Cartesian and Spherical Coordinate System

Daily irradiation duration calculation on a sloping terrain can be appreciated using a spherical coordinate system, which is very suitable for observing the passage from

a horizontal to a sloping surface. The mutually orthogonal unit directions ur, ue, and u, in a spherical coordinate system are shown in Fig. 3.26. Hence, an oblique view can be obtained for the solar noon meridian and of the meridian containing the sloping surface point A of latitude в and hour angle ф. Here, the latitude is positive (negative) for the northern (southern) hemisphere.

Geographically, the longitude and latitude (hour angle) can be expressed very conveniently as the components of a spherical coordinate system with its origin at the earth’s center. Hence, any point on the earth will have geographically its latitude, в, on the line that connects this point to the earth’s center and perpendicular to this line at the same geographic point the longitude, ф, from the solar noon half meridian. The first axis in the spherical coordinate system is the radial line from the earth’s center to the location of the point on the earth’s surface; it falls on the radius of the earth and is denoted by r with its unit vector as ur. This is the direction of the horizontal plane normal vector. On the same point perpendicular to this axis there is the second axis of the spherical coordinate system that lies within the meridian and it is tangential to the earth’s surface. The unit vector of this axis is from the point A

toward the north (south) with positive (negative) angle values. Its unit vector is ue as shown in Fig. 3.26. The completion of the spherical coordinate system requires the third axis perpendicular to the previous two axes and it is tangential at the same earth point to the latitude circle and its unit vector is щ.

On the other hand, there is another coordinate system in the form of Cartesian axes that go through the earth’s center with the z axis directed toward the north along the earth’s rotational axis with unit vector k. The x axis with its unit vector i, goes through the earth’s center and it constitutes the intersection line between the equator plane and the solar noon half meridian plane. The third Cartesian coordinate axis is perpendicular to these two axes and has unit vector j. The change of the geographic point on the earth changes the spherical system accordingly, but the Cartesian system remains as it is. In solar energy calculations, it is necessary to refer to the constant coordinate system, which is the defined as the Cartesian system and hence all the directions must be expressed in terms of (i, j, k) unit vectors.

The unit vector ur is radial outward at point A and is perpendicular to the hori­zontal plane tangential at point A where ue is in the direction of increasing absolute value of the latitude and it is tangential to the meridian containing point A and иф has the direction of increasing hour angle and in the mean time it is perpendicular to both щ and ur. Hence, these unit vectors can be related to a Cartesian coordinate system unit direction vectors i, j, and k by considering the earth’s center point O.

In Fig. 3.25, N indicates the north pole, and I is the direct solar radiation vector. The k axis coincides with the direction of the line segment O-N, which is part of the earth’s rotation axis, and the other two unit vectors, i and j, fall on the equatorial plane. The ur, u, and uo unit vectors can be expressed in terms of the latitude, hour angle, and the Cartesian unit vectors i, j, and k, by considering from Fig. 3.26 the projections of spherical coordinates on the Cartesian coordinate system as follows:

ur = (cos в cosф)і + (cosв sin ф’^ + (sin 0)k, (3.35)

u = (-sin в cos ф)і + (-sin в sin ф) + (cos e)k, (3.36)


^ = (-sinф)і + (cosф)j. (3.37)

These equations are valid for horizontal planes with its normal vector that falls on to the ur direction. Hence, there is no need for further calculation in order to define the position of the horizontal plane. It is convenient to remember at this point that, in the solar radiation and energy calculations, so far as the planes are concerned their positions are depicted with the normal vectors. For the sake of argument, let us define a plane with its three dimensions as the thickness, Tr, length, Lф, and width, We, where each subscript indicates the direction of each quantity. In other words, any horizontal plane is defined by its latitudinal width, longitudinal length, and radial thickness, as shown in Fig. 3.27.

This plane is horizontal in the sense that its normal direction coincides with the radial axis of the spherical coordinate system (Fig. 3.26). Hence, the rotation of this plane around the radial axis causes the plane to remain horizontal whatever the

Fig. 3.27 Solar horizontal plane

rotation angle. On the other hand, the horizontal plane becomes inclined in one of the three rotations:

1. Rotation around the ф axis: It gives the plane a tilt angle, hence the new axis as r’, в’, and ф’ have the configuration in Fig. 3.28.

On the other hand, in the case of a sloping surface a view perpendicular to the great circle containing the solar noon meridian appears as in Fig. 3.29.

Such a two-dimensional view shows several of the geometric factors governing the solar radiation of a sloping surface where solar irradiance, I, falls at the base of a slope, which can be downward or upward from the horizontal plane tangen­tial at A’. Herein, a positive (negative) sign to a downward (upward) slope is considered. The downward slope in this figure is rotated a°, which is the critical angle, and if exceeded, would result in a shaded slope during solar noon. If the view in Fig. 3.25 represents the northern hemisphere summer solstice then the solar declination would be 8 = 23.45°, and day-long irradiation would be experi­enced at latitudes 66.55° < в° < 90°. On the other hand, Fig. 3.30 is a graphical

summary of a rotation of the coordinate vectors ur, ue, and иф exerted to achieve a downward (positive) slope. The third unit vector иф = иф is perpendicular onto the plane and serves as the axis of rotation in this instance. An upward (negative) slope would be achieved by making the rotation in a clockwise direction.

Here, a is the tilt angle or the slope angle, which is counted as positive toward the north as upward slope. The new position of the plane has u. as the new normal and ив axis perpendicular to it. This rotation will leave the unit vector of the ф axis the same, hence, иф = иф. The relationship between (и. and ив) and the original axes unit vectors can be written as

u. = (cosa)ur – (sina)ue, (3.38)

ив = (sina)ur + (cosa)u, (3.39)


иф = иф. (3.40)

Hence, the substitution of Eqs. 3.35 and 3.36 conveniently into the first two equations gives the inclined plane expressions with respect to longitude and latitude as follows:

u. = (cos a cos в + sin a sin в) cos фі + (cos a cos в + sin a sin в) sin фj

— (cos a sin — sin a cos в)к, (3.41)

Ue = (sin a cos в — cos a sin 0)cos ф i + (sin a cos в — cos a sin в) sin ф

+ (sin a sin в + cos a sin 0)k, (3.42)


иф = иф, (3.43)

or more succinctly

u’r = cos(0 — a) cos ф i + cos(0 — a) sin фj + sin (в — a)k, (3.44)

u0 = sin (в — a) cos фі + sin (в — a) sin фj + cos(0 + a)k, (3.45)


иф = иф. (3.46)

2. Rotation around the в axis: It gives to the plane an aspect angle of ^ (see Fig. 3.31). In this case, the в axis remains the same and the plane can be de­fined with its new normal direction along the и" and perpendicular axis to it as


Similar to the previous case the Eqs. 3.41-3.43 remain the same except a will be replaced by ^ and finally the relevant expressions are expressed succinctly as

и" = cos(0 — ^) cos фі + cos(0 — ^) sin фj + sin^ — ^)k,

Fig. 3.31 в axis rotation

иф = sin (в — Щ cos фі + sin(0 — £2) sin фj + cos(0 + £)k, (3.48)


ив’ = ив. (3.49)

3. Two successive rotations, first around the ф axis and then subsequently around the в axis: In this manner both a and £ angles will be effective as in Fig. 3.32. Figure 3.33 shows how a slope can be rotated £° west (east) of north to achieve aspects ranging from 0° west (east) of north to 180° west (east) of north. The accepted convention is to make £ positive (negative) if the rotation is west (east) of north.

The rotations in Fig. 3.33 with respect to the unit vectors є’ф and eв (see also Fig. 3.32) lead to doubly rotated unit vectors иф'(= ur), ив", and иф’. These doubly rotated unit vectors provide the coordinate system with which to describe the geometry of a sloping surface in full generality, and they are given by the following equations:

и"= [cos a cos в cos ф — sin a cos £ sin в cos ф + sin a sin £ sin ф] i+

[cos a cos в sin ф — sin a cos £ sin в sin ф — sin a sin £ cos ф] j+ , (3.50) [cos a sin в + sin a cos £ cos в ] k

Fig. 3.32 ф and в axes rota­tion

u” = [— sin a cos в cos ф — cos a cos ^ sin в cos ф + sin a sin ^ sin ф] i+

— [sin a cos в sin ф + cos a cos ^ sin в sin ф + cos a sin ^ cos ф] j+ , (3.51) [— sin a sin в + cos a cos ^ cos в] k


ифф’ = — [— cos ^ sin ф + sin ^ sin в cos ф] i+

[cos ^ cos ф — sin ^ sin в sin ф] j+ (3.52)

[sin ^ cos в] k.

In the case of a = ^ = 0 the unit vectors in Eqs. 3.50-3.52 revert to those in Eqs. 3.35-3.37, respectively.

Updated: June 16, 2015 — 2:14 pm