**2. Determining the directions of light vectors in dynamics under direct solar irradiation**

The process of modeling solar irradiation is a complex system problem in defining many variables in time and space. Among them, primary and derivative ones are distinguished. The primary includes the movement of the Earth around its axis during the day (determines the change in time) and the movement of the Earth around the Sun during the year, which determines the position of the Earth's axis of rotation relative to the Sun; shape and position in space of a given object and objects shading it (azimuth, shape, and position parameters); position and shape of the solar receiver (azimuth, shape, and position parameters). A change in primary factors in time and space leads to a change in derivative factors that directly determine the flow of solar energy into the solar receiver. Derived factors include the direction of sunlight, which is given by the coordinates of the Sun (azimuth, height), which change with latitude, time of day and day of the year; the intensity of sunlight, which depends on the angle of incidence— the angle between the normal to the surface of the solar receiver and the direction of sunlight and duration of solar exposure. Calculations of the conditions of variable solar irradiation in architecture do not need great accuracy.

Therefore, they can be modeled based on a number of assumptions that simplify the geometric model. A basic model of solar radiation in the form of a diurnal cone of solar rays of variable geometry is proposed based on the following assumptions:


During the transition to the next day, the Earth rotates around the Sun by an angle γ:

$$
\gamma = \frac{360^{\circ}}{365},
\tag{1}
$$

and occupies a new daily position.

6.Calculation of time is carried out on the basis of mean solar time, which corresponds to the conditions for the uniform motion of the Earth in a circular orbit.

With such assumptions, many rays during the year break up into many rays that are formed during the movement of the Sun every day. The complex helical guiding cone breaks up into 365 circular guiding cones. In this regard, there is no need to use modeling and calculation of solar irradiation conditions by astronomical data or the results of instrumental observations or graphic materials created on this basis.

The formation of a cone, called the diurnal cone of solar rays, can be represented as a result of the rotation of the direction to the Sun around the axis per day when considering the movement of the Sun in relation to the Earth, which during the day is in one point of the orbit. Due to the parallelism of the sun's rays on a given day, the diurnal cone will be the same at any point in the vicinity of the Earth and have an axis parallel to the Earth's axis. The plane of the horizon, drawn through the top of a plane parallel to the Earth, divides the cone into day and night parts. This cone is variable in time since during the year the angle α of intersection of the direction to the Sun with the Earth's axis changes daily. Let us consider how the cone changes when the Earth moves on the arcs AB, BC, CD, and DA of the orbit g (see **Figure 1**).

**Figure 1.** *Change in the diurnal cone of sunlight as the earth moves around the sun.*

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

At point A, the angle α between the generatrix and the axis of the cone is equal to the angle φ = 66.55° since the orthogonal projection of the axis onto the plane of the orbit Ω coincided with the direction to the Sun.

At point B, the direction to the Sun is perpendicular to the axis of the cone. Therefore, during rotation, the sunbeam SB will describe the plane. Consequently, when moving from point A to B, the angle α increases from 66.55° to 90°, and the surface of the diurnal cone straightens and becomes a plane at point B.

At point C, the continuation of the sun's ray SC makes an angle α = φ with the axis. When the sun ray SC rotates around the axis, a conical surface also appears. In fact, the conical surfaces at points A and C have different cavities (summer and winter) of the same conical surface.

At point D, the sunbeam, as at point B, makes a right angle with the axis. Therefore, when the beam SD rotates, a plane is also formed. With further movement, this plane again turns into a conical surface when the angle α changes to a value of 66.55° at point A.

Thus, for any chosen point in the vicinity of the Earth due to the change in angle α during the year, a family of coaxial diurnal cones of solar rays is formed with a common apex at this point. The area of this family is limited by the summer and winter cavities of the solstice cone, the generatrix of which makes an angle α = φ = 66.55° with the axis. For any intermediate point G1 of the orbit g, the angle α is determined depending on the angle γ of the rotation of the Earth relative to the Sun [11].

Let us consider the features of determining the coordinates of the light vector s under direct solar irradiation, the dynamics of which are related to the geometry of the motion of the Sun. Using the geometric model of the daytime solar cone (Φ), the movement of the sun rays is determined for such initial conditions as the latitude of the area and the day of the year. The desired coordinates of the unit vector s at some point in time are determined by the values of the angles Н° <sup>s</sup> and А° s. If to take into account only the daytime part of the cone Φ, that is, the upper one in relation to the horizontal plane P1, then the value of the applicate (Z) of the vector s will depend only on the magnitude of the angle Н° s. The values of the coordinates (X; Y) change in accordance with the position of the projection s1 and its angle А° <sup>s</sup> with the north direction (N).

For further calculations, in addition to the direction of sunlight, it is important to take into account the dynamics of quantitative changes in illumination from the Sun. Thus, the values of the average illumination Е┴ с of the surface element, which is located perpendicular to the direction of the sun's rays, change for a given latitude in accordance with the day of the year and the hour of the day. The values of Е┴ с according to are calculated by the formula:

$$E\_c^\perp = \frac{E\_c^\*}{\Delta^2} \cdot p^M,\tag{2}$$

where Δ – the distance from the Sun at a given moment, determined from astronomical tables, Е° <sup>с</sup> – the light solar constant, approximately equal to 135,000 lux, р – air transparency, and M – air mass, determined according to the table, depending on the value of Hs ° .

If to present the value of Е° с in graphical form, then it will be displayed as a guide segment attached to the calculated point. To determine the direction of such a segment, it is advisable to use the geometric model of the diurnal cone Φ, which sets the dynamics of changes in the direction of the sun's rays. Then the lengths of the generators of the diurnal cone Φ will correspond to the values of Е° с on the selected scale.

In **Figure 2**, the change in the hourly directions of the sun's rays by 50° n.l. for 22.06 is determined by the daytime part of the diurnal cone of solar rays Φ. Variable values of the lengths of the segments on the hourly Φ correspond to the average values of Е┴ с at a given time of the day.

For example, Е┴ <sup>с</sup> = 68,200 lux at 12 o'clock, Е┴ <sup>с</sup> = 66,600 lux – 10, 14 hours; Е┴ <sup>с</sup> = 60,400 lux – 8, and 16 hours with a gradual decrease in their values, which correspond to a zero value during sunrise and sunset (at 400; 2000 hours).

For graphical construction of the indicated lengths, let us set aside them in full size on the contour generatrix of the daily cone Ф and transfer them by rotation to the hour generatrix. A curve k is drawn through the upper ends of the guide segments constructed in this way, illustrating the dynamics of changes in the quantitative values of normal illumination Е┴ с during the day at a given latitude. With its help, it is possible to determine graphically the intermediate hourly values of Е┴ с. The guide

#### **Figure 2.**

*The dynamics of changes in the quantitative values of normal illumination <sup>Е</sup>┴<sup>с</sup> at a given latitude by 50˚ n.l. for 22.06.*

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

segment, graphically depicting the value of normal solar illumination for given parameters (latitude, day, hour), is recorded with its spatial coordinates. For example, at 4 p.m., the direction of the sunbeam is determined by the unit vector s with coordinates {cosН° <sup>s</sup> sinА° s; cosН° <sup>s</sup> cosА° s; sinН° s}. With this in mind, the coordinates of the guide segment that sets the value of normal solar illumination for the same hour with its length d corresponding to the value of Е┴ <sup>с</sup> = 60,400 lux are: {dcos Н° ssinА° s; dcos Н° scosА° s; dsinН° s}.

According to the well-known formula, the amount of direct solar illumination of a surface element with its different inclination and orientation, given by the direction of its normal n, will be determined:

$$\mathbf{E} = \mathbf{E}\_c^\perp \cos \theta,\tag{3}$$

where θ – the angle between the vectors s and n, which is calculated by the corresponding formula in analytical geometry, with known values of the coordinates of the vectors.

Thus, the geometric model of the daily cone of solar rays with the constructed curve k — the curve of variable quantitative values of normal illumination Е┴ с makes it possible to graphically simulate and analytically solve all the parameters that determine the amount of direct solar illumination arriving at a given surface element in dynamics.
