**3. Determining the directions of light vectors in dynamics under scattered solar irradiation**

For geometric modeling of surface irradiation under conditions of scattered sky light, it is necessary to determine the direction of the light flux. As is known, the source of natural light in the light field in this case is the atmosphere, which in geometric modeling is taken as a hemisphere. The atmosphere scatters direct sunlight, respectively, not having a single direction of incidence. That is, the solution of the problems of modeling the irradiation of the sky with scattered light becomes more complicated due to the different directions of the light rays.

To solve this issue, let us assume that when irradiated with scattered sunlight, the light flux has a main direction for each individual point of a given surface. Using the provisions of the theory of the light field [6], let us take the light vector as the predominant direction of the incident light. The light vector characterizes the value and direction of the "pressure" of light on a spherical body (which radius approaches zero) located at a given point on the surface. According to [6], the direction of the light vector coincides with the direction of the solid angle vector, the absolute value of the solid angle vector determines the absolute value of the light vector.

Thus, for modeling spherical illumination at points of a given surface, each of them is conditionally taken as a sphere (radius approaching zero). In **Figure 3**, the center of the sphere O coincides with the point A under study.

The rays of diffuse light coming from the points of the sky open for a given point A of the surface k lie inside the solid angle dω. It has a vertex at a given point on the surface A and rests on the contour of a part of the surface of the hemisphere of the sky, which illuminates a given point on the surface A and is determined using the tangent plane τ to it.

**Figure 3.** *Geometric modeling of irradiation of point a with scattered sky light.*

Therefore, the solid angle dω is defined as the part of the celestial hemisphere (CH), which is formed by at least two planes intersecting P1 and τ (**Figure 3**). The horizontal plane П<sup>1</sup> (XOY), as is well known, is given by the eq. Z = 0 and is perpendicular to the coordinate axis ОZ. The plane τ, with known values of the angle of its inclination ωτ to the horizon plane and the azimuth of the plane Аτ, passes through the origin, its equation:

$$A\mathbf{x} + B\mathbf{y} + \mathbf{C}\mathbf{z} = \mathbf{0},\tag{4}$$

where A, B, and C — the coordinates of the vector n to which the plane τ is perpendicular. The coordinate values are determined in accordance with the accepted coordinate system and the specified north direction N.

It should be noted that when solving irradiation problems, it is important to take into account both the "upper" and "lower" sides of the tangent plane τ, which differ from each other in the direction of the vector n and the values of А<sup>τ</sup> and ωτ. So, in **Figure 4**, when considering the irradiation of the "upper side" of the plane τ, the unit vector n<sup>B</sup> has coordinates {�sinω<sup>в</sup> τcosA<sup>в</sup> τ;sinA<sup>в</sup> τsinω<sup>в</sup> τ;sinω<sup>в</sup> <sup>τ</sup>}. When determining the irradiation of the lower side of the plane τ<sup>н</sup> , the value of the azimuth A<sup>н</sup> <sup>τ</sup> and the angle of inclination ω<sup>н</sup> <sup>τ</sup> = 180°-ω<sup>в</sup> <sup>τ</sup>. Are taken into account. The perpendicular <sup>τ</sup><sup>н</sup> vector n<sup>н</sup> has the opposite direction from n<sup>в</sup> and coordinates {�sinA<sup>н</sup> τsinω<sup>в</sup> τ;cosA<sup>н</sup> τsinω<sup>в</sup> τ; sinω<sup>в</sup> τ}.

Therefore, the tangent plane τ to a surface element gives a geometric representation of the amount of light coming from the celestial hemisphere to it, dividing the celestial hemisphere into two parts. The direction of the normal specifies the orientation of a given surface element in space and determines the part of the sky that takes part in the irradiation.

According to the above provisions, the coordinates of the vector dω coincide with the coordinates of the axis of the solid angle dω, which is the bisector of the linear angle μ, which measures the dihedral angle between the planes P1 and τ. So, in **Figure 4**, when determining the irradiation of the "upper" side of the plane τ, the angle μ<sup>в</sup> = (180° - ω<sup>в</sup> <sup>τ</sup>), and for the "lower" side of the plane <sup>τ</sup>, the angle <sup>μ</sup><sup>н</sup> <sup>=</sup> <sup>ω</sup><sup>в</sup> <sup>τ</sup>. Thus, *Geometric Modeling of Parameters of Variable Natural Light during the Integration… DOI: http://dx.doi.org/10.5772/intechopen.112134*

#### **Figure 4.**

*Geometric modeling of sky irradiation with scattered light of a given point O of the surface, taking into account the characteristics of the tangent plane τ.*

the coordinates of the vectors *dω<sup>в</sup>* {sinA<sup>в</sup> τcosA<sup>в</sup> <sup>τ</sup>; sin (180°- <sup>ω</sup><sup>в</sup> <sup>τ</sup>)/2}, and *<sup>d</sup>ω<sup>н</sup>* {�sinA<sup>н</sup> τ;, cosA<sup>н</sup> <sup>τ</sup>; sin <sup>ω</sup><sup>в</sup> <sup>τ</sup>/2}.

The obtained coordinates of the vectors *dω<sup>в</sup>* , and *dω<sup>н</sup>* determine the direction and position of the light vectors in space when a given plane is irradiated with scattered sky light.

According to the provisions of the theory of the light field [6], the length of the light vector ε with a uniform distribution of brightness in the sky changes in accordance with the change in dω, the vector of the solid angle dω, to which it is equal in magnitude. The absolute value of the light vector ε makes it possible to determine the quantitative value of irradiation at a given point on the surface with a scattered sky. Irradiation will change in proportion to the value of cos θ — the angle between the normal n and the light vector ε at a given point:

$$\mathbf{E} = \boldsymbol{\varepsilon}\cos\theta,\tag{5}$$

In the case, when the irradiated element is perpendicular to the direction of the light vector, that is, the directions of the normal and the light vector coincide, and the angle θ = 0°, there is a position that corresponds to the greatest transfer of light energy. Accordingly, with an increase in the angle between the normal and the light vector at a given point, there is a decrease in exposure. All this makes it possible to analyze the dynamics of irradiation of a surface element according to the change in its real position in space and the parameters of the light source— scattered sky light. When determining the irradiating area of the sky, it is important to take into account the possible shading of a surface element by other objects. In this case, the change in the value of the solid angle dω, and, accordingly, the position of the light vector ε is set by determining the center of gravity of the sky area through which the vector ε passes, or by determining and compiling the vectors ε of simple parts of a given sky area.
