John F. Cotton College of Architecture & Environmental Design California Polytechnic State University San Luis Obispo, California JOHN F.

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1 SO L I DMO D E L I N GAS A TO O LFO RCO N S T RU C T I N SO G LA REN V E LO PE S by John F. Cotton College of Architecture & Environmental Design California Polytechnic State University San Luis Obispo, California JOHN F. COTTON is Associate Professor of Architecture in the College of Architecture and Environmental Design, California Polytechnic State University, San Luis Obispo, CA. A B S T R AC T This paper presents a method for constructing solar envelopes in site planning using a 3D solid-modeling program as the tool. The solar envelope for a building site is a mechanism for ensuring that planning regulations on the solar access rights of other sites are observed. In this application, solid modeling offers the practical advantage of being a general-purpose tool having the capability to handle sets of site conditions that are quite complex. The paper reviews the concept of solar envelopes and demonstrates the method of application of solar-envelope construction to a site defined to avoid overly simplifying conditions. Techniques for displaying the constraints on building sections imposed by a solar envelope are presented as well. INTRODUCTION With the growing importance of solar energy to the operation of buildings, tools have been sought to assist in the design and regulation of development to ensure that appropriate solar access is maintained for all. One such tool is the solar envelope pioneered by R. L. Knowles:... the solar envelope is a set of volumetric limits that govern building on a particular site.... The solar envelope provides the largest developable volume within time constraints. The envelope accomplishes this by defining the largest container of space that would not cast shadows off-site at specified times of the day. We can call these specified times of the day the cut - off times. [1] 253

2 The key to understanding the geometry of solar envelopes is recognition of the absence of measurable solar parallax over typical dimensions of a site. As a consequence of this absence, the intersection of an edge of a site with a line of sight to the sun at a cutoff time defines one plane of the envelope: any object whose height is constrained to remain below the plane Fig. 1. will not cast a shadow beyond that edge at the specified time. The ensemble of planes defined in this way by all edges of the building site for each of the cutoff times thus generates a multifaceted limiting surface that satisfies Knowles definition of the solar envelope. As an example, consider the application of a solar envelope to a simple rectangular site oriented with its long dimension in the north-south direction on flat terrain. Regulatory conditions governing the solar envelope might require that no shadows be cast off-site between the times of 9:00 AM and 3:00 PM (local solar time) on the 21st of December. If this were the case, the solar envelope would have the form of one half of a hip-roof structure as shown in Figure 1. The triangular plane on the north face defines the limit on heights for the north boundary of the site at both cutoff times. The eastfacing trapezoidal face sets the height limit for afternoon shadows at the east boundary of the site; the west-facing trapezoidal face performs a corresponding Fig. 2. role for the west boundary of the site. The two hip rafters on the envelope point from the northwest and northeast corners of the site to the sun s positions at 9:00 AM and 3:00 PM respectively. In this simple symmetric case, the position

3 of the intersection of the hip rafters with the north end of the envelope s ridge line can be determined from trigonometry: where: D = L/[2*Tangent(Solar Azimuth)] H = (L/2)*Tangent (Solar Altitude)/Sine(Solar Azimuth) D is the distance south in plan view from the midpoint of the north site line, H is the elevation above ground level, and L is the length of the north site line Analytical solutions quickly become impractical as complexities of non-rectangular site boundaries and sloping terrain are introduced and when asymmetric cutoff times are Fig. 3. applied to the solar envelope. In order to address these more complex conditions, numerous methods for constructing solar envelopes have been devised, including: flag or pole techniques for mapping shadow patterns, computer programs for calculating the vertical heights of an envelope on a grid superimposed over the site, and descriptive geometry techniques [2, 3, 4, 5]. Solutions for some of the more common site geometries have been tabulated as well [6]. The advent of 3D solid-modeling programs on microcomputers has opened the path to new methods for generating solar envelope forms. The real value in using solid modeling for this purpose is the ability of the site planner to integrate the analysis of solar-access constraints into the overall design process. This avoids the necessity of turning to special-purpose applications to conduct the analysis. The particular method described in this paper makes use of form Z, a 3D solid-modeling program available on the Apple 680X0 and PPC computer platforms. Essential to the method described here are tools in form Z that allow orthogonal projections of objects onto planes, extrusions of prisms, terrain modeling and Boolean operations on surfaces and solids. A SOLID MODELING SOLAR ENVELOPE EXAMPLE Figure 2 shows in plan view the site conditions used in the example of the application of solid modeling to solar envelopes presented in this paper. 255

4 The latitude of the site in the example is 36 degrees north. The approximate plan dimensions of the terrain topographical model are 200 feet by 200 feet; the maximum change in site elevation within the model boundaries is approximately 45 feet. The boundary of the solar envelope for the building site was Fig. 4. chosen deliberately to have no cardinal orientations for its segments and to fall on terrain that slopes non uniformly and not in a cardinal direction. Cutoff times were specified for both winter and summer limits to force consideration of solar access constraints on all boundaries of the building site. Further, the daily time constraints were chosen to be asymmetric between morning and afternoon: 10:00 AM and 3:00 PM (local solar time) on June 21 and on December 21. These conditions were imposed to demonstrate the general applicability of the method described here. PROCEDURE The steps in the method as illustrated by the example are as follows: A terrain topographical model of the overall site is generated in the form of a solid having an upper surface laid out on an 8-foot grid mesh with elevations referred to an arbitrary base datum. The form Z terrain mesh model fits a surface smoothly to the site contours; the surface does not necessarily contain the contour curves. The form Z program does provide an option that allows the upper surface of the terrain solid to be constructed as triangulated facets connecting all contours directly where strict conformity to the mapped contour curves is important. The sur- Fig. 5.

5 Fig. 6. face-fitting model is not fundamental to the method for generating solar envelopes described in this paper. In general, a site boundary will be defined by a set of points in three dimensional space with prescriptions for the line segments or curves connecting these points. In the method considered in this paper it is assumed that the building site can be defined with sufficient accuracy by specifying the x, y, z coordinates of the ends of a connected series of straight-line edge segments. In the specific example used here, the four corners of the building site shown in Figure 2 are assumed to lie on the mesh surface of the terrain model. However it may be defined in the model, the boundary of the building site is projected orthogonally onto a horizontal plane located at the reference datum. This creates a base profile for the solar envelope construction. Fig. 7. envelope terrain block shown in Figure 3. The base profile of the solar envelope is used for extruding a solid prism vertically from the horizontal plane to a height greater than the maximum elevation of the surface of the terrain model. The resulting prism is then applied in a Boolean intersection with the solid terrain model to create the solar The top surface of this solar envelope terrain block provides a three-dimensional template for constructing a more refined 3D polyline boundary for the solar envelope itself. 257

6 Fig. 8. Variations in the slope of the site can be built into the boundary of the solar envelope by making use of object intersection snap control while drawing the 3D polyline boundary. The 3D polyline boundary is constructed with as many segments as are judged necessary to achieve the needed accuracy for the final envelope. Seven segments are used in the example. Line segments oriented to the correct altitude and azimuth solar angles for each of the four cutoff times are then mapped onto the vertices of the 3D polyline boundary. The relevant line segments for the 10:00 AM, December 21 solar angles are shown in Figure 4. The five pairs of adjacent solar angle line segments define four constraining polygons that set the envelope limits imposed by the 10:00 AM, December 21 cutoff time. These polygons are displayed as translucent shaded areas connecting the adjacent solar angle line segments in Figure 4. A similar set of constraining polygons would be constructed for each of the remaining cutoff times. The boundary of each of the constraining polygons is used as the profile for a vertically extruded solid prism. These prisms (numbering 14 in total in the example presented in

7 this paper) are applied as difference objects in Boolean subtractions from a master prism. This master prism itself is a duplicate of the solid prism used earlier in the Boolean intersection to create the solar envelope terrain block. The procedure is illustrated in Figure 5 for 10:00 AM, December 21. This same procedure is applied to the constraining polygons for each of the cutoff times specified at the outset. When all of the Boolean subtractions have been completed, the solar envelope will be a prism having a multifaceted top surface. The final shape of this surface is determined by those sets of constraining polygons that are most limiting. It is not necessary to ascertain beforehand which of the cutoff times will be limiting on the final form of the envelope. Either the block will be carved away progressively or those subtractions that do not limit the final shape will result in null Boolean operations. The solar envelope block constructed using this procedure is shown in plan view in Figure 6. The controlling constraining polygons are identified by differing shaded areas on the top of the solar envelope block. The form Z software provides a contour-cutting routine for solids that is very useful for visualizing the form of the solar envelope as it sits on the terrain. This routine allows a solid to be sectioned repeatedly at specified intervals. Applying two series of these section cuts at right angles on the solar envelope block creates an egg crate representation. This representation of the solar envelope, similar to the method of portraying solar envelopes by Knowles [7], is shown on the terrain model in Figure 7. A useful 2-dimensional representation for analysis can be extracted from the solid models. Section cuts of the solid terrain model that coincide with the earlier section cuts made of the solar envelope block allow the construction of section cuts for terrain and envelope together. Surface Boolean subtractions are used to integrate the two section cuts into one profile for each section. Integrated section cuts made at 16-foot intervals in both the north-south and east-west directions over the building site are shown in Figure 8. These can be plotted at any desired scale for accurate analysis of the envelope constraints on building sections. CONCLUSIONS A 3D solid-modeling program with the appropriate tools presents an elegant solution to the general problem of constructing solar envelopes. The method described in this paper is not inherently limited by any complexity that might appear in a specific case. As shown 259

8 in the example, variations in the slope and orientation of site edges present no problem. Interior site angles greater than 180 degrees can be accommodated; curved boundaries can be approximated by discrete polyline segments to the accuracy needed. The accuracy of the solid terrain model itself can be extended by incorporating more tightly spaced and more accurately defined contour curves. Overall, the governing consideration in any case will be the accuracy demanded of the solar envelope in relation to the computing time and memory available in the system used and the amount of operator time to be devoted to a project. The example described in this paper could easily be constructed on a Macintosh II computer having 8 MB of RAM, requiring less than two hours of operator and computing time from start to finish. The value of this method for handling complex site situations using systems of modest computing power is enhanced further by the fact that the tool can be used as an integrated part of a general-purpose design modeling program. ENDNOTES 1. Knowles, R. L Sun Rhythm Form. Cambridge: the MIT Press Ibid Matus, Vladimir Design for Northern Climates. New York: Van Nostrand Reinhold Company Knowles, R. L Sun Rhythm Form. Cambridge: the MIT Press Ibid Brown, G. Z Sun Wind, and Light: Architectural Design Strategies. New York: John Wiley & Sons Knowles, R. L Sun Rhythm Form. Cambridge: the MIT Press