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Shading: Shadow Angles

When attempting to shade a window, the absolute azimuth and altitude of the Sun are not as important as the horizontal and vertical shadow angles relative to the window plane. These can be calculated for any time if the azimuth and altitude of the Sun are known.

Horizontal Shadow Angle (HSA)

This is the horizontal angle between the normal of the window pane or the wall surface and the current Sun azimuth. The normal to a surface is basically the direction that surface is facing - its orientation.

Figure 1 - The derivation of horizontal shadow angle (HSA).
Figure 1 - The derivation of horizontal shadow angle (HSA).

If the orientation of the surface is known, then HSA is simply given by:

Vertical Shadow Angle (VSA)

The VSA is more difficult to describe. It is best explained as the angle that a plane containing the bottom two points of the wall/window and the centre of the Sun makes with the ground when measured normal to the shaded surface.

Figure 2 - Creating the plane of Vertical Shadow Angle from points on the shaded surface and the current Sun position.
Figure 2 - Creating the plane of Vertical Shadow Angle from points on the shaded surface and the current Sun position.

The VSA can the be determined by the altitude of a line taken exactly normal to the shaded surface (window or wall) running along this plane.

Figure 3 - The derivation of the VSA based on the plane containing the Sun.
Figure 3 - The derivation of the VSA based on the plane containing the Sun.

Once the HSA is calculated, the VSA is given by:

Shading Depth for Equator-Facing Surfaces

An equator-facing surface is one that faces due South in the Northern hemisphere and due North in the Southern hemisphere. Surfaces at other orientations are substantially more complex and are dealt with in the section following below.

However, for equator-facing surfaces, it is the VSA that determines the depth of any required shade. In order to shade to the very bottom of the window, the required shade must extend out at least as far as this plane. This applies to horizontal, angled and drop-down shades - as shown in Figure 4 below.

Figure 4 - To shade the bottom of the window, the depth of any shading device will be determined by the VSA plane.

Important Seasonal Characteristic

The VSA is an important aspect of shading design because of one very useful characteristic. For the Summer half of the year, the path of the Sun through the sky is such that the lowest VSA on equator-facing surfaces occurs at solar noon. At all other times in the morning and afternoon, the VSA is always greater. The Sun actually rises and sets behind the surface and is only in front of it for part of the day.

Figure 5 - Over Summer, the lowest VSA always occurs at solar noon.
Figure 5 - Over Summer, the lowest VSA always occurs at solar noon.

This means that the noon solar altitude alone can be used to calculate the depth of a shading device in knowledge that, if it is wide enough, it will definitely shade the surface all day.

On the Spring and Autumn Equinoxes, the VSA is actually constant throughout the day. This means that, to design a shading device to fully shade for exactly 1/2 the year, you can use the VSA for any time on or around mid-day on the 21st of March or the 21st September.

Figure 6 - At the Equinox, the lowest VSA always occurs at solar noon.
Figure 6 - At the Equinox, the lowest VSA always occurs at solar noon.

In Winter, the highest VSA occurs at solar noon. This means that, at all other times, the Sun is at a lower altitude in the sky. Thus, as the lowest VSA actually occurs at sunrise or sunset - where its value has to be zero. Thus, to fully shade a window throughout the day in Winter would require an infinitely large shade or one that completely covered the window.

Figure 7 - In Winter, the highest VSA occurs at solar noon.
Figure 7 - In Winter, the highest VSA occurs at solar noon.

Horizontal Shade Dimensions

Once calculated, the HSA and VSA can be used to determine the size of horizontal shading device required for a window. If the height value refers to the vertical distance between the shade and the window sill (or from the shade to the bottom of the surface requiring shade), then the depth of the shade and its width from each side of the window can be determined using relatively simple trigonometry.

Figure 8 - Width, depth and height as they apply to this simple method of horizontal shading device design.
Figure 8 - Width, depth and height as they apply to this simple method of horizontal shading device design.

The depth of the horizontal shade is given by:

The width from each side of the window is given by:

The width simply refers to the additional projection from each side of the window. Exactly which side and which direction will be obvious from the time of day and which side of the window the Sun is on.

The depth and width of angled shades can also be calculated. In this case each edge simply runs up and down an imaginary line from the corner of the window to the position of the Sun at the cut-off time and date. The following animation shows how you can base a range of shades at different angles on this method.

Figure 9 - Using the same width, depth and height method, you can design shades at any angle.
Figure 9 - Using the same width, depth and height method, you can design shades at any angle.

Non-Equator-Facing Surfaces

Unfortunately, no similarly simple shading depth rules apply to surfaces that are not directly equator-facing. The shading situation depends very much on the exact combination of surface orientation and site latitude. Figure 10 illustrates this by showing variations in VSA for a surface offset from due South by 45deg. It is clear to see in this situation that, even in Summer, the Sun rises almost directly in front of the surface - meaning that a horizontal or angle shading device would not be effective until later in the day when the Sun is at a much higher altitude.

Figure 10 - The hourly shadow angles for a surface offset from due South by 45deg.
Figure 10 - The hourly shadow angles for a surface offset from due South by 45deg.

To illustrate this further, and to show that you can calculate shadow angles for any surface at any orientation, the animation in Figure 11 shows the effects of rotating a surface's orientation through a full 360deg. In this case the orientation is applied by rotating the Sun path relative to the model. However, the exact same effect would be true if you transformed the surface.

Figure 11 - How shadow angles vary for a surface at different orientations. In this animation, the sun-path rotates around the object whilst the date and time remains constant.
Figure 11 - How shadow angles vary for a surface at different orientations. In this animation, the sun-path rotates around the object whilst the date and time remains constant.

The Solution ?

The design of shading for non-equator-facing surfaces is more complex because you must first determine exactly when in the day (and over the year) that shading is required. Then there will usually be a trade-off between shading times and the size of the resultant shade.

There are several more complex shading design methods you can use for this purpose, each outlined here in subsequent topics.

Shading: Overshadowing
Shading: Special Glasses

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