When looking at a single point, it is either in shade or it isn't. Thus, the shading mask for a single point is hard-edged as it will instantaneously move from 100% shaded to entirely unshaded as we consider different parts of the sky. However, if we look at a surface, it could be entirely shaded, entirely unshaded or partially in shade. Thus, whilst it is possible to draw the exact shape of shading for a single point, the shading for a surface is always soft-edged as it needs to fade gradually from 100% in shade to 0% in shade.

Figure 1 - The locations at which shading was calculated.

To illustrate this, consider the overshadowing situation shown in Figure 1 and what the shading diagrams might look like when calculated for:

• Each of the three points on the window sill,
• The entire line of the window sill, and
• The entire window surface.

Calculated Shading Masks

Point 1 is shaded by the thickness of the wall and the window reveal as well as the horizontal shade above and the vertical to the east. Its shading can be accurately displayed showing the exact dates and time is comes into and goes out of shade, as shown in Figure 2.

Figure 2 - Shading calculated at Point 1.

In order to consider the shading at two points simultaneously, Points 1 and 2, there will be some times when they are both in shade (100%), sometimes when only 1 is in shade (50%) and times when neither is in shade (0%). Hence we can add their two shading masks together, each being worth 50% shading. This shows three different regions representing 100%, 50% and 0% shading.

Figure 3 - Cumulative shading calculated at Points 1 and 2.

If we add in Point 3, now there are times when all are shaded (100%), only two are shaded (66.66%), only one is shaded (33.33%) and none are shaded (0%). This results in 4 different regions as shown in Figure 4.

Figure 4 - Cumulative shading calculated at Points 1, 2 and 3.

To extend this further, consider the line running along the window sill. We can effectively model a line by simulating it as a large number of discrete points running along it. If we do that, depending on the level of accuracy we require, we may have to calculate and sum hundreds of shading masks.

Alternatively, if we divided the sky into a number of segments, we could check to see how much of the line was 'visible' from each sky segment. In this way we could display the shading as a series of radial squares with each showing the percentage shading. Figure 5 shows the sky divided into 5°x5° segments and the percentage of the window sill line visible from each sky segment. As you can see, there are now many different and smoother degrees of shading.

Figure 5 - Shading calculated for all points along the window sill line.

Surface Overshadowing

This same procedure can be carried out for any number of lines or any number of surfaces, still based on the same number and size sky segments. Figure 6 shows it applied to the entire window surface. In this case, because some points near the very top of the window are pretty well always in shade, some degree of shading extends over the whole diagram.

Figure 6 - Shading calculated for all points over the entire window surface.

If you have not already done so, you should have a look at the interactive shading mask selector to view a range of different shading configurations and their associated shading masks.