When attempting to shade a window, the absolute azimuth and altitude of the Sun are not as important as the horizontal and vertical shadow angles relative to the window plane. These can be calculated for any time if the azimuth and altitude of the Sun are known. Horizontal Shadow Angle (HSA)
This is the horizontal angle between the normal of the window pane or the wall surface and the current Sun azimuth. The normal to a surface is basically the direction that surface is facing - its orientation.
Figure 1 - The derivation of horizontal shadow angle (HSA). If the orientation of the surface is known, then HSA is simply given by:
Vertical Shadow Angle (VSA)
The VSA is more difficult to describe. It is best explained as the angle that a plane containing the bottom two points of the wall/window and the centre of the Sun makes with the ground when measured normal to the shaded surface.
Figure 2 - Creating the plane of Vertical Shadow Angle from points on the shaded surface and the current Sun position. The VSA can the be determined by the altitude of a line taken exactly normal to the shaded surface (window or wall) running along this plane.
Figure 3 - The derivation of the VSA based on the plane containing the Sun. Once the HSA is calculated, the VSA is given by:
Shading Depth for Equator-Facing Surfaces
An equator-facing surface is one that faces due South in the Northern hemisphere and due North in the Southern hemisphere. Surfaces at other orientations are substantially more complex and are dealt with in the section following below. However, for equator-facing surfaces, it is the VSA that determines the depth of any required shade. In order to shade to the very bottom of the window, the required shade must extend out at least as far as this plane. This applies to horizontal, angled and drop-down shades - as shown in Figure 4 below.
Figure 4 - To shade the bottom of the window, the depth of any shading...