**Appendix A**

In the discussion that we sustained at point 3 about the spherical cap; we demonstrated that the necessary form factor was (Equation (4)),

$$F\_{12} = \frac{a^2}{a^2 + h^2} \,. \tag{A1}$$

If we may define a fraction β, composed of the squared heights and radiuses, expression A1 is reduced as

$$
\beta = \frac{h^2}{a^2} \tag{A2}
$$

$$F\_{12} = \frac{1}{\beta^2 + 1} \tag{A3}$$


**Figure A1.** Prolate spheroid.

Element 1 is an oblong ellipsoid and element 2 is the circle that serves as a base to the curved figure, h > a.

Previously, we defined the dimensionless relation m:

$$m = \sqrt{1 - \frac{a^2}{h^2}}\tag{A4}$$

Employing the first law,

$$F\_{12} = \frac{a \times m}{a \times m + h \times \arcsin(m)}\tag{A5}$$

$$F\_{21} = 1\tag{A6}$$

$$F\_{11} = \frac{h \times \arcsin(m)}{a \times m + h \times \arcsin(m)}\tag{A7}$$

Making it as before in the spherical cap

$$\beta = \frac{h^2}{a^2};\ m = \sqrt{1 - \frac{1}{\beta^2}}\tag{A8}$$

$$F\_{12} = \frac{\sqrt{1 - \frac{1}{\beta^2}}}{\sqrt{1 - \frac{1}{\beta^2}} + \beta \times \arcsin\left(\sqrt{1 - \frac{1}{\beta^2}}\right)}\tag{A9}$$

• Paraboloid of revolution (Figure A2).

**Figure A2.** Paraboloid of revolution.

In this case, element 1 is the paraboloid and element 2 is the circle that serves as a base to the upper curved surface

$$F\_{12} = \frac{6 \times a \times h^2}{\left[\left(a^2 + 4 \times h^2\right)^{3/2} - a^3\right]} \times F\_{21} = 1\tag{A10}$$

$$F\_{11} = 1 - \frac{6 \times a \times h^2}{\left[\left(a^2 + 4 \times h^2\right)^{3/2} - a^3\right]}\tag{A11}$$

$$\beta = \frac{h}{a} \times F\_{12} = \frac{6 \times \beta^2}{\left[ (1 + 4 \times \beta^2)^{3/2} - 1 \right]} \tag{A12}$$

In Figure A3, we plotted the evolution of factor F12 for some usual geometric bodies. The cone presents higher values of emitted energy while the sphere and the ellipsoid are less prone to di ffuse radiation internally. Other forms such as the conoid or the hyperboloid could be included in the future.

**Figure A3.** Comparison of four common curved geometries (non-dimensional).
