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# Sun-Path: Projections

Whilst there are many different types of sun-path diagram, there are only really two types of projection system used; Polar and Cartesian.

### Polar Projections

Polar projections produce circle-shaped diagrams in which each data point is represented by an angle around and distance from the centre. In the case of a polar sun-path diagram, the solar azimuth is plotted as an angle from North (0-360) whilst the solar altitude is given as the distance from the centre (0-90).

A simple linear projection of altitude lines around the sky dome straight down onto a flat surface would make angles near the horizon very close together and those near the zenith quite far apart. If you were interested in the effect of an overhead shading device, this would be entirely appropriate. However, in many architectural situations the bulk of overshadowing from surrounding buildings and vegetation occurs when the Sun is much nearer the horizon. Thus, alternate projections are necessary in order to increase the level of accuracy and detail for angles nearer the horizontal.

#### Polar Co-ordinate Projections

The differences between the three main polar projections are explained in the following animation.

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Interactive animation showing how different polar projections differ in thier distribution of altitude lines. Click on the words Spherical, Stereographic or Equidistant to activate an explanation.

##### Spherical Projection

In this method, the radial distance from the centre is simply the cosine of the altitude angle. As shown above, the relative change in radius between 75° and 90° is very much greater than between 75° and 90°. This makes such diagrams very good for considering overhead shading or very tall surrounding buildings.

##### Equidistant Projection

Using this method the radial distance is simply a linear factor of the altitude angle. Thus the relative change in radius between all angles is the same, so there is no bias towards either the zenith or the horizon.

##### Stereographic Projection

This is a more complex projection in which azimuth lines are first projected back to a reference point located a distance of 1 radius beneath the circle centre. The point where each of these lines intersects the zero axis gives the radial distance. The primary advantage of this method is that it increases the resolution of the diagram at low solar altitudes making it more suitable for the majority of surrounding building overshadowing situations.

### Cartesian Projections

The Cartesian coordinate system is used to uniquely determine any point in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point. Each coordinate is given relative to a universal origin point and measured along two perpendicular axis. This typically results in a rectangular-shaped diagram with solar azimuth along the horizontal axis and solar altitude along the vertical axis.

#### Axial Scaling

Not all parts of the sky contribute equally to the daylight that arrives at a surface or passes through a window. Different sky conditions can result in variations in the distribution of luminance over the sky dome, with some areas brigher or darker than others. For a flat surface or a window, light from some parts of the sky will arrive at normal incidence whilst others will arrive at almost grazing incidence, with the cosine law determining their relative contribution.

These effects can be accommodated within cartesian sun-path diagrams by applying non-linear scales to one or both of the axis, as shown in the interactive animation immediately below. This can be done solely for visualisation purposes or, with the addition of distributed points or a grid of squares, used as a basis for manually taking off measurements for daylighting potential or overshadowing effect.

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Cartesian projection showing the scaling effects of sky illuminance distribution and surface incidence.

##### Orthographic Projection

This refers to a straight mapping of azimuth and altitude angles onto the X and Y axis (respectively) of a cartesian graph. The key to an orthographic projection is that, much like plan view or elevation of a 3D object, the data on each axis is mapped linearly, with no attempt to include modifiers for surface incidence or sky luminance.

##### Waldram Diagram

Waldram Diagrams are intended for use in the manual calculation of potential daylight levels in buildings. As such, the scale in the altitude axis needs to consider important effects such as sky luminance distribution, surface incidence and sometimes even corrections for angular variations in transmission through the glass.

Daylight design is typically driven by worst-case conditions, that being a dark overcast sky in winter. The CIE-Overcast-Sky model defines the zenith as being 3 times as bright as the horizon. This means that the equivalent angular difference in altitude in these two parts of the sky dome should result in squares with areas differing by a factor of 3.

Sun-Path: Components
Sun-Path: Sun Positions