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Annual patterns of overshadowing can be visualised by plotting the outlines of obstructing buildings and vegetation on to a sun-path diagram. This produces what is known as the solar aperture - being that area through which a particular point can 'see' the sky.

Imagine lying on your back looking through a full 180° fish eye lens pointing straight up towards the zenith of the sky. The image you see would be very close to that shown immediately below. If the path of the Sun through the sky was then overlaid, it would be possible to tell when in the year you would be in direct Sun and when you would be in shade by simply determining when the Sun was obstructed by objects in the surrounding environment.

Figure 1 - Animation showing a photograph of the entire sky dome taken through a 180° fish-eye lens and how the sun-path and solar aperture is defined.

To construct such a diagram, consider an imaginary hemisphere surrounding some point - usually termed the Focus Point. Areas of Sun blockage are determined by projecting lines out from this point to vertexes on the silhouette of each object and determining where they intersected this imaginary hemisphere. The resulting shapes on this hemisphere can then be transferred to a sun-path diagram. Any area of shadow in the resulting diagram represents an area when the Focus Point is in shade, where an object would block light from the Sun when it is behind it.

Figure 2 - Diagram of a simple solar window projected onto an imaginary hemisphere around an object.

The diagram below shows the resulting sun-path diagram when the Sun blockages from above are transferred. When the Sun path for the particular location is overlaid, the designer can quickly determine, from a single diagram, both the times of day and days of the year that the Focus Point would be in direct Sun or in shadow.

Figure 3 - The same solar window as in Figure 2, projected onto a sun-path diagram.

The process of projecting shadow blocks onto a sun-path diagram is not particularly complex, just laborious. You can download ECOTECT and the SolarTool and use them to automatically generate these types of overshadowing diagrams for complex 3D models, however you will undoubtedly find it useful to have an understanding of the actual process.

However, note in the diagram above that all the vertical lines in the model run radially straight from the outer horizon line into the centre at the zenith of the sky. However, the horizontal lines at the top of each object are actually slightly curved.

If you were to trace points running along an infinite horizontal line somewhere above you, their altitude angles would gradually reduce the further away each point got, until they finally disappeared over the horizon - at which point their altitude would be zero. This results in the curvature of all horizontal lines when plotted on a stereographic sun-path diagram. This curvature actually follows lines of constant vertical shadow angle.

If you imagine that you had a semicircular plate lying flat on the ground with the same radius as the sky hemisphere, the shape of its perimeter would run around the perimeter of the stereographic diagram when viewed from directly above.

If the plate was then tilted about its centre point, its perimeter would still be on the surface of the sky hemisphere, but when viewed from directly above it would no longer appear to be exactly circular. As its tilt angle increased, it would gradually change from a circular line when lying flat, to a single straight line running through its centre when fully vertical. The shapes this makes at various tilt angles in plan view represent the curvature of horizontal lines at the same relative shadow angle.

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Figure 7 - Use the slider at the bottom to tilt the semicircular plate to various angles. The dotted lines represent the shape at each 10° angular step.

The diagram below shows this curvature as a series of shadow angles on a protractor, displayed as a set of rotating dotted white lines. To line up the protractor, the straight line through the middle of the diagram must run parallel to each horizontal line in the model. This means rotating it around as you draw in the top of each surface. You should try this out by using the slider at the bottom of the diagram to rotate the shadow angle protractor. See if you can pick what angle it must be for each of the curved horizontal lines in the model.

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Figure 8 - Use the slider at the bottom to rotate the shadow angle protractor to see if you can line it up with each of the curved horizontal lines in the model.

The following interactive animation illustrates the procedure by which the overshadowing block of a building can be translated onto a stereographic sun-path diagram.

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Figure 9 - Use the buttons at the bottom of the animation to step through the process of creating an overshadowing block for a buildng.

• Step 1 - Select and mark the Focus Point at which the overshadowing is to be determined.
• Step 2 - Determine the azimuth angle between true North and the first vertical corner of the building from the selected point (positive angles are always taken in a clockwise direction). Draw a line from the very centre of the stereographic diagram out to the azimuth angle just calculated. This line represents the line of the vertical corner if it were infinitely high.
• Step 3 - Determine the altitude angle from the ground plane to the very top of the vertical corner. Using the concentric altitude lines, work out and mark the measured altitude on the radial azimuth line you just drew.
• Step 4 - Repeat Steps 1-3 for the next corner to form the sides of the first surface.
• Step 5 - If it is only a short wall you can simply draw a line between the two altitude points. If not, the horizontal line between the two marked point will actually be curved within the sun-path diagram. To accurately draw this curve, rotate the shadow angle protractor so that it faces the normal of the surface being projected. If it is a truly horizontal line, both altitudes should line on the same shadow angle - in which case you simply trace the interpolated curve between the two. If it is not, such as the corner of a hip roof, you will have to interpolate between the two. Normally this can be done by eye, however if you need more accuracy you can divide the wall with a number of vertical lines and repeat steps 1-3 on each to get a series of guide point between them.
• Step 6 - Fill in the polygon defined by the horizon line, the two vertical side lines and the horizontal top. This region represents the overshadowing block of the first surface. When you display the sun-path, any time the sun falls within this block, the focus point will be in shadow.
• Step 7 - Repeat this procedure for all surfaces of the building that are directly visible from the focus point.

## Useful References

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.
Sun-Path: Sun Positions