to the case considered (Figure3), we find that if Y (the distance) is a constant,Applying the integral this proposition equation yields:to the case considered (Figure 3), we find that if Y (the distance) is a constant, the integral equation yields:  L b y b =  1 1  n Ltan− b p y tan− p b  (27) n=2  tan y −− a2 + y2 tan a2 + y2  (27) 2 y a + y a + y

Figure 3. ConﬁgurationConfiguration factor between a rectangle consisting of two annexed triangles (A11 and A2)) and a point aligned with the normal to a vertex andand contained in a plane perpendicular toto thethe ﬁgure.figure.

By virtuevirtue of of the the former, former, we havewe have exposed exposed how complexhow complex it becomes it becomes to find an to analytical find an expressionanalytical expressionto solve the to problem solve the if problem only we alterif only the we coordinate alter the coordinate plane of calculation, plane of calculation, not to mention not to figures mention of figuresgeometry of geometry other than other those than limited those by limited straight by lines.straight That lines. is whyThat is we why concluded we concluded that there thatwas there a wassignificant a significant need for need a graphical for a graphical solution solution based on ba parametricsed on parametric design as design this would as this imply would a universal imply a universalsolution and solution an extraordinary and an extraordinary simplification simplification of the issue. of the issue.

2.4. Calculation of Integral Equation for a ‘Rectangle’ Over a Horizontal Square 2.4. Calculation of Integral Equation for a ‘Rectangle’ Over a Horizontal Square Previously (Equation (27)) we had found the solution in terms of arctangent for the entire rectangle. Previously (Equation (27)) we had found the solution in terms of arctangent for the entire The adaptation of this formula to an arbitrary movement of points on a horizontal surface (Figure4), rectangle. The adaptation of this formula to an arbitrary movement of points on a horizontal surface (Figure 4), requires taking into account various singularities depending on the position of the rectangle’s base, the situations of symmetries and what a unique “form factor algebra” permits.

Mathematics 2020, 8, 2176 7 of 25 requires taking into account various singularities depending on the position of the rectangle’s base, theMathematics situations 2020, of 8, x symmetries FOR PEER REVIEW and what a unique “form factor algebra” permits. 7 of 29

Figure 4. Outline of a ‘typical’ rectangle to be used for the calculations. Figure 4. Outline of a ‘typical’ rectangle to be used for the calculations.

For example, in the Y axis and at level –H1, the adapted solution would give: For example, in the Y axis and at level –H1, the adapted solution would give:       D  M  D  M   D 1 M  D 1 M  n =n=LL tantan−  − tantan−    (28)(28)  qH +D  qH +D qH +D qH +D   2 2  2 2  − 2 2  2 2  H1 + D H1 + D H0 + D H0 + D

3.3. Methodology

3.1.3.1. Solid Angle Projection Law ToTo obtainobtain thethe energyenergy exchangeexchange betweenbetween aa givengiven surfacesurface and a point P,P, wewe wouldwould simplysimply tracetrace aa conecone whosewhose vertexvertex isis thethe point considered P,P, andand itsits basebase isis thethe surfacesurface inin questionquestion (d(dΩΩ),), andand proceedproceed toto intersectintersect the the said said cone cone with with a sphere a sphere of unit of radius unit (Figureradius5 ).(Figure The enclosure 5). The within enclosure this intersection,within this projectedintersection, on projected any reference on any plane reference that weplane may that require we may and require divided and bydivided the horizontal by the horizontal area of area the aforementionedof the aforementioned unit sphere unit sphere (that is, (that divided is, divided by π), willby π equate), will equate to the valueto the ofvalue the configurationof the configuration factor determinedfactor determined by virtue by ofvirtue the analytical of the analytical methods methods of exact of integration exact integration described described in the previous in the previous chapters. chapters.If, by means of what we have discussed in detail in the above sections, we were trying to find certain geometric factors or proportions, it seems equally reasonable to achieve these same values through graphic procedures such as the ones employed in Descriptive Geometry. The advantages are obvious to the project engineer, because if, due to the difficulty of the mathematical problem, hesitation appears, we are allowed to somehow ‘visualize’ the entire process. In summary, what has been proven is that the configuration factor is a mere proportion, a dimensionless number, consisting of a ratio of areas, one of them the area of a certain projection and the other the total area involved expressed in circular terms, usually π [6]. The most important consequence of the former is that the problem, regardless of its complexity, will always present a unique non-trivial solution since the projection will have a univocal value and cannot be non-existent.

Mathematics 2020, 8, 2176 8 of 25 Mathematics 2020, 8, x FOR PEER REVIEW 8 of 29

σ FigureFigure 5.5. The cone of radiationradiation ofof thisthis figurefigure cutscuts thethe areaarea σ0’ onon thethe unitunitsphere. sphere. TheThe orthogonalorthogonal σ σ π 2 projectionprojection ofof σ0’ overover thethe irradiatedirradiatedplane plane produces produces the the area area σ00’’.. SuchSuch area area divided divided by by πrr2(r(r= = 1) equates toto thethe configurationconfiguration factor.factor. In addition, as the configuration factors are actually projections, they hold an additive property, If, by means of what we have discussed in detail in the above sections, we were trying to find which is useful to know when dealing with sundry and simultaneous emitting surfaces, because it certain geometric factors or proportions, it seems equally reasonable to achieve these same values is possible to add their effects. The average over the receiving surface of the configuration factors through graphic procedures such as the ones employed in Descriptive Geometry. thus obtained will constitute the form factor that had been so laboriously sought for by means of The advantages are obvious to the project engineer, because if, due to the difficulty of the mathematical formulation and solving of fourfold integral equations. mathematical problem, hesitation appears, we are allowed to somehow ‘visualize’ the entire process. Alternatively, graphic methods and analytical methods or a combination thereof can be used. In summary, what has been proven is that the configuration factor is a mere proportion, a This is the gist of our procedure. dimensionless number, consisting of a ratio of areas, one of them the area of a certain projection and It also seems gratifying to assume that we have solved the fundamental problem of radiation with the other the total area involved expressed in circular terms, usually π. [6] geometric methods. Why is that? Because it means that ‘form’ is very important in this type of problem The most important consequence of the former is that the problem, regardless of its complexity, and that not all forms act in the same way from a radiative point of view; some geometries favor will always present a unique non-trivial solution since the projection will have a univocal value and exchanges more than others and the mission of the heat transfer analyst is to adequate these forms. cannot be non-existent. predicting their impact. Such repercussion can also be verified by observers, that is, manufacturers, In addition, as the configuration factors are actually projections, they hold an additive property, society, etc., using, as we shall see, a simple tool. which is useful to know when dealing with sundry and simultaneous emitting surfaces, because it is However, the sole hindrance that we identified at the time was that numerical validation of the possible to add their effects. The average over the receiving surface of the configuration factors thus simple rectangle over a horizontal plane (Figure4) was not readily available, thus casting a shadow obtained will constitute the form factor that had been so laboriously sought for by means of over the viability of our approach. Fortunately, Cabeza-Lainez was able to work out an exact solution mathematical formulation and solving of fourfold integral equations. and this relevant finding is presented in AppendixB. Alternatively, graphic methods and analytical methods or a combination thereof can be used. Then a similar problem arose with circular emitters in a free position. Cabeza-Lainez has completely This is the gist of our procedure. solved this anew [7], as described in AppendixC, as it may help to validate other curved geometries. It also seems gratifying to assume that we have solved the fundamental problem of radiation 3.2.with Algorithms geometric Aided methods. Design Why is that? Because it means that ‘form’ is very important in this type of problem and that not all forms act in the same way from a radiative point of view; some geometries Although today, most Computer Aided Design (CAD) programs incorporate what is known favor exchanges more than others and the mission of the heat transfer analyst is to adequate these as parametric design (which implies that it is not necessary to redraw the objects that we design if forms. predicting their impact. Such repercussion can also be verified by observers, that is, manufacturers, society, etc., using, as we shall see, a simple tool.

Mathematics 2020, 8, x FOR PEER REVIEW 9 of 29

However, the sole hindrance that we identified at the time was that numerical validation of the simple rectangle over a horizontal plane (Figure 4) was not readily available, thus casting a shadow over the viability of our approach. Fortunately, Cabeza-Lainez was able to work out an exact solution and this relevant finding is presented in Appendix B. Then a similar problem arose with circular emitters in a free position. Cabeza-Lainez has completely solved this anew [7], as described in Appendix C, as it may help to validate other curved geometries.

3.2. Algorithms Aided Design Mathematics 2020, 8, 2176 9 of 25 Although today, most Computer Aided Design (CAD) programs incorporate what is known as parametric design (which implies that it is not necessary to redraw the objects that we design if they theychange change shape, shape, position position or size), or size), we weshould should not not consider consider all all of of these these as as Algorithm Algorithm Aided Aided Design programs (AAD). An algorithm is a logical expression that leads to procedures to be perfor performedmed through finite finite sets of closed basic instructions. Algorithms Algorithms simulate simulate human human abilities abilities to to break break down a problem into a series of simple steps that can be easily executed executed and, although although they they are are closely closely related related to to computers, computers, algorithms can be defineddefined independently of programmingprogramming languageslanguages andand applications.applications. However, not just any procedure can be consideredconsidered properly as an algorithm, since many procedures areare far far from from being being well well defined defined or containor contain ambiguities, ambiguities, logical logical contradictions contradictions or infinite or infinite loops. loops.Some important properties that are required for a procedure to be considered an algorithm are: AnSome algorithm important is aproperties well-defined that set are of required properly for concise a procedure instructions. to be considered an algorithm are: • • An algorithm is demands a well-defined a defined set setof properly of inputs concise that may instructions. or may not come from the output of a • • previousAn algorithm algorithm. demands a defined set of inputs that may or may not come from the output of a Anprevious algorithm algorithm. generates a precise output (Figure6): • • An algorithm generates a precise output (Figure 6):

FigureFigure 6.6. Algorithm programmed inin GrasshopperGrasshopper®® toto obtain obtain all all the the straight straight lines. lines.

• Finally, anan algorithm can produce warnings and error messages through through the the appropriate appropriate editor. editor. • If thethe inputsinputs are are not not appropriate, appropriate, e.g., e. ifg. we if we enter enter text text instead instead of numbers, of numbers, the algorithm the algorithm will return will anreturn error an message error message instead instead of the expected of the expect output,ed output, via the via appropriate the appropriate editor [ 8editor]. [8].

In addition,addition, throughthrough parameterization, parameterization, we we have have not no merelyt merely obtained obtained a straight a straight line, line, (Figure (Figure6) but, 6) bybut, simply by simply modifying modifying the value the value of the of coordinates, the coordi movingnates, moving the mouse the overmouse the over sliders the we sliders can alterwe can the ‘line’alter the output ‘line’ and output draw and all thedraw straight all the lines straight contained lines contained in the Euclidean in the Euclid spaceean with space a single with gesture a single of thegesture tracer. of the tracer.

3.3. Calculation by Algorithms Aided Design through Finite Element Method

The main problem of analytical calculation is that, although we may employ a convenient algebra of form factors [6] to find the radiance received by a surface, issuing from basic figures limited by straight lines, the number of shapes of the emitters, whose direct integration is very difficult or impossible to perform, remains numberless. To solve such an impasse, we have successfully developed a Grasshopper® algorithm that allows us to calculate the configuration factor subtended by an arbitrary point of any surface (i.e., irrespective of its shape and orientation) from an emitter or ‘three-dimensional figure’, with manifold geometry or inclination (including those whose integration is deemed impossible). Mathematics 2020, 8, x FOR PEER REVIEW 10 of 29

3.3. Calculation by Algorithms Aided Design through Finite Element Method The main problem of analytical calculation is that, although we may employ a convenient algebra of form factors [6] to find the radiance received by a surface, issuing from basic figures limited by straight lines, the number of shapes of the emitters, whose direct integration is very difficult or impossible to perform, remains numberless. To solve such an impasse, we have successfully developed a Grasshopper® algorithm that allows us to calculate the configuration factor subtended by an arbitrary point of any surface (i.e. irrespective of its shape and orientation) from an emitter or ‘three-dimensional figure’, with manifold geometry Mathematics 2020, 8, 2176 10 of 25 or inclination (including those whose integration is deemed impossible). To calculate the incident radiation n (W/m2), we will only have to add up all the configuration factorsTo that calculate we may the incidentrequire radiationand multiply n (W them/m2), weby the will luminance only have tothat add departs up all thefrom configuration the emitter (W/sr·mfactors that2). we may require and multiply them by the luminance that departs from the emitter (W/srThem2 ).steps which we have followed to develop the algorithm can be summarized below: · The steps which we have followed to develop the algorithm can be summarized below: 3.3.1. Geometric Definition of Both the Irradiated Surface and the Emitter and, Subsequently, Division3.3.1. Geometric of the Irradiated Definition Surface of Both into the a Irradiated Grid of N Surface × (N − 1) and Squares, the Emitter Locating and, Its Subsequently, Corners Division of the Irradiated Surface into a Grid of N (N 1) Squares, Locating Its Corners Let us consider for example, a rectangle 2 m× wide− by 2 m high whose base is 5 m above the ground.Let usThe consider rectangle’s for example, vertical aaxis rectangle coincides 2 m widewith bythe 2 mmiddle high whose point baseof the is 5horizontal m above the receiving ground. Thesurface rectangle’s of 10 × 10 vertical m. axis coincides with the middle point of the horizontal receiving surface of 10 10 m. × First we establish the plane of the irradiated surface defining, by means of sliders, the componentsFirst we of establish its normal the plane vector of and the irradiatedusing the surface‘Vector defining, XYZ’ function, by means constr of sliders,ucting the the components said plane withof its normalthe help vector of the and ‘Plane using Normal’ the ‘Vector function XYZ’ as function, shown constructingin the upper the part said of plane the following with the helpfigure of (Figurethe ‘Plane 7). Normal’ function as shown in the upper part of the following figure (Figure7).

Figure 7. First part of the algorithm.

Subsequently, wewe divide divide the the irradiated irradiated surface surface into into N N (N× (N 1)− 1) squares squares (ﬁnite (finite elements) elements) using using the × − ‘Square’the ‘Square’ function function that that generates generates a bi-dimensional a bi-dimensional grid grid with with square square cells cells and and retains retains the pointsthe points at the at thecorners corners of the of squares.the squares. Note that (in this case) the first ﬁrst row of corner points has been cancelled, since in the plane of the ‘rectangle’,‘rectangle’, thethe radianceradiance willwill necessarilynecessarily be equal to zero and the size of the squares is divided by its number for a better performance of the algorithm (Figures7 7 and and8 ):8): The result is detailed in FigureFig.8: 8:

3.3.2. Calculate the Conﬁguration Factors To perform this, it is convenient to select a ‘rectangle’ (arbitrary), draw the spheres centered in each corner of the divisions (through the ‘Sphere’ command) and extrude ‘light cones’ based on the ‘rectangle’ and vertex positioned at the corners of the divisions (‘Extrude point’ command).

In the ensuing operation, we have to draw the intersections between the ‘light cones’ and the spheres, (Boundary representation vs. Boundary Representation: ‘Brep|Brep’ command), project these intersections on the selected plane (‘Project’ command) and calculate the areas involved (‘Area’ command). Finally, we normalize the results to obtain dimensionless conﬁguration factors (Figure9): Mathematics 2020, 8, x FOR PEER REVIEW 11 of 29

Mathematics 2020, 8, 2176 11 of 25

Mathematics 2020, 8, x FOR PEER REVIEW 11 of 29

Figure 8. An example of a rectangle 2 m wide by 2 m high, and whose base lies 5 m above the ground. (We assume for convenience that the horizontal surface is a square of 10 × 10 m).

3.3.2. Calculate the Configuration Factors To perform this, it is convenient to select a ‘rectangle’ (arbitrary), draw the spheres centered in each corner of the divisions (through the ‘Sphere’ command) and extrude ‘light cones’ based on the ‘rectangle’ and vertex positioned at the corners of the divisions (‘Extrude point’ command). In the ensuing operation, we have to draw the intersections between the ‘light cones’ and the spheres, (Boundary representation vs. Boundary Representation: ‘BrepǀBrep’ command), project these intersections on the selected plane (‘Project’ command) and calculate the areas involved (‘Area’ Figurecommand). 8. AnFigure example Finally, 8. An ofweexample a normalize rectangle of a rectangle 2the m results2 wide m wide by toby 2 obtain2 m high, high, dimensionless and and whose whose base lies baseconfiguration 5 m liesabove 5 the m ground. abovefactors the (Figure ground. (We9): assume for(We convenience assume for convenience that the that horizontal the horizontal surface surface is is a asquare square of 10 of × 10 10 m). 10 m). × 3.3.2. Calculate the Configuration Factors To perform this, it is convenient to select a ‘rectangle’ (arbitrary), draw the spheres centered in each corner of the divisions (through the ‘Sphere’ command) and extrude ‘light cones’ based on the ‘rectangle’ and vertex positioned at the corners of the divisions (‘Extrude point’ command). In the ensuing operation, we have to draw the intersections between the ‘light cones’ and the spheres, (Boundary representation vs. Boundary Representation: ‘BrepǀBrep’ command), project these intersections on the selected plane (‘Project’ command) and calculate the areas involved (‘Area’ command). Finally, we normalize the results to obtain dimensionless configuration factors (Figure 9): FigureFigure 9. Algorithm 9. Algorithm to draw to draw N N ×(N (N − 1) spheres spheres in inthe the corner corner of the of subdivisions. the subdivisions. × − The result, as expected, is depicted below (Figure 10): TheMathematics result, as2020 expected,, 8, x FOR PEER is REVIEW depicted below (Figure 10): 12 of 29

Figure 9. Algorithm to draw N × (N − 1) spheres in the corner of the subdivisions.

The result, as expected, is depicted below (Figure 10):

Figure 10. Visualization of the intersections between the ‘radiant cones’ and the unit spheres and their Figure 10. Visualization of the intersections between the ‘radiant cones’ and the unit spheres and their correspondingcorresponding projections projections on the on the chosen chosen plane. plane.

3.3.3. Exporting3.3.3. Exporting the Data the Data to Excel to Excel and and Representing Representing them by by Color Color Maps Maps ® To transferTo transfer the data the data to Excel to Excel,® we, we ﬁrstfirst sort sort them them using using the ‘Partition the ‘Partition List’ command List’ command and then the and then the ‘GhExcel’‘GhExcel’ plugin plugin that that allows allows usus to export export an an ar arrayray of ofdata data to the to thesaid said spreadsheet spreadsheet through through the the ‘ExcelWrite’‘ExcelWrite’ command, command, (Figure (Figure 11). 11).

Figure 11. Algorithm to sort and export the data to Excel®.

Finally, we have designed a ‘renderer’ which assigns to each value obtained a color of the spectrum to display the results, (Figure 12):

Figure 12. Algorithm designed to graph the results.

Mathematics 2020, 8, x FOR PEER REVIEW 12 of 29

Figure 10. Visualization of the intersections between the ‘radiant cones’ and the unit spheres and their corresponding projections on the chosen plane. Figure 10. Visualization of the intersections between the ‘radiant cones’ and the unit spheres and their corresponding projections on the chosen plane. 3.3.3. Exporting the Data to Excel and Representing them by Color Maps

3.3.3.To Exporting transfer the dataData to to Excel Excel® ,and we firstRepresenting sort them them using by the Color ‘Partition Maps List’ command and then the Mathematics 2020, 8, 2176 12 of 25 ‘GhExcel’ plugin that allows us to export an array of data to the said spreadsheet through the To transfer the data to Excel®, we first sort them using the ‘Partition List’ command and then the ‘ExcelWrite’ command, (Figure 11). ‘GhExcel’ plugin that allows us to export an array of data to the said spreadsheet through the ‘ExcelWrite’ command, (Figure 11).

Figure 11. Algorithm to sort and export the data to Excel®. Figure 11. Algorithm to sort and export the data to Excel®. Figure 11. Algorithm to sort and export the data to Excel®. Finally, we have designed a ‘renderer’ which assigns to each value obtained a color of the Finally,spectrum we have to display designed the results, a ‘renderer’ (Figure 12): which assigns to each value obtained a color of the spectrum to display theFinally, results, we (Figure have designed 12): a ‘renderer’ which assigns to each value obtained a color of the spectrum to display the results, (Figure 12):

Figure 12. Algorithm designed to graph the results. FigureFigure 12. Algorithm12. Algorithm designed designed to to graph graph the theresults. results. Mathematics 2020, 8, x FOR PEER REVIEW 13 of 29

The outcomeThe outcome for such for ansuch algorithm an algorithm is is shown shown below (Figure (Figure 13). 13 ).

Figure 13.FigureColor 13. Color map map of theof the conﬁguration configuration factor factor obtain obtaineded by the bydiscretization the discretization for each meter for (110 each meter (110 values)values) of a of horizontal a horizontal areaarea of of 10 10 × 10 m102, illuminated m2, illuminated by a ‘rectangle’ by a ‘rectangle’ of 2 × 2 m2, 5 m of above 2 2 the m ground2, 5 m above the × × ground andand whose whose vertical vertical axis axis coin coincidescides with the with middle the point middle of the point horizontal of the surface. horizontal surface.

3.3.4. Numerical3.3.4. Numerical Results Results ® ® When exporting the values obtained by the Grasshopper algorithm® to an Excel sheet, it® is Whenrelatively exporting easy to the sort values them in obtained a matrix and by the the figures Grasshopper obtained are displayedalgorithm in Table to an1. Excel sheet, it is relatively easyIn to this sort manner, them the in form a matrix factor andF12 is theobtained figures as the obtained average of are all displayedthe configuration in Table factors1. f12, In thiswhich manner, yields thea value form (for factora discretization F12 is obtained of 11 × 11 = as110 the points) average of F12 = of 0.00556888. all the configuration factors f12, If we consider the point with the highest configuration factor (point 5,3), where f12 = 0.01144, and which yields a value (for a discretization of 11 11 = 110 points) of F12 = 0.00556888. we want to know the radiation of that point, as×suming a radiant power in the rectangle of 1000 W/m2, If we consider the point with the highest configuration factor (point 5,3), where f = 0.01144, we will obtain this by simply multiplying the value of f12 by 1000, that is, n = 11.44 W/m2. 12 and we wantThe to array know presented the radiation in Table 1 ofcoincides that point, with that assuming found by the a analytical radiant procedures power in (Table the rectangle2) of 2 2 1000 W/mdescribed, we will in obtainthe references this by [6]. simply In this way multiplying the procedure the is valuevalidated. of f12 by 1000, that is, n = 11.44 W/m . The array presented in Table1 coincides with that found by the analytical procedures (Table2) described in the references [6]. In this way the procedure is validated.

Mathematics 2020, 8, 2176 13 of 25

Table 1. Conﬁguration factors of an area of 10 10 m2 illuminated by a ‘rectangle’ of 2 2 m2 located 5 m from the ground at the middle point of the horizontal surface. × × Distance in the X-Axis to the Rectangle’s Plane (m) 0 1 2 3 4 5 6 7 8 9 10

Dimensionless Conﬁguration Factor f12 (i.e., Divided by π) 0 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00203 0.00279 0.00371 0.00469 0.00546 0.00577 0.00546 0.00469 0.00371 0.00279 0.00203 2 0.00368 0.00497 0.00650 0.00807 0.00930 0.00977 0.00930 0.00807 0.00650 0.00497 0.00368 Distance in the Y-axis to the 3 0.00473 0.00624 0.00796 0.00966 0.01096 0.01144 0.01096 0.00966 0.00796 0.00624 0.00473 Rectangles Plane (m) 4 0.00519 0.00665 0.00825 0.00977 0.01090 0.0132 0.01090 0.00977 0.00825 0.00665 0.00519 5 0.00517 0.00644 0.00778 0.00901 0.00989 0.01021 0.00989 0.00901 0.00778 0.00644 0.00517 6 0.00486 0.00590 0.00695 0.00788 0.00852 0.00876 0.00852 0.00788 0.00695 0.00590 0.00486 7 0.00440 0.00521 0.00600 0.00668 0.00715 0.00732 0.00715 0.00668 0.00600 0.00521 0.00440 8 0.00389 0.00451 0.00510 0.00559 0.00592 0.00604 0.00592 0.00559 0.00510 0.00451 0.00389 9 0.00339 0.00385 0.00429 0.00464 0.00488 0.00496 0.00488 0.00464 0.00429 0.00385 0.00339 10 0.00293 0.00328 0.00360 0.00386 0.00402 0.00408 0.00402 0.00386 0.00360 0.00328 0.00293

Table 2. Conﬁguration factors of an area of 5 10 m2 (the values that are omitted are obviously symmetrical) illuminated by a ‘rectangle’ of 2 2 m2. Values obtained × × by analytical methods ([6], pp. 183–184).

Distance in the X-Axis to the Rectangle’s Plane (m) 5 6 7 8 9 10

Dimensionless Conﬁguration Factor f12 0 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 1 0.005765 0.00546298 0.00468552 0.00371248 0.00278731 0.00202982 2 0.00976818 0.00929669 0.00806846 0.00649857 0.00496683 0.00367902 3 0.01144492 0.01095576 0.00966096 0.00795806 0.00623622 0.00473341 Distance in the Y-axis to the Rectangle-s Plane (m) 4 0.01131701 0.0108993 0.00977478 0.00824998 0.00664699 0.00518805 5 0.01020997 0.00988871 0.00901007 0.00778368 0.00644494 0.0051747 6 0.00875831 0.00852443 0.00787588 0.0069471 0.00589812 0.00486368 7 0.0073163 0.00715037 0.0066848 0.00600327 0.0052104 0.00440123 8 0.00603659 0.00591986 0.00558912 0.00509595 0.00450753 0.00388887 9 0.00496121 0.00487895 0.004644 0.00428824 0.00385462 0.00338694 10 0.00408173 0.00402333 0.00385542 0.00359789 0.00327834 0.00292612 Mathematics 2020, 8, x FOR PEER REVIEW 15 of 29

Table 2. Configuration factors of an area of 5 × 10 m2 (the values that are omitted are obviously symmetrical) illuminated by a ‘rectangle’ of 2 × 2 m2. Values obtained by analytical methods ([6], pp. 183–184).

Distance in the X-axis to the Rectangle’s Plane (m) 5 6 7 8 9 10 Dimensionless Configuration Factor f12 0 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 1 0.005765 0.00546298 0.00468552 0.00371248 0.00278731 0.00202982

-axis to the the to -axis 2 0.00976818 0.00929669 0.00806846 0.00649857 0.00496683 0.00367902 Y 3 0.01144492 0.01095576 0.00966096 0.00795806 0.00623622 0.00473341 4 0.01131701 0.0108993 0.00977478 0.00824998 0.00664699 0.00518805 5 0.01020997 0.00988871 0.00901007 0.00778368 0.00644494 0.0051747 6 0.00875831 0.00852443 0.00787588 0.0069471 0.00589812 0.00486368

Rectangle-s Plane (m) (m) Plane Rectangle-s 7 0.0073163 0.00715037 0.0066848 0.00600327 0.0052104 0.00440123

Distance in the the Distance in 8 0.00603659 0.00591986 0.00558912 0.00509595 0.00450753 0.00388887 9 0.00496121 0.00487895 0.004644 0.00428824 0.00385462 0.00338694

Mathematics 2020, 108, 21760.00408173 0.00402333 0.00385542 0.00359789 0.00327834 0.0029261214 of 25

4. Discussion 4. Discussion 4.1. Convergence 4.1. Convergence To test the convergence of the method we need to gradually decrease the size of the elements and showTo test how, the convergence by doing so, of thethe methodform factors we need converge to gradually towards decrease a certain the size value of thewhich elements must and be increasinglyshow how, by closer doing to so, that the calculated form factors analytically. converge towards a certain value which must be increasingly closerOne to thatof the calculated advantages analytically. of our algorithm is that by simply modifying the value of the sliders to increaseOne the of thepartitions advantages we ofobtain our algorithmthe new results is that byrapidly, simply and modifying proof of the convergence value of the is sliders readily to available.increase the partitions we obtain the new results rapidly, and proof of convergence is readily available. In thethe ensuingensuing ﬁgure figure (Figure (Figure 14 14),), we we found found the the results results by raisingby raising the numberthe number of partitions of partitions or ﬁnite or elements from 10 10 = 100 elements to 100 100 = 10,000 elements, with a computing time of only finite elements from× 10 × 10 = 100 elements to× 100 × 100 = 10,000 elements, with a computing time of 136only s 136 (conventional s (conventional computer). computer).

Figure 14. 14. TheThe color color map map of the of theconfiguration conﬁguration factor factor obtained obtained by discretization by discretization for each for 10 each cm (10201 10 cm 2 2 (10201values) values) over an over area an of area10 × of10 10m2 radiated10 m radiated by a ‘rectangle’ by a ‘rectangle’ of 2 × 2 ofm2 2, 5 m2 mover, 5 the m over horizontal the horizontal surface Mathematics 2020, 8, x FOR PEER REVIEW × × 16 of 29 atsurface its middle at its axis. middle axis.

TheThe following following figure ﬁgure (Figure (Figure 15)15) shows shows the the values obtained obtained for for the the form form factor by discretizing thethe surface surface of of 10 10 × 1010 m min in121, 121, 441, 441, 961, 961, 1681, 1681, 2601 2601,, 3721, 3721, 5041, 5041, 6561, 6561, 8281, 8281, 10201, 10,201, 12321, 12,321, 14641, 14,641,17161, 19,881 and 22,801× points, respectively (the value obtained for 22,801 points is F12 = 0.006165497), 17,161, 19,881 and 22,801 points, respectively (the value obtained for 22,801 points is F12 = 0.006165497), showing a clear convergence towards the value F12 = 0.0062 analytically calculated [6]. showing a clear convergence towards the value F12 = 0.0062 analytically calculated [6].

0.007 0.006 0.005 0.004 0.003

Form factor 0.002 0.001 0 0 5000 10000 15000 20000 25000 Number of points

Figure 15. Convergence of the algorithm for the previously studied case. Figure 15. Convergence of the algorithm for the previously studied case.

4.2. Versatility of the Method As noted above, the difficulties in obtaining analytical solutions for emitters with forms other than the trivial are enormous, and even in many cases impossible. However, the method presented here allows solutions with high accuracy, limited only by the performance of the computer, for any type of emitters and irrespective of the orientation. As a more complex example, we have calculated the configuration factors on a 10 × 10 m horizontal surface, radiated by an elliptical emitter whose major and minor axes are 2 m and 1 m respectively and whose lowest point is located at a height of 5 m. As before, vertical axis of the ellipse coincides with the middle point of the horizontal surface in 5 m (Figure 16). Notice that, for the time being, the elliptical shape cannot be solved analytically due the appearance of elliptic integrals in the process [9].

Figure 16. The color map of the configuration factor obtained by the discretization per each 20 cm (2601 values) of an area of 10 × 10 m2 illuminated by an Ellipse of 2 m of major semi-axis and 1 m of minor semi-axis located 5 m from the ground in the middle point of the side of the horizontal surface.

Mathematics 2020, 8, x FOR PEER REVIEW 16 of 29

The following figure (Figure 15) shows the values obtained for the form factor by discretizing the surface of 10 × 10 m in 121, 441, 961, 1681, 2601, 3721, 5041, 6561, 8281, 10201, 12321, 14641, 17161, 19,881 and 22,801 points, respectively (the value obtained for 22,801 points is F12 = 0.006165497), showing a clear convergence towards the value F12 = 0.0062 analytically calculated [6].

0.007 0.006 0.005 0.004 0.003

Form factor 0.002 0.001 0 Mathematics 2020, 8, 2176 0 5000 10000 15000 20000 25000 15 of 25 Number of points

4.2. Versatility of the MethodFigure 15. Convergence of the algorithm for the previously studied case.

As noted4.2. Versatility above, of thethe Method diﬃculties in obtaining analytical solutions for emitters with forms other than the trivialAs arenoted enormous, above, the difficulties and even in in obtaining many casesanalytical impossible. solutions for However, emitters with the forms method other presented here allowsthan solutions the trivial are with enormous, high accuracy, and even in limited many cases only impossible. by the performance However, the method of the presented computer, for any type of emittershere allows and solutions irrespective with high of accuracy, the orientation. limited only by the performance of the computer, for any As atype more of complexemitters and example, irrespective we of have the orientation. calculated the conﬁguration factors on a 10 10 m horizontal As a more complex example, we have calculated the configuration factors on a 10 ×× 10 m surface, radiatedhorizontal by surface, an elliptical radiated emitterby an elliptical whose emitter major whose and minormajor and axes minor are 2axes m andare 2 1m m and respectively 1 m and whose lowestrespectively point and is located whose lowest at a height point is of located 5 m. at As a height before, of 5 vertical m. As before, axis vertical of the ellipseaxis of the coincides ellipse with the middle pointcoincides of the with horizontal the middle surfacepoint of the in horizontal 5 m (Figure surface 16 ).in Notice5 m (Figure that, 16). for Notice the that, time for being, the time the elliptical shape cannotbeing, be the solved elliptical analytically shape cannot be due solved the analytically appearance due of the elliptic appearance integrals of elliptic in integrals the process in the [9]. process [9].

Figure 16.FigureThe 16. color The mapcolor map of the of the conﬁguration configuration factor factor ob obtainedtained by the by discretization the discretization per each 20 per cm each 20 cm (2601 values) of an area of 10 × 10 m2 illuminated by an Ellipse of 2 m of major semi-axis and 1 m of (2601 values) of an area of 10 10 m2 illuminated by an Ellipse of 2 m of major semi-axis and 1 m of minor semi-axis located 5× m from the ground in the middle point of the side of the horizontal surface. minor semi-axis located 5 m from the ground in the middle point of the side of the horizontal surface.

Mathematics 2020, 8, x FOR PEER REVIEW 17 of 29 Finally, we have simulated the conﬁguration factors on a 10 10 m horizontal surface, given by an × arbitrary emitterFinally, in the we shapehave simulated of a scalene the configuration triangle of factors the dimensions on a 10 × 10 m depicted horizontal in surface, Figure given 17. by The graphic results appearan arbitrary in Figure emitter 18 inand the theshape numeric of a scalene values triangle in Tableof the 3dimensions. depicted in Figure 17. The graphic results appear in Fig.18 and the numeric values in Table 3.

Figure 17. Sketch of the dimensions of a vertical scalene triangle radiating over a horizontal area of Figure 17. Sketch of the dimensions of a vertical scalene triangle radiating over a horizontal area of 10 × 10 m2. 10 10 m2. ×

Figure 18. Map of the configuration factors obtained by the discretization per each 20 cm (2601 values) of an area of 10 × 10 m2 illuminated by an arbitrary scalene triangle with dimensions expressed in Fig. 17.

Mathematics 2020, 8, 2176 16 of 25

Table 3. Conﬁguration factors of an area of 10 10 m2 illuminated by a ‘scalene triangle’. × Distance in the X-Axis to the Triangle’s Plane (m) 0 1 2 3 4 5 6 7 8 9 10

Dimensionless Conﬁguration Factor f12 (i.e., Divided by π) 0 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00329 0.00593 0.01103 0.02039 0.03471 0.04974 0.05761 0.05139 0.03030 0.01256 0.005371 Distance in the Y-axis to the 2 0.00554 0.00942 0.01614 0.02677 0.04031 0.05151 0.05363 0.04433 0.02858 0.01539 0.00799 Triangles Plane (m) 3 0.00642 0.01014 0.01581 0.02349 0.03169 0.03715 0.03682 0.03064 0.02171 0.01371 0.00825 4 0.00627 0.00921 0.01318 0.01789 0.02229 0.02476 0.02415 0.02067 0.01578 0.01106 0.00740 5 0.00559 0.00769 0.01027 0.01303 0.01535 0.01650 0.01604 0.01413 0.01141 0.00862 0.00623 6 0.00474 0.00619 0.00782 0.00943 0.01069 0.01126 0.01097 0.00989 0.00833 0.00664 0.00509 7 0.00392 0.00490 0.00593 0.00689 0.00761 0.00791 0.00773 0.00710 0.00618 0.00513 0.00411 8 0.00321 0.00387 0.00453 0.00513 0.00554 0.00572 0.00560 0.00552 0.00465 0.00399 0.00332 9 0.00262 0.00307 0.00351 0.00388 0.00414 0.00424 0.00416 0.00393 0.00357 0.00314 0.00268 10 0.00214 0.00246 0.00275 0.00299 0.00315 0.00322 0.00316 0.00302 0.00278 0.00249 0.00218 Notice that, in the case of the ellipse and the scalene triangle unlike the rectangle, it is not possible nowadays to calculate the conﬁguration factors by any other procedure. Mathematics 2020, 8, x FOR PEER REVIEW 17 of 29

Finally, we have simulated the configuration factors on a 10 × 10 m horizontal surface, given by an arbitrary emitter in the shape of a scalene triangle of the dimensions depicted in Figure 17. The graphic results appear in Fig.18 and the numeric values in Table 3.

MathematicsFigure2020 17., 8 Sketch, 2176 of the dimensions of a vertical scalene triangle radiating over a horizontal area of17 of 25 10 × 10 m2.

Figure 18. Map of the configuration factors obtained by the discretization per each 20 cm (2601 values) Figure 18. Map of the configuration factors obtained by the discretization per each 20 cm (2601 values) of an area of 10 10 m2 illuminated by an arbitrary scalene triangle with dimensions expressed in of an area of 10 ×× 10 m2 illuminated by an arbitrary scalene triangle with dimensions expressed in Fig. Figure 17. 17. 5. Conclusions and Future Aims Given the accurate convergence of the values obtained for the particular case of rectangular emitters and receivers within a cuboid, we consider that the graphical interface program that we have proposed is fully accomplished. However, for the time being, we have to warn that there is no feasible way to validate the other figures presented, such as ellipse and scalene triangle. Nevertheless, we believe that a complex physical-mathematical problem, which has occupied the mind of scholars for decades if not centuries, is solved with perfect ease in a semi-automatic manner. Ensuing research will extend the procedure to other types of three-dimensional spaces where analytical calculations were proposed but still remain underdeveloped, for instance curvilinear sources in non-cuboidal domains affected by interferences and obstacles [9]. A much wider repertoire of possible radiative exchanges would be achieved in this fashion, with relative simplicity and accuracy and without alternative. The repercussions of such findings would be wide-ranging in fields as diverse as aerospace technologies [10], Light Emitting Diodes (LED), radiotherapy medicine, risk prevention, acoustics and architecture [11]. On an industrial basis, we consider that the vast field of artificial lighting and design of luminaries will immediately benefit from our methods. Whereas the findings hereby exposed would represent a major achievement in Optics, precisely for the realms of Heat Transfer and Lighting, it is understood that our contribution has been timely aided by the enormous potential of Algorithms Aided Design in the numerical resolution of physical-mathematical problems that, when approached directly, are extremely difficult to accomplish (if not altogether impossible).

Author Contributions: Conceptualization, J.-M.C.-L.; data curation, F.S.-A.; software, F.S.-A.; investigation, J.-M.C.-L.; writing–original draft, F.S.-A. and J.-M.C.-L. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Mathematics 2020, 8, 2176 18 of 25

Acknowledgments: The authors would like to warmly thank Enrique Salguero-Jiménez for his unselfish collaboration in the preparation of most of the figures that appear in this work. Cabeza-Lainez wants to honor with the article the figure of the late Engineer Jose Ramon Perez de Lama. Our piece of writing has been composed in praise of Tomomi Odajima San. Conflicts of Interest: The authors declare none.

Nomenclature

Fij Form Factor (surface to surface, dimensionless) fij Conﬁguration factor (surface to point, dimensionless) 2 Ei,Mi Emitted Radiant Power (W/m ) 2 Ni Received Radiant Power (W/m ) 2 ni Unit Received Radiant Power (point from surface, W/m ) Φi Radiant ﬂux (W) L Emitted Luminance or radiance (W/sr m2) i · I Radiant intensity (W/sr) 2 Ai Surface area (m ) r distance (m) θi angle with the normal (radian) Ω solid angle (sr)

Appendix A. Further Details on the Calculations of Basic Conﬁguration Factors

Appendix A.1. Triangles and Shapes Composed of Triangles In the ﬁrst case, which is also the most elementary, we would substitute the value of the cosine as expressed in Equation (16) and ﬁnd the fundamental double-integral equation,

Z b Z z y2 E = L dzdx (A1) 2 0 0 (x2 + y2 + z2)

Note that we write again E (surface-to-point received power) instead of the previous magnitude n because this nomenclature is more usual. We have chosen an instant of time t in which the luminance or radiance L can be considered constant, keeping in mind that Lπ = M. We will solve it as an example of the general ensuing procedure to attest to its diﬃculty. The upper line of triangle A1 is deﬁned by equation z = x(a/b), and therefore, replacing the new value of z:

Z b Z a x 2 b y E = L dzdx (A2) 2 0 0 (x2 + y2 + z2)

Solving the integral of the second member of Equation (A2), (the details of the resolution of this and all other integrals presented in this article can be consulted in [6]) we obtain:     L  b a  E =  tan 1  (A3) 2  q − q   y2 + b2 y2 + b2 

Respectively, the upper triangle, surface A2 in Figure2, would have yielded:     L  a b  E =  tan 1  (A4) 2  q − q   y2 + b2 y2 + b2  Mathematics 2020, 8, 2176 19 of 25

Appendix A.2. Rectangle The full rectangle is the sum of Equations (A3) and (A4), that is:     L  a 1 b b 1 a  E =  p tan− p + q tan− q  (A5) 2  2 2 2 2   y + a y + a y2 + b2 y2 + b2 

We would ask the reader to understand that, with all this complexity, we have only obtained the value of radiative exchange at a single point located precisely under the vertex of the rectangle and in a particular direction at right angles to the surface source.

Appendix A.3. Square For example, in a square whose side is d = 2a = 2b; the ﬁnal expression is the result of multiplying by eight the basic triangle previously obtained, which yields:      d d  E = 4L tan 1  (A6)  q − q   4y2 + d2 4y2 + d2 

Appendix A.4. Equilateral Triangle If we contemplate the ﬁgure of the equilateral triangle (Figure A1), the expression will now be six times the rotated triangle, and we receive: Mathematics 2020, 8, x FOR PEER REVIEW 21 of 29      d 1 d √3  E = 3L q tan− q  (A7)  d d√3  E=3L 12y2 + d2 tan 12y2 + d2  (A7) 12y +d 12y +d

Figure A1. Conﬁguration factor between an equilateral triangle composed of six rotated triangles and Figure A2. Configuration factor between an equilateral triangle composed of six rotated triangles and a point in a parallel plane. a point in a parallel plane. Appendix A.5. Hexagon A.5. Hexagon In the speciﬁc case of the hexagon (Figure A2), we would need 12 times the rotated triangle, which, expressedIn the as specific a function case of the of sidethe d,hexagon gives us: (Figure A3), we would need 12 times the rotated triangle, which, expressed as a function of the side d, gives us:     d √3 1 d  E = 6L d 3 tan− d  (A8)  q √ q  E=6L 4y2 + 3d2 tan 4y2 + 3d2  (A8) 4y +3d 4y +3d

Figure A3. Configuration factor between a hexagon composed of 12 rotated triangles and a point under the center in a parallel plane.

A.6. Circle Finally, the limiting case of all simple figures composed of triangles is a regular n→∞ sided polygon (Figure A4) whose limit is obviously a circle and in it: L b a E= ·limn · tan (A9) 2 →;→ y +b y +b

Mathematics 2020, 8, x FOR PEER REVIEW 21 of 29

d d√3 E=3L tan (A7) 12y +d 12y +d

Figure A2. Configuration factor between an equilateral triangle composed of six rotated triangles and a point in a parallel plane.

A.5. Hexagon In the specific case of the hexagon (Figure A3), we would need 12 times the rotated triangle, which, expressed as a function of the side d, gives us: d√3 d E=6L tan (A8) Mathematics 2020, 8, 2176 4y +3d 4y +3d 20 of 25

Figure A2. Conﬁguration factor between a hexagon composed of 12 rotated triangles and a point under Figure A3. Configuration factor between a hexagon composed of 12 rotated triangles and a point the center in a parallel plane. under the center in a parallel plane. Appendix A.6. Circle A.6. CircleFinally, the limiting case of all simple ﬁgures composed of triangles is a regular n sided polygon (Figure A3) whose limit is obviously a circle and in it: →∞ Finally, the limiting case of all simple figures composed of triangles is a regular n→∞ sided    polygon (Figure A4) whose limit is obviously a circle and in it:     L   b 1 a  E =L lim n  q b tan− q a  (A9) 2 ·a 0;n  ·  E= ·lim→ →∞n · 2 + 2 tan 2 + 2  (A9) 2 →;→ yy +bb yy +bb  Mathematics 2020, 8, x FOR PEER REVIEW 22 of 29

Figure A3.A4. ConfigurationConﬁguration factor between a circle consisting of n rotated triangles and a point from the normal to the center that is locatedlocated inin aa planeplane parallelparallel toto thethe circle.circle.

Since a 0 we can substitute the angle (arc of tangent) for its tangent, ﬁnding: Since a→→0 we can substitute the angle (arc of tangent) for its tangent, finding:   L  baba  E=E = limlim nn  A10(A10) 2 n  2 2  ( ) 2 →→∞ yy ++bb but whenwhen n n→∞,, the the product product n a n·a is equivalent is equivalent to 2π bto or 2 2ππbr (theor 2 lengthπr (the of length the circumference) of the circumference) and, therefore, and, in a moretherefore, usual in formulation→ a ∞ more usual the amount formulation· of energy the as amount a function ofof energy radius as and a distancefunction is of found radius [7]: and distance is found [7]:  2  " 2 #  b  r E = Lπ  = Lπ (A11)  2 b+ 2  y2 +rr2 E=Lπb y =Lπ (A11) b + y y +r

A.7. Calculations in a Plane Perpendicular to the Figure Imploring the patience of the reader, we need to remark that in the previous paragraphs we have limited ourselves to establishing the component of radiation that was perpendicular to the emitting surface considered (the one that is received onto a parallel plane), at a particular point under the vertex or the center of the figure. Imagine if you can the complexity that arises when trying to find the solution for points moving freely in space, which is almost unfathomable. Even so, as is logical, radiance should be treated like a vector, thus, if we want to complete the problem we will have to also find the values of the two components that are missing and remain parallel to the area considered. The defining integral then turns out to be (if we multiply by the corresponding cosine): zy E=L dxdz (A12) (x + y +z) Applying this proposition to the first case considered (Figure A5) we would find that if Y (the distance) is a constant, the first part of the integral becomes not lengthy and the equation adopts the value: L b b √a +b E= tan − tan (A13) 2 y √a +b y

Mathematics 2020, 8, 2176 21 of 25

Appendix A.7. Calculations in a Plane Perpendicular to the Figure Imploring the patience of the reader, we need to remark that in the previous paragraphs we have limited ourselves to establishing the component of radiation that was perpendicular to the emitting surface considered (the one that is received onto a parallel plane), at a particular point under the vertex or the center of the figure. Imagine if you can the complexity that arises when trying to find the solution for points moving freely in space, which is almost unfathomable. Even so, as is logical, radiance should be treated like a vector, thus, if we want to complete the problem we will have to also find the values of the two components that are missing and remain parallel to the area considered. The defining integral then turns out to be (if we multiply by the corresponding cosine):

ZZ zy E = L dxdz (A12) (x2 + y2 + z2)

Applying this proposition to the first case considered we would find that if Y (the distance) is a constant, the first part of the integral becomes not lengthy and the equation adopts the value:  p   2 + 2  L  1 b b 1 a b  E = tan− p tan−  (A13) 2  y − a2 + b2 y 

For the entire rectangle, the boundaries would lie between a and 0, instead of using a function of x. Grouping the sum of the two terms, it turns out that:   L  1 b y 1 b  E = tan− p tan− p  (A14) 2  y − a2 + y2 a2 + y2 

To find the amount of the term due to the upper triangle in Figure3, we would need to change the integration limits (from z = a to z = ax/b) and consequently:  p   2 + 2 y  L  b 1 a b 1 b  E =  p tan− p tan− p  (A15) 2  a2 + b2 y − a2 + y2 a2 + y2  which could have been deduced by logic by subtracting Equation (A13) from Equation (A14). Adding and subtracting rectangles would allow us to obtain the result for a rectangular shape over an unlimited horizontal surface as described in Figure4 and Equation (28). Only a few months ago Cabeza-Lainez obtained a similar solution for a triangle with a horizontal side and also for circular sectors if the center of the vertical circle lies on the horizontal surface. [11] In this quite elaborate fashion, we have discovered how complex it becomes to find an analytical expression to solve the problem if we only alter the coordinate plane of calculation not to mention figures of geometry other than the regular triangle or rectangle. That is why we concluded that a graphical solution based on parametric design would imply a much more general solution and extraordinarily simplify the issue. This is the reasoning behind our method.

Appendix B. Particulars of the Numerical Validation of the Projected Solid-Angle Principle

Numerical Validation for a Rectangle Suppose that the emitter is a rectangle 1 m. high and 3 m. wide and we try to calculate the conﬁguration factor at a point P that lies in the plane of symmetry of the rectangle at a longitudinal distance of 2 m and is located 1 m below the bottom edge of the said rectangle (Figure A4). So, D = 2, H0 = 2, H1 = 1 and M = 1.5. With this data, Equation (28) yields the well-known result: " # 2 1 1.5 2 1 1.5 2 n = L tan− tan− = 0.18368L W/m √5 √5 − √8 √8

However, if we try to check this value by means of the projected solid-angle law, it becomes extremely diﬃcult, as we shall see. Mathematics 2020, 8, 2176 22 of 25 Mathematics 2020, 8, x FOR PEER REVIEW 24 of 29

Figure A4. Scheme of a ‘rectangle’ type to employ in calculations. Figure A6. Scheme of a ‘rectangle’ type to employ in calculations. Since f12 must be a dimensionless conﬁguration factor, we ought to divide by the area of the projected unit sphere, that is πr2 = π, from which we would arrive at: So, D = 2, H0 = 2, H1 = 1 and M = 1.5.

With this data, Equation (28) yields fthe12 = well-knownn/(Lπ) = 0.0584 result: (A16) 2 1.5 2 1.5 We can geometrically obtainn=L the lengthstan of the minor− semi-axestan of = the 0.18368L( projectionW of/m the) two ellipses involved which are, respectively, b1 = 0.8941 and√ b50 = 0.7071.√5 The√8 major semi-axis√8 is logically 1 m. Strangely, the only way to solve the problem is by changing the coordinates to (very rare) generalized polar coordinates,However, and thusif we we try would to check have: this value by means of the projected solid-angle law, it becomes extremely difficult, as we shall see. x = a ρ cos θ; y = b ρ cos θ, Since f12 must be a dimensionless configuration factor, we ought to divide by the area of the 2 thus,projected the di ﬀuniterential sphere, element that yields is πr dA = π=, froma b ρ dwhichρ dθ. we would arrive at: This is, in fact, the only possible procedure to obtain the area of an elliptic sector: f12 = n/(Lπ) = 0.0584 (A16)

Z θ Z Z θ " #1 Z θ We can geometrically1 1 obtain the lengths 1ofρ 2the minor semi-axes1 1 of the(θ projectionθ ) of the two A = a b ρ dρ dθ = a b dθ = a b dθ = a b 1 − 2 (A17) ellipses involved whichθ2 are,0 respectively, b1 =θ 0.89412 2 0 and b0 = 0.7071.θ2 2 The major 2semi-axis is logically 1 m. Now it will suffice to find the values of θ1 and θ2 under the new coordinate system. Since the angles are equalStrangely, with respect the to the only axis way of symmetry to solve of the the ‘rectangle’,problem is the by previous changing expression the coor candinates be transformed to (very into rare) the simplergeneralized product: polar a b coordinates,θi. and thus we would have: For the larger· ellipse· y/x = 2/1.5 (by proportion); a = 1, b = 0.8944. x = a ρ cos θ; y = b ρ cos θ, Then θ = tan 1(2/(1.5 0.8944)), which is 0.9799; and the complementary with respect to π/2 is 0.5908 1 − · pseudo-radiansthus, the differential (a new localelement unit yields only valid dA=a for ‘each’ b ρ ellipse). dρ dθ. The area of the larger ellipse is 0.5984 0.8944 = 0.5284 and accordingly for the small ellipse the area would be 0.3447;This therefore, is, in fact, the final the areaonly yields possible 0.5284 ·procedure0.3447 = to0.1837. obtain the area of an elliptic sector: − This quantity divided by π provides f = 0.0584.ρ 1 (θ − θ) (A18) A= a b ρ dρ dθ =a b12 dθ = a b dθ =a b (A17) 2 2 2 ‘Quod erat demonstrandum’. It is important to stress that the values obtained for the configuration factors through the application of the principleNow of theit will projected suffice solid to find angle the are values not approximations of θ1 and θ to2 under those obtainedthe new by coordinate analytical calculationssystem. Since but arethe identical.angles are Such equal a demonstration, with respect to simple the axis as it of is, symmetr due to they extremeof the ‘rectangle’, difficulties arisingthe previous with the expression integration can of ellipticbe transformed geometries, into is a the finding simpler solely product: attributed a·b· toθ Cabeza-Lainez.i. In this way he has dissipated the clouds that once spread over the validity of such a convenient principle, for our method and beyond. For the larger ellipse y/x = 2/1.5 (by proportion); a = 1, b = 0.8944. AppendixThen C.θ The=tan Recurrent2/(1.5 Problem ∙ 0.8944) of, which Circular is 0.9799; Emitters and the complementary with respect to π/2 is 0.5908 pseudo-radians (a new local unit only valid for ‘each’ ellipse). Circular emitters can be treated in the same way as any type of emitter by virtue of the law of the projected solid angleThe area (Figure of A5the). larger Nonetheless, ellipse explicit is 0.5984·0.8944 solutions were = 0.5284 unassailable and accordingly [7,9]. for the small ellipse the area would be 0.3447; therefore, the final area yields 0.5284 − 0.3447 = 0.1837. This quantity divided by π provides f12 = 0.0584. (A18). ’Quod erat demonstrandum’ .

Mathematics 2020, 8, x FOR PEER REVIEW 25 of 29

It is important to stress that the values obtained for the configuration factors through the application of the principle of the projected solid angle are not approximations to those obtained by analytical calculations but are identical. Such a demonstration, simple as it is, due to the extreme difficulties arising with the integration of elliptic geometries, is a finding solely attributed to Cabeza- Lainez. In this way he has dissipated the clouds that once spread over the validity of such a convenient principle, for our method and beyond.

Appendix C. The Recurrent Problem of Circular Emitters

MathematicsCircular2020 ,emitters8, 2176 can be treated in the same way as any type of emitter by virtue of the law23 of 25of the projected solid angle (Figure A7). Nonetheless, explicit solutions were unassailable. [7,9]

Figure A5. Projection on the horizontal plane of the intersection of an oblique cone and a unit Figure A7. Projection on the horizontal plane of the intersection of an oblique cone and a unit radius radius sphere. sphere. Cabeza-Lainez proposed that, through a so-called “pair of dejection” (Figure A6), we could determine the parametersCabeza-LainezMathematics of the cone 2020 proposed that, 8, x FOR subtends PEER that, REVIEW the through emitting circlea so-called of radius “pair R, at pointof dejection” P. (Figure 26A8), of 29 we could determine the parameters of the cone that subtends the emitting circle of radius R, at point P.

FigureFigure A6. Procedure A8. Procedure to to ﬁnd find the the mainmain angles angles of the of theradiant radiant cone. cone.

The resultingThe equationresulting equation of the elliptical of the elliptical cone is: cone is: x y z 2 + 2 = 2 A19 x y z ( ) sin+α cos αtan β =cos β (A19) 2 α cos2 α tan2 β cos2 β The intersection between sinthe oblique cone of parameters α and β and a sphere of unit radius defines itself by the expression: The intersection between the oblique cone of parameters α and β and a sphere of unit radius deﬁnes itself by the expression: x + y +z =1 2 2 2 2 2 2 x x y y z z (A20) x + y + z = 1 + + = = (A20) sinsinα 2 cosα αtancos2βα tancos2ββ cos2 β Thus, we arrive at the miraculous equation of the intersection between the former cone and a sphere of unit radius with its center in the vertex of the cone (Equation (A20)); this curve that we have found is called Tomomi by Cabeza-Lainez [10] and yields the following parametric equations for its coordinates (Figure A9), x = sin α cos t.

y=sinβsint (A21) z=cos α cos t + cos βsin t

Mathematics 2020, 8, 2176 24 of 25 Mathematics 2020, 8, x FOR PEER REVIEW 27 of 29

Thus, we arrive at the miraculous equation of the intersection between the former cone and a sphere of unit radius with its center in the vertex of the cone (Equation (A20)); this curve that we have found is called Tomomi by Cabeza-Lainez [10] and yields the following parametric equations for its coordinates (Figure A7), q x = sin α cos t.y = sin β sin tz = cos2 α cos2 t + cos2 β sin2 t (A21) Mathematics 2020, 8, x FOR PEER REVIEW 27 of 29

Figure A9. Axonometric view of Tomomi curve.

Looking for the planesFigure of common A7. Axonometric symmetry view between of Tomomi the sphere curve. and the elliptical cone, and applying elementary theorems Figureof intersections A9. Axonometric of viewquadrics, of Tomomi we curve. can observe the Tomomi curve of Cabeza-LainezLooking for as the a planesbranch of of common hyperbola symmetry or as an between ellipse the(Figure sphere A10). and the elliptical cone, and applying elementary theoremsLooking offor intersections the planes of of common quadrics, symmetry we can observe between the the Tomomi sphere curveand the of Cabeza-Lainezelliptical cone, and as a branch of hyperbolaapplying or as elementary an ellipse (Figuretheorems A8 of). intersections of quadrics, we can observe the Tomomi curve of Cabeza-Lainez as a branch of hyperbola or as an ellipse (Figure A10).

Figure A10. (Top). Projection of the intersection parallel to the axis of the cone: Hyperbola. (Bottom). Projection of the intersection normal to the axis of the cone: Ellipse. Figure A8. (Top). Projection of the intersection parallel to the axis of the cone: Hyperbola. (Bottom). ProjectionFigure A10. of ( theTop intersection). Projection normal of the intersection to the axis of paralle the cone:l to Ellipse.the axis of the cone: Hyperbola. (Bottom). Projection of the intersection normal to the axis of the cone: Ellipse.

Mathematics 2020, 8, x FOR PEER REVIEW 28 of 29 Mathematics 2020, 8, 2176 25 of 25 Although it is actually a fourth degree curve as shown (Figure A11) in the ensuing representation, even so it is not possible in most cases to find its enclosed area by numerical Although it is actually a fourth degree curve as shown (Figure A9) in the ensuing representation, even so it is notprocedures. possible in [12] most cases to ﬁnd its enclosed area by numerical procedures [12].

Figure A9. Perspective view of the Tomomi’s curve. Figure A11. Perspective view of the Tomomi’s curve. References Author Contributions: J.M.C.-L. has developed the analytical part of the work, while F.S.-A. has dealt with the 1.programmingYamauchi, work J. The in amount Grasshopper of flux incident® and the to rectangularcomparison floor of throughthe results rectangular obtained windows. numericallyRes. with Electrotech. those obtainedLab. 1929by J.M.C.-L., 250. All authors have read and agreed to the published version of the manuscript. 2. Moon, P.H. The Scientific Basis of Illumination Engineering; Dover Publications: New York, NY, USA, 2008. Funding: This research received no external funding. 3. Moon, P.H.; Spencer, D.E. The Photic Field; The MIT Press: Cambrige, MA, USA, 1981. 4.Acknowledgments:Lambert, J.H. Photometry, The authors or, on would the Measure like andto warmly Gradations thank of Light, Enrique Colors, Salguero-Jiménez and Shade: Translation for fromhis unselfish the Latin collaborationof Photometria, in the Sive, preparation De Mensura of most et Gradibus of the Luminis,figures that Colorum appear et Umbrae, in this withwork. Introductory Cabeza-Lainez Monograph wants and to honor Notes with bythe David article L. DiLaurathe figure; Illuminating of the late Engineering Engineer Jose Society Ramon of North Perez America: de Lama. New Our York, piece NY, of USA,writing 2001. has been 5.composedCabeza-Lainez, in praise of J.M. Tomomi Radiative Odajima performance San of louvers. Simulation and examples in Asian Architecture. ConflictsIn Proceedings of Interest: of The the 6thauthors International declare none. Conference on Indoor Air Quality, Ventilation & Energy Conservation in Buildings (IAQVEC), Sendai, Japan, 28–31 October 2007; Volume 3, ISBN 978-4-86163-072-9 C3025 4762E. \ 6.NomenclatureCabeza-Lainez, J.M. Fundamentos de Transferencia Radiante Luminosa o la Verdadera Naturaleza del Factor de Forma y sus Modelos de Cálculo; Netbiblio: Seville, Spain, 2010. Fij Form Factor (surface to surface, dimensionless) 7. Cabeza-Lainez, J.M.; Pulido-Arcas, J.A. New configuration factors for curved surfaces. J. Quant. Spectrosc. fij Configuration factor (surface to point, dimensionless) Radiat. Transf. 2013, 117, 71–80. [CrossRef] Ei, Mi Emitted Radiant Power (W/m2) 8. Tedeschi, A. AAD_Algorithms-Aided Design. Parametric Strategies Using Grasshopper®; Le Penseur Publisher: Ni Received Radiant Power (W/m2) Potenza, Italy, 2014; pp. 22–23. ni Unit Received Radiant Power (point from surface, W/m2) Z. Phys. 1924 28 9. Φi Fock, Radiant V.A. Zur flux Berechnung (W) der Beleuchtungsstärke. , , 102–113. [CrossRef] 10.Li Sasaki, Emitted K.; Sznajder, Luminance M. Analytical or radiance view (W/sr·m factor solutions2) of a spherical cap from an infinitesimal surface. Int. J. I Heat MassRadiant Transf. intensity2020, (W/sr)163, 120477. [CrossRef] 11.Ai Salguero Surface Andujar, area (m F.;2) Rodriguez-Cunill, I.; Cabeza-Lainez, J.M. The problem of lighting in underground r domes,distance vaults, (m) and tunnel-like structures of antiquity; an application to the sustainability of prominent Asian θi heritage angle (India, with Korea,the normal China). (radian)Sustainability 2019, 11, 5865. [CrossRef] 12.Ω Gershun, solid A.angle The (sr) Light Field (translated by P. Moon and G. Timoshenko). J. Math. Phys. 1939, 18, 51–151. [CrossRef] References Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional a1.ffi liations.Yamauchi, J. The amount of flux incident to rectangular floor through rectangular windows. Res. Electrotech. Lab. 1929, 250. 2. Moon, P.H. The Scientific© 2020 byBasis the of authors. Illumination Licensee Engineering MDPI,; Basel,Dover Switzerland. Publications: This New article York,is NY, an openUSA, access2008. 3. Moon, P.H.; Spencer,article D.E. distributed The Photic under Field; theThe terms MIT Press: and conditions Cambrige, of MA, the CreativeUSA, 1981. Commons Attribution 4. Lambert, J.H. Photometry,(CC BY) licenseor, on the (http: Measure//creativecommons.org and Gradations of/ licensesLight, Colors,/by/4.0 and/). Shade: Translation from the Latin of Photometria, Sive, De Mensura et Gradibus Luminis, Colorum et Umbrae, with Introductory Monograph and Notes by David L. DiLaura; Illuminating Engineering Society of North America: New York, NY, USA, 2001. 5. Cabeza-Lainez, J.M. Radiative performance of louvers. Simulation and examples in Asian Architecture. In Proceedings of the 6th International Conference on Indoor Air Quality, Ventilation & Energy Conservation