The aim of most overshadowing analysis is to determine when a particular point in a model is exposed to direct sunlight and when it isn’t. This can be done manually by projecting shadows over the model at various times of the day / year and recording whether the point is in shade or not, as shown below. However this can be quite laborious and even then gives information only for the shading from direct solar radiation.

Shadows at 9am

Shadows at 3pm

Figure 1 - Diagram showing shadows projected at two different times of the day.

Visually in the one image, the above method displays shadows over the whole site for a single instance in time. An alternate method is to project the surfaces of all potential obstructions back towards the point of interest and onto a theoretical hemisphere centred at the point of interest. If the resulting spherical shapes are rendered on a sun-path diagram, the individual regions obstructed by each object can be quickly determined as shown in Figure 2 below.

If the annual path of the sun is overlaid on the diagram, it is possible in the one image to display the overshadowing for the whole year for a single point in space. Moreover, the diagram also shows the relative area of the sky that is obstructed at that point, thus providing information on the availability of diffuse radiation as well.

Figure 2 - Site obstructions projected back onto an imaginary hemisphere centred at the point of interest.

When generated and projected this way, it is possible to derive a series of individual transformed polygons, as shown in Figure 3 below.

Figure 3 - Projected site obstructions shown in a sun-path diagram centred at the point of interest.

However, using these in any meaningful way requires the application of complex trigonometric procedures to geometrically sum the areas of spherical polygonal regions. A much simpler and easier method would be to divide the sky up into a series of discrete segments, as detailed in the Sky Subdivision topic. It would then be simply a matter of determining where each sky segment was 'visible' to the point of interest of not.

Figure 4 - Two example shading masks for a shadowed point – the left shading generated polygons for each obstructing surface projected and drawn in the sun-path diagram, whilst on the right the sky dome is divided into a series of segments, each either shading the poijnt of interest or not.

The benefit of using the segmented sky approach is that the resulting data is easy to access and immediately useful in a whole range of numerical analysis. Determining if the point is in shadow at any time simply requires the altitude and azimuth of the sun, from which the specific sky segment can be determined and the appropriate shading value referenced. Similarly, the fraction of unobstructed sky can be quickly found by simply summing the shading values for each segment and dividing by the total number of segments.

Partial Shading Effects

As shown in the subdivided sky in Figure 4 above, the shading mask for a single point is hard edged - it is either in shade at a particular time or it isn’t. The shading mask for a planar surface however is more often soft-edged as only a portion of the surface may be in shade at certain times. This important concept is discussed in detail in the Partial Shading topic.

Point

Surface

Figure 5 - The shading masks are very different for points and surfaces due to partial shading effects.

Incidence Angle Effects

Once the sky has been subdivided, the orientation and inclination angle of the surface can be used to determine which parts of the sky it is actually exposed to. For example a vertical surface, no matter which way it faces, will only ever 'see' at best one half of the sky dome - meaning that it will only ever receive a maximum of one half of the available diffuse component. A horizontal surface that faces upwards, on the other hand, could see it all.

However for a horizontal surface, any light from the zenith of the sky arrives normal to that surface whilst light from the horizon arrives at grazing incidence. For a vertical surface the reverse is true – light from the zenith arrives at grazing incidence whilst light from the horizon arrives along its normal. This means that the area of the sky that contributes most to any surface depends greatly on its inclination angle.

Figure 6 below shows a surface at different inclinations and its corresponding surface incidence effect mapped over a mask. This is simply the cosine of the incidence angle the geometric centre of each sky segment makes with the surface normal.

Figure 6 - Different areas of the sky dome 'visible' to surfaces at various tilt angles, together with their corresponding diffuse incidence masks.

To consider this effect, an extra layer is added to the shading mask of all surfaces. Each segment in this layer is assigned the cosine of the angle that its' centre point makes with the surface's normal. To calculate the relative contribution of any sky segment, its fractional exposure (1 - overshadowing) and incidence value are multiplied together and converted back to a shading percentage. When done for each segment in the mask, the total effect can then be displayed, as shown in Figure 7 below.

Figure 7 - The moderation of overshadowing effects by incidence angle to produce a surface shading mask.

Reflected Solar Radiation

Using shading masks it is also possible to accommodate the effects of transparent and reflective surfaces (Marsh, 2005). Basic ray-tracing techniques can be used to continue tracing rays through any number of transparent surfaces, with each surface reducing the total contribution of the ray. If the ray hits a reflective surface, it can be forked to produce a reflected ray which is then traced in the same way as the original ray. The direction of this spawned ray is determined by mirroring the origin of the first ray about the plane of the reflective surface, as show in Figure 8 below.

Preparation of reflected ray.

Determining which sky segment the reflected ray passes through.

Figure 8 - The process of tracking the contribution of solar reflections in a shading mask.

Once reflected, the closest sky segment in the direction of the new ray is determined. If the reflected ray is not obstructed before hitting this sky segment, then it contributes an additional fraction to this segment equal to the original ray’s fractional value when striking the reflective surface multiplied by both the reflectivity and specularity of its assigned material.

The sky segment that the reflected fraction is added to is usually in a completely different part of the sky than the original segment being tested.

It is important to note that reflected radiation is in addition to unobstructed radiation. Also, it is possible for several reflective surfaces to contribute reflections within the same sky segment. Thus, in such cases it is even possible with focusing effects in the surrounding environment that the fractional contribution of a sky segment may end up being greater than 1.

To consider reflection effects, an additional layer is added to the shading mask containing the reflected radiation data. The total contribution of each segment in the mask is then found by simply adding the reflected fraction to the exposure value (1 - overshadowing) and then converting back to a shading percentage, as shown in Figure 9 below.

Figure 9 - The effect of a reflective surface is to add an extra fraction of solar radiation. This can be accommodated by simply adding the shading and reflection layers of the shading mask together.

Overall Shading and Reflection

Bringing all these effects together, it is possible to calculate an overall surface shading mask first adding the exposure and reflected layers together and then multiplying the result by the surface incidence layer, as outlined in the following image.

Figure 10 - The sequence of effects resulting in a total shading mask.

Important Benefits

One major benefit of pre-calculating and storing shading masks for each surface is that they need only be regenerated when the physical geometry, transparency or reflectivity of the model changes. Additionally, by storing different aspects of the shading data in separate layers, it is possible to re-use elements in each array that do not change between calculations, or that can be simply transformed to match different objects.

References

1. Marsh, A.J., 2005, The Application of Shading Masks in Building Simulation, Building Simulation 2005, Nineth International IBPSA Conference, Montreal, Canada, 2005. (view as pdf)

Useful Links

• The Solar Pathfinder:
  http://www.solarpathfinder.com/
  An incredible device that provides a panoramic reflection of a site with overlaid local sun-paths, providing a full year of accurate solar/shade data.