# Innovative Tool to Determine Radiative Heat Transfer Inside Spherical Segments
## 1 Introduction  
View or form factors are a relevant entity in all the phenomena that involve radiation [1]. Since this particular process of heat transfer, unlike the common belief, does not imply any kind of continuity or contact between the affected bodies, pure form and position become pivotal for the transmission of energy [2]. Diffuse radiation is inextricably correlated to surface area [3]. In basic elementary shapes (cuboid), no particular problem appears with such a concept [3], but when it is required to extend the well-known differential equations to concrete and finite forms arbitrarily deployed, we have to deal with two sets of different and often complex double integrals [4, 5]. It is known that to obtain an exact fourfold primitive, even of the simplest elements, is a mathematical feat and often a serendipity, to say the least [6].  
Even recent studies have failed to overcome this severe hindrance for the completion of important problems in radiative transfer in the void [3]. Experts have frequently resorted to numerical or statistical approaches such as Monte Carlo methods [7], albeit they are not conclusive in most cases, not only because of the many errors that arise in the process but also due to the impossibility of validating their accuracy as they are in fact the only method available to try to surmount this shortcoming. In the case that we are addressing, MonteCarlo methods are not available due to the presence of double curvatures which invalidate any form of regular or coherent tiling of the surfaces.  
The present author was able to perform the said quadruple integration in an exact manner solely for three kinds of simple forms: rectangles (whether perpendicular or parallel), parallel coaxial circles, and the same for a sphere and a circle [8; 9]. It seems puzzling that if the rectangles simply form an angle different to \(\pi/2\) or zero between them, or if the circles are slightly offset or placed in a perpendicular fashion, we cannot even reach an exact solution to the problem of finding the form factor between these rather common elements.  
If the forms are incomplete and fragmented, then the problem increases significantly because we need to remark that, until the author discovered it, there was no exact solution available even for the form factor due to a triangle [10]. Only the simpler expression of the configuration factor (a double integral entity which regulates the exchange on a point to point basis, but not including the whole surface) had been identified and solved for particular points [11].  
Furthermore, another degree of complexity arises when the exchanges take place in a three-dimensional environment or in other words, the surfaces involved are not planar but curved or warped [12].  
It is clearly improbable for the time being to find the form factor of a set of arbitrarily curved surfaces by integration alone [3]. However, other numerical methods based on discretization of the said areas in differential units are also not viable since such kinds of surfaces cannot be divided into any form of regular, concentric, or symmetrically patterned tiles. Once again we arrive at the previous impasse. With enough fortune and considerable effort, the author has discovered ways to advance in the finding of exact solutions by means of innovative principles of their own creation [13], suitable for algebraic treatment and eventually offering exact expressions that altogether avoid integration and are currently being programmed in algorithms. In the following sections we will explain and develop one such property, perhaps the most relevant of all.