PH-2521 -- The Application of Homogeneous Coordinates to Shadowing Calculations

When assessing heat gains in buildings due to solar radiation, it is necessary to know how much of each part of the building is shaded and how much is in direct sunlight. The sunlit area of each surface changes as the position of the sun changes during the day. The purpose of a shadow algorithm is to compute these sunlit areas.

The ASHRAE Load Algorithms Manual (1) describes three shadow algorithms. Each was judged to have different strong points and weak points. SHADOW is an algorithm developed by Groth and Lokmanhekim which gives a general method of treating shadow receiving and shadow casting surfaces. As originally presented it is quite time consuming, especially when accuracy is desired, because of the discrete element analysis method which it uses. SHADOW 1 is described as "an algorithm to find the ratio of the sunlit and shaded areas of a given window where the shadows are cast by various combinations of overhang and side fins." It is fast and accurate but is limited to shadows from a few rectangular surfaces placed in a limited number of ways relative to the shaded surface. SHADOW 2 is described as "an algorithm to determine whether or not a given window is shaded by a remote object such as an adjacent building." The algorithm is fast but it is approximate and applicable only when the window is small compared to the shadowing object.

The shadowing algorithm described hereinafter starts from the work of Groth and Lokmanhekim (2) as described in the ASHRAE Load Algorithms Manual (1) and used in NECAP (3). It uses the same coordinate transformation and projection techniques but replaces the discrete element analysis with a method using homogeneous coordinates. Because of this, it is fast and accurate while retaining generality. The algorithm was developed at the U.S. Army Construction Engineering Research Laboratory, Champaign, IL, as part of the Building Loads Analysis and System Thermodynamics (BLAST) program (4) (the algorithm is, therefore, referred to as the BLAST shadowing algorithm in the text that follows).