**4. Determining the directions of light vectors in dynamics when surfaces are irradiated with complex natural light (with the total exposure to direct and diffused sunlight)**

Under conditions of complex light, each of the sources, independently of the others, creates an irradiation that can be represented using light vectors. The sum of the light vectors determines the total irradiation at the calculated point, which, in geometric modeling, is represented by the total light vector εΣ. Its direction is determined by the light vectors εс and εн graphically applied to a given point, corresponding to the transfer of direct sunlight εс and scattered sky light εн to the calculated point for given irradiation conditions. The desired light vector εΣ is obtained by the parallelogram rule when the light vectors εс and εн are replaced by one light vector, the coordinates of which are given by the sum of the corresponding coordinates of the light vectors ε<sup>с</sup> and εн. The direction of the light vector E<sup>Σ</sup> determines the predominant direction of light energy transfer to the calculated point, and its length, according to (5), sets the quantitative value of natural irradiation EΣ entering the calculated point. The use of the above provisions on the geometric modeling of light vectors in direct sunlight and scattered sky light makes it possible to take into account the dynamics of changes in irradiation conditions when determining and geometric modeling the direction of complex light and its quantitative value.

For example, let us determine the total value of exposure to point O, with clear sky at 12:00 22.06 at 50° north latitude. Point O refers to the vertical plane τ (XOZ), the normal n to which coincides with the direction of the OY axis (**Figure 5**). The direction of the sun's rays for the given conditions will be determined by the angles δ = 50°, φ = 66.6°, the height of the sun Но <sup>s</sup> = 180°-(δ + φ) = 63,4°, and the azimuth of the sun А° <sup>s</sup> = 180°. Let us determine the amount of irradiation by direct sunlight E<sup>с</sup> arriving at a given surface element. According to (5), at Е┴ <sup>с</sup> = 68,200 lux, E<sup>с</sup> = Е┴ сcos (180°-(δ + φ)) = 68200cos63 and 4 εС ° = 30,537 lux. Thus, the transfer of direct sunlight to point O is characterized by the light vector εс with coordinates: {0; Eсcosho = 13,673; Eсsinho = 27,304}.

#### **Figure 5.**

*Geometric modeling of the total irradiation value of the point O belonging to the vertical plane τ (XOZ) at 12:00 22.06 at 50° north latitude.*

*Geometric Modeling of Parameters of Variable Natural Light during the Integration… DOI: http://dx.doi.org/10.5772/intechopen.112134*

Let us consider the definition of the geometric parameters of the light vector εн, which characterizes the value and direction of the "pressure"at the point O of scattered sky light with a uniform distribution of brightness L. The light vector εн has a direction along the axis of the solid angle dω within which the light flux is transferred.

The length of the light vector corresponds to the amount of scattered light illumination E, determined by the formula:

$$\mathbf{E}\_{\rm H} = Ldo \cos \theta,\tag{6}$$

where θ — the angle between the normal n and the light vector εн.

For the case under consideration (**Figure 5**), let us assume that with a uniform distribution of the brightness of the clear sky at noon, L = 4000 cd. The solid angle dω formed by two mutually perpendicular planes τ, and P1 is equal to π. Thus, for the given irradiation conditions at the calculated point corresponding to the length of the vector εн, it is equal to Eн = cos 45°4000 π =8881 lux. Consequently, the coordinates of the desired light vector εн {0; Eнcosθ = 6280; Eнsinθ = 6280}.

The light vector of total exposure εΣ has coordinates {0; 13673 + 6280; 27304 + 6280}, and its length is found from a right triangle whose legs are equal to the coordinates of the light vector εΣ:

$$\mathbf{E}\_{\Sigma} = \sqrt{\mathbf{Y}^2 + Z^2} = \sqrt{\mathbf{1}\mathbf{9}\mathbf{9}\mathbf{53}^2 + \mathbf{33}\mathbf{584}^2} = \mathbf{39064}, \text{lux} \tag{7}$$

For comparison, let us determine the total natural irradiation entering the horizontal plane (XOY) at the calculated point O for the same conditions, that is,. at 12:00 22.06 (**Figure 6**).

The direction of the light vector εс, characterizing the irradiation from direct sunlight, will remain unchanged, but the length of the vector will change, which will be equal to Eс = Е┴ сcos(90°-(180°-(δ + φ))) = 68200cos26,6° = 60,981 lux. In this case, the vector ε<sup>с</sup> has coordinates: {0; Escos Нs ° = 27,304; EssinНs° = 54,526}. The amount of scattered light and the direction of the light vector εн will also change. Thus, the direction of the light vector εн coincides with the OZ axis, the solid angle dω for the given conditions is 2π, and the length of the vector ε<sup>н</sup> is E<sup>н</sup> = cos0°4000лк2π = 25,120 lux.

The coordinates of the vector εн {0;0;25,120}, and the coordinates of the total vector εΣ {0;27,304;79,646}. The length εΣ, corresponding to the value of the total exposure of the calculated point from direct sunlight and scattered sky light, is determined using the obtained coordinates of the light vector εΣ and is equal to:

$$
\sqrt{\mathbf{Y}^2 + \mathbf{Z}^2} = \sqrt{27 \mathbf{3} \mathbf{0} \mathbf{4}^2 + 7 \mathbf{9} \mathbf{6} \mathbf{4} \mathbf{6}^2} = \mathbf{84} \mathbf{19} \mathbf{6} \text{ lux} \tag{8}
$$

The obtained values of the irradiation parameters from the total action of direct and scattered sunlight are summarized in **Table 1** and allow us to draw conclusions regarding the nature of the variability of the complex light irradiating a given point on the surface of the heliosystem.

Thus, the shift of complex light irradiating a given point on the surface is determined by the uneven weight of the components of direct and diffused sunlight. It should be noted that, in these cases, the change in the parameters of the complex light occurred due to a change in the position of the tangent plane, which determines the position of the calculated point in space. This proves the possibility of modeling the irradiation of planes by changing their position in space for given light conditions.

#### **Figure 6.**

*Geometric modeling of the total irradiation value of the point O belonging to the horizontal plane τ (XOY) at 12:00 22.06 at 50 º north latitude.*


#### **Table 1.**

*Variability values of the irradiation parameters on the surfaces by changing their position in space.*

*Geometric Modeling of Parameters of Variable Natural Light during the Integration… DOI: http://dx.doi.org/10.5772/intechopen.112134*

The use of the above provisions on the geometric modeling of light vectors in direct sunlight and scattered sky light makes it possible to take into account the dynamics of changes in irradiation conditions when determining and geometric modeling the direction of complex light and its quantitative value. The direction of the light vector EΣ determines the predominant direction of light energy transfer to the calculated point, and its length sets the quantitative value of natural irradiation E<sup>Σ</sup> entering the calculated point. Determining the length of the vector εΣ according to formulas (6, 7) allows you to calculate a rather complex combined effect of direct and diffused light entering the calculation point for simulating its irradiation conditions.
