A number of people have recently requested information on how to complete an overshadowing exercise I set some Masters students at a couple of different Universities. As a result, I have added some more descriptive text and reformatted the original answer sheet to describe both the exercise problem and an example method that can be used to generate the answer.
The aim of this exercise was to introduce the concept of shadow volumes and to try to get students to really think through some relatively complex solar geometry issues. The exercise itself required the design of an optimum shading device to fully shade a complex concave polygonal site boundary. The actual question was as follows:
Using the site model provided, construct an elevated sun shade that will protect the entire red area from the Sun each day from 09:00 to 17:00 from 1st April to 10th September. Your task is to use the minimum amount of shading material whilst ensuring that no part of your shade comes closer than 2.4m to the shaded area. Groups will be competing to find out who's design has the lowest total surface area of shading material.
The exact site requiring shading is shown in Figure 1 below. If you load the ECOTECT model, you can see that this is located in Cardiff, with a latitude of 51.5°, a longitude of -3.0° and it's time zone being GMT+0. A zipped copy of this ECOTECT model can be downloaded using the Attached Files link at the bottom of this page.
To solve such a problem, you have to start somewhere - so why not begin with a horizontal plane hovering some arbitrary distance above the site - say about 10m. As the Sun's position moves through the sky and can be quite low in the morning and evening, we'll make the plane quite a bit bigger than the actual site to give us some tolerance for this, as shown in Figure 2.
A Single-Point Shading Polygon
The first thing we have to do is determine the exact shape of a shade that would perfectly shade a single point over this date/time period. If you imagine a laser beam located at one of the site vertexes that always points exactly towards the Sun, what we are after is the shape it would describe as it hits the shading surface if we tracked the Sun at each extreme position within the date and time range specified. Obviously the Sun is highest in the sky in Summer, so one of these lines will need to follow the Sun on the Summer Solstice. The other line would track the Sun on that date within the range with the Sun lowest in the sky. Then, the two lines are joined by tracking the Sun at 09:00 and 17:00 between the two dates.
To do this, first locate the Transform Origin at one of the vertexes of the site polygon - any one will do to start with. The origin is shown as small red axis with X, Y and Z at the end of each line. Next, make sure that the large horizontal plane above the site is currently selected. Finally, select the Calculate > Shading and Shadows > Project Transform Origin > Solar Shading Profile menu item. This will display the dialog box shown in Figure 3.
Set the required date range from 1st April to 10th September and the time range from 9am to 5pm. When you have finished, select the OK button.
You should now see the exact shape required to shade the origin point for the selected dates and times projected onto the shading surface, as shown in Figure 4.
A Shading Polygon for Each Vertex
Your next job is to create the same shading polygon for each vertex in the site. Whilst you can do this by repeating the same process as described above, the Sun's rays are so close to parallel by the time they reach the Earth's surface that the required shape is going to be exactly the same. Thus, you can simply copy the original shape from vertex to vertex with the interactive move command and the Apply to Copy option selected.
After doing this for each vertex that makes up the perimiter of the site, you should end up with a series of 17 shapes as shown below.
An Overall Shading Polygon
The next step is to join up all the extreme points to form the required total enclosing shading polygon. This appears relatively simple, however you will need to study the animation in Figure 7 below very carefully in order to see why only a small number of connecting lines were used.
This process involves determining where the extremities of the overall shape boundary needs to move from an extreme point on one of the vertex shading polygons to the next. These two extreme points must then be joined by a line parallel to and the same length as the line segment on the site joining those two vertexes. I've looked at various algorithms to automatically generate this overall shape from multiple polgons, but it is not a simple convex hull or anything like that. There is definitely a research project in this if anyone is interested.
The result is the required shape for a flat shade at the specified offset above the site. The animation in Figure 8 below shows shadows cast into the site each hour of the day on the 10th of September (the same as 1st April) and the 21st June (the Summer Equinox). There may look to be excess shade, but there really isn't.
Once you have this shape, it actually represents a single slice through a 'conical' volume which represents the area of actual required shading. Any slice within this volume will equally shade the site.
To obtain a slice at any other angle, simply create a plane at the angle you want, first select and tag it as the cutting plane. Next select the projected 'conical' surfaces and use the Modify > Cutting Plane > Trim Selection... menu item. Make sure you choose Profile not Trim so as to leave the planes untouched.
Unfortunately there is no way around it - you need to physically trace each point on the resulting profile line to obtain a planar shading surface.
To find the shade with the smallest surface area, you will have to try a range of planes at different angles. The two examples below show that the requires area can be substantially different depending on the plane angle.
Obviously the closer the plane is to the site, the smaller it will be. Thus, rather than tell you the actual answer, you'll still have to find out for yourself.