The geometry of shadows
This is best illustrated by considering the dimensions of the shadow cast on a vertical window by a reveal and an overhanging lintel.
Figure 7.11(a) shows in perspective a window which is recessed from the wall surface by an amount R. The front elevation of the window, seen in Figure 7.11(b), shows the pattern of the shadow cast by the sunlight on the glass. The position of the point P’ on the shadow, cast by the sun’s rays passing the comer point P in the recess, is the information required. In other words, if the co-ordinates x and у of the point P’ on the glass can be worked out, the dimensions of the shaded area are known.
Figure 7.11 (c) shows a plan of the window. In this, the wall-solar azimuth angle is n and co-ordinate x is immediately obtainable from the relationship:
X = R tan n (7.10)
To see the true value of the angle of altitude of the sun, we must not take a simple sectional elevation of the window but a section in the plane of the sun’s rays; that is, along A-A, as shown by Figure 7.1 (d).
If the hypotenuse of the triangle formed by R and x in Figure 7.11(c) is denoted by M,
Then the value of M is clearly R sec n. Reference to Figure 1.11(d) shows that the value of
The co-ordinate у can be obtained from
У = M tana
= R sec n tan a (7.11)
The CIBSE Guide publishes data in tabular form on the shading cast on a recessed window.
Calculate, by means of equations (7.10) and (7.11), the area of the shaded portion of a
View of window
Plan of window
Fig. 7.11 Determining the co-ordinates of the shadow cast on a window by its reveal.
Window which is recessed by 50 mm from the surface of the wall in which it is fitted, the following information:
Altitude of the sun 43°30′
Azimuth of the sun 66° west of south
Orientations of the window facing south-west
Dimensions of the window 2.75 m high x 3 m wide
The angle n is 66° — 45° = 21° and so, by equation (7.10)
X = 50 tan 21°
= 50 x 0.384 = 19 mm
By equation (7.11)
Y = 50 sec 21° tan 43°30′
= 50 x 1.071 x 0.949 = 51 mm
It is to be noted that the depth of the recess is the only dimension of the window which governs values of x and y. The actual size of the window is only relevant when the area of the shadow is required. It follows that equations (7.10) and (7.11) are valid for locating the position of any point, regardless of whether it is the corner of a recessed window or not. Thus, as an instance, the position of the end of the shadow cast by a flagpole on a parade ground could be calculated.
Continuing the answer to example 7.6,
Sunlit area = (2750 — y) x (3000 — x)
= 2699 x 2981 = 8.05 m2
Total area = 2.75 x 3.00 = 8.25 m2
Shaded area = 8.25 — 8.05 = 0.2 m2
This does not seem to be very significant, but it must be borne in mind that it is the depth of the recess which governs the amount of shading.
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