Spatial Relations and Architectural Plans Layout problems and a language for their design requirements

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1 Spatial Relations and Architectural Plans Layout problems and a language for their design requirements Can BAYKAN * Abstract: Programs for the solution of space layout problems are not widely used in practice. In order to be usable, such systems need to be intuitively understandable by the designers and enable them to formulate problems in a natural way. In this paper, we discuss the characteristics of space layout, such as the levels of organization, the types of objects and the constraints associated with them. The representations used for problem requirements include constraints on dimensions, areas and aspect-ratios; binary constraints, i.e. rectangle relations, bounded-difference constraints, orientation constraints; and global constraints. Design requirements should be formulated declaratively, independent of the solution methods that may be used. Specifying design requirements independently of the solution methods is useful for comparing layout programs in terms of the design requirements they can use, in terms of the set of significantly different solutions produced, and in terms of search efficiency. Keywords: space layout, spatial relation, rectangle algebra, constraint 1 Introduction Space layout has been studied as a research problem since the 1960 s. Site planning, arrangement of rooms in a building, and arrangement of furniture or equipment in a room have been solved by computer programs. In spite of being studied for a long time, the impact of this research on practice in terms of providing usable systems to professionals has been very limited [BAY 95]. Some reasons for this are discussed below. Design constraints and objectives used by automated layout systems are formally defined in mathematical terms. Using such systems requires knowledge and skills that are not commonly attained by designers. Formulating a layout problem as an optimization problem is not how designers intuitively define it. Layout problems are highly combinatorial and change from being under-constrained to being overconstrained with a simple change to problem specifications. The layout problem is sometimes used as a test bed for algorithms and programming approaches rather than with the aim of building a program to solve realistic problems. The importance given to problem formulation by some researchers may be secondary to the importance given to solution techniques. Space layout is not just problem solving, the designer is trying to bring order and achieve a composition. Problem definition needs to be interactive and under the guidance of the designer, and should be expressive enough to enable the designer to search for a composition. Layout problems are intuitively defined by designers as satisficing problems; automated systems should be formulated to work similarly. The program should give adequate explanation and feedback to let the designer * Faculty of Architecture, Middle East Technical University, Turkey. baykan@metu.edu.tr

2 understand what is happening and why. In this paper, we start from a consideration of space layout. A task analysis of spatial layout classifies layout problems into levels and discusses the characteristics of each level in section 2. The types of constraints that will enable a designer to define layout requirements is described in section 3, and some methods of reasoning that can be used with this representation are given in section 4. Section 5 discusses how these can be brought together in the user interface. 2 Space Layout Configuration in space is the subject of various professions, such as, urban design, architecture, interior design and furniture design. These professions are defined according to the scale of the objects they design, and the scales define levels of organization and control in the environment. The levels are street level, building level, partitioning level and furniture level [HAB 98]. At the street level, roads and the city blocks defined by them are the elements. At the building level, columns, load-bearing walls, floors and roof are the elements. Non-load-bearing elements that define space constitute the partitioning level. Where all floors and walls constitute a solitary system, as in some masonry houses, the partitioning level does not exist. The furniture level concerns the organization of furniture and equipment in space. Each level controls the configuration at the level below it. The wholes we experience in daily life combine two levels, such as a room combines the partitioning level and the furniture level. We can take the dimensions and location of doors in a room as fixed and design the configuration of equipment and furniture inside it, or determine both together. In the building level, we can take the building dimensions and the location of the load-bearing elements as fixed or determine these together with the organization of spaces in the building. When dealing with the arrangement of rooms, a distinction between partitions and load-bearing elements needs to be made, especially in systems that can consider multi-floor layouts. The relationships and the types of objects that are considered are different at the building and room levels. At the room level, the objects being configured are spaces and the partitions are implicitly defined by the edges of the spaces. Thus the partitions have no thickness, and load bearing elements such as columns and walls are usually not considered. Load-bearing elements need to be solid objects, different from the rooms which are spaces, and their position inside the rooms or at the room boundaries need to be controlled. The room configuration problem is abstracted as the configuration of rectangles in a rectangular envelope. Adjacency between spaces for access, adjacency to outside for getting light, view and natural ventilation is important. Also proximity and zoning some rooms together in a part of the building may be of concern. Dimensions, area and aspect-ratio of a space need to be controlled. Since a room is perceived only from inside, we are not easily aware of the global relations of alignment and order in a layout of rooms. At the furniture level, there is a need to consider both solids and use-areas. Solids may not overlap but use-areas can. All use-areas should be connected to provide access. This is a global constraint. Other global constraints may specify the alignment of fronts of solids in a kitchen, or the visibility of a TV from all seats in a livingroom. The latter constraint involves not only a seat and the TV but restricts the location of every solid in the room such that it is not between the TV and seat. A

3 solid has a front, left and right sides, and a back. Its orientation has to be considered alongside geometrical and topological relations. Although it is possible to formulate furniture level problems using only solids and distances between them, it does not permit controlling accessibility. In order to ensure that all use-areas are connected, they have to be explicitly represented [BAY 01]. Two and three dimensional configurations are not conceptually different, but three dimensional configurations are combinatorially much more complex. As a result, almost all space layout programs deal with two dimensions. We can also define an intermediate level between two and three dimensions that covers a large subset of actual problems encountered as two-and-a-half dimensional configurations. If there are vertical layers in a configuration, such that organization within a layer is vertically uniform and fully determined by the plan, than that configuration is twoand-a-half dimensional. An example is a building where all partitions extend vertically from floor to ceiling, but are at different locations in each floor. There may be spaces spanning more than one layer, such as, elevators, stairs, etc. A multi-story layout can be treated as two separate configurations with constraints propagated from one configuration to the other to enforce stairs to be in the same location in all [MY 00]. Another alternative is to superimpose all rooms and other elements in a single configuration. Rooms at the same level cannot overlap but rooms at different floors can. Load-bearing elements and stairs that span more than one level occur once. 3 Specifying Design Requirements Design knowledge can be formulated in terms of relationships between objects. The relations can be functional, topological, or geometric. The objects are finite, nonzero rectangles without holes, oriented parallel to the axes of an orthogonal coordinate system. Dealing with rectangles is much simpler due to the fact that a rectangle can be split into its x and y components, and some types of reasoning can take place independently in each dimension. Constraints are a natural way of representing and maintaining relationships. In mathematical terms, a constraint defines a relation among a set of variables. Constraints are declarative and independent of the techniques that are used to solve them. Constraints are classified according to the number of rectangles involved. 3.1 Unary Constraints Unary constraints involve one rectangle. Upper and lower bounds on the dimensions, area and aspect ratio of a rectangle are unary constraints. Aspect ratio has been an important consideration in some periods and styles, which used only the golden ratio or integer ratios such as 1:1, 1:2, 2:3, etc. Area and aspect-ratio constraints are not linear and relate the horizontal and the vertical dimensions. Some objects may have a front, back and sides. Spaces and use-areas are usually directionally uniform, but furniture and equipment is not. A refrigerator has a front, a back, etc. that needs to be taken into account in a layout. Orientation is used to specify the alignment of the long side or the front of a rectangle. Orientation is discrete as there are four possible orientations for a rectangle; its front may be facing North, South, East or West.

4 3.2 Binary Constraints A binary constraint involves two objects. It is natural to define layout problems using spatial relations to indicate the location of an object with respect to another. Some spatial relations are purely topological, such as inside and non-overlap. Others, such as distance, may involve a dimension. Adjacency is a geometrical relation if the length of the common border is specified, topological otherwise. Figure 1. Relations of the rectangle algebra Rectangle Relations Relations of the rectangle algebra, used for specifying design constraints between two rectangles, is shown in figure 1. Rectangle 1 is shown in grey and rectangle 2 in white. There are 13x13=169 possible qualitative relations between two rectangles. These are derived by the cross product of Allen s 13 relations between two intervals [ALL 83; CON 00]. The relations of the rectangle algebra are exhaustive and

5 mutually exclusive. The 13 relations in the first row of figure 1 cover all possible qualitative relations; <, =, >, between the vertical edges of two rectangles. It may be said that having 169 different relations is too much, but they enable a designer to make distinctions that are significant in a layout domain, and simplify the specification of design requirements by collecting all possible relations between two rectangles in one matrix. The relations in figure 1 need to be augmented by the addition of dimensions and orientations to express the binary requirements of space layout Distance Constraints The relations of the rectangle algebra are qualitative, that is dimensions and distances are not considered. There are three types of dimensions between two rectangles as shown in figure 2. These are inside-distance shown at left, overlapdistance shown in the middle, and distance shown at right. Figure 2. Types of distance between two rectangles It can be determined from a rectangle relation which if any of these dimensional parameters apply to it. Thus it is sufficient to specify them for each pair of objects together with the rectangle relation matrix. They do not have to be specified separately for each relation in the matrix Orientation Constraints There are four possibilities for the relative orientation of two objects; depending on the difference in their orientations. These are parallel, clockwise-from, opposite, and counter-clockwise-from. It is necessary to specify relative orientation together with some rectangle relations Complex Constraints The spatial relationships between objects may be quite complicated. The relation between two counters in a kitchen is that they can be adjacent side to side with their fronts and backs aligned and both facing the same way; or side to front with the fronts of the counters facing the interior of the 90 angle with the back of one aligned with the side of the other; or if they are not adjacent, the distance between them should be at least 120 cm. It is possible to specify this relationship using the combination of rectangle relations, orientations and dimensional parameters. There may be relations involving three or more rectangles, i.e. the bathroom is adjacent to the corridor or entrance. Global relations about access or visibility may involve all use-ares or all solids in a problem. In this case, a design requirement is represented by more than one rectangle relation matrix. Instead of requiring that a relation from each matrix be instantiated for the solution of the problem, one relation

6 from two matrices may need to be instantiated or only some combinations of relations from different matrices are acceptable. 3.3 Global Constraints Global constraints are relations on all or a large number of objects. The constraint that there should not be any holes in a layout is global. A global constraint can be represented by rectangle relations between all pairs of objects involved in the relation. Whereas a binary constraint is satisfied when one of the relations in the matrix is satisfied, a combination of relations from different matrices need to be satisfied in case of a global constraint Holes At the partition level, the rectangles being configured are rooms arranged in the building envelope. Preventing holes or empty areas in the building envelope can be expressed as a global constraint. Figure 3. Non-trivial, trivial, internal and external holes Holes that can be eliminated by extending one or more spaces is termed a trivial hole. Holes that require some spaces to extend and others to shrink in order to be eliminated are non-trivial holes. Non-trivial holes are always internal and are surrounded by a pin-wheel configuration, whereas trivial holes can have zero, one, two or three sides adjacent to the building envelope. as shown in figure 3, where the black area is a non-trivial hole and the gray areas are internal or external trivial holes Access A global constraint at the furniture level specifies that all use-areas or empty spaces between furniture or equipment in a room should be accessible from each other. The kitchen shown in figure 4 violates access constraint because the use-areas in the kitchen are in two pieces with no passage between them. Figure 4. A kitchen layout violating access requirement.

7 4 Operations on Rectangle Relations 4.1 Consistency checking It is possible to represent allowed spatial relations between two objects by placing 1 s and 0 s in a 13x13 matrix of spatial relations as shown in figure 4. Each matrix can completely specify the allowed relations between two rectangles LivingRoom - Apartment o Apartment - Vestibule = LivingRoom - Vestibule LivingRoom - Vestibule ' = LivingRoom - Vestibule '' Figure 5. Composition and intersection of rectangle relations The matrix at the top left in figure 5 shows the allowed relations between a livingroom and apartment. The livingroom can be inside and in the south-west corner of the apartment, indicated by the lower left 1 in the matrix. The living room can be equal to the apartment, indicated by the top right 1; it can also be in the north-west or south-east corners in addition to the south-west corner. The second matrix shows the relations between the apartment and vestibule. Eight different relations are possible, all of which have the characteristic that the vestibule is inside and adjacent to the east boundary of the apartment. Composing these two matrices results in the third matrix, which contains the relations that are allowed between the livingroom and the vestibule by the relations between livingroom-apartment and apartment-vestibule. The fourth matrix at bottom left shows the previously defined constraints between livingroom-apartment that they have to be adjacent. Intersecting the third and fourth matrices results in the last on at the bottom row in figure 4, where the 1 s specify that the livingroom can be adjacent to the vestibule at its west or south.

8 Relations that are inconsistent with others can be eliminated as a result of applying the composition operator such that path-consistency is achieved. Path-consistency means that every remaining relation takes place in some solution. Individual solutions can be enumerated by selecting a relation from each matrix. Other operations that are defined on rectangle relations are union and difference. [CON 00] 4.2 Levels of Abstraction Three levels of abstraction can be defined in the rectangle algebra. At the base level, the 169 relations are considered separately. At the intermediate level, the relations are grouped into >1 groups, and at the top level, all the relations of a matrix are considered together. The intermediate level is created by grouping the relations in a matrix in natural way. At this point only an intuitive description of this rather than a mathematical description will be given. For an example, consider the bottom left matrix in figure 4, which shows the adjacency relation between the livingroom and vestibule. At the second level, there are four groups, consisting of the relations in row 2, row 12, column 2 and column 12. These correspond to the livingroom being adjacent to the vestibule at its north, south, west and east. In his way, it is a natural way of dividing the relations into groups which can be automated. This level is not meaningful for all relations. It may be non-existent. The top level of abstraction for the same matrix considers all relations in it. In this case, it specifies that the two objects are overlapping or touching. A relation at this level does not exist if the relations span all rows or columns. Table 1. Relations between the same directional lines of two rectangles l1 l3 > >= <= < l4 > >= <= < l2 > >= <= < > >= <= < Other quantitative parameters are the length and width of the objects. These are the distances from one edge of the rectangle to the other. All the qualitative rectangle relations, together with the quantitative parameters above can be uniformly represented as algebraic inequalities by bounded difference constraints [BF 97]. A bounded difference constraint is an inequality of the form l1 - l2 <= d. The bounded difference constraints are between lines in the same direction, i.e., horizontal or vertical. Thus reasoning about distances in the two directions are separate. There are 16 equations between the lines of two rectangles when the equations are in the form of bounded difference equations. The 16 possible relations are shown in table 1. Let l1 and l2 be the horizontal lines of rectangle 1 and l3 and l4 be the horizontal lines of rectangle 2. The possible relations between them that can be represented as bounded difference constraints are shown in the cells of the table. Thus all the rectangle relations in the 13x13 matrix can be represented algebraically by the 16 horizontal and 16 vertical equations. The

9 abstractions at the higher levels, when they exist, are also described by the same inequalities. Topology and geometry, i.e., distances and dimensions, can be uniformly represented as algebraic constraints. It is more efficient to be able to use them together for reasoning, rather than to split topology and geometry [BF 97]. 5 A User Interface for Design Requirements The mode of operation is that the designer defines the constraints and the program finds the solutions. The designer should understand the implications of the constraints on the solutions through feedback from the layout program. The design process proceeds iteratively as two searches, the search for the right problem specification and the search for the solution. In order for this to work, we need more expressive constraints to enable the designer to specify what he/she wants, and explanation and feedback from the program concerning its reasoning. One mechanism we found very useful is to enable the program to evaluate a configuration drawn by the user in terms of a set of constraints. The proposed graphical user interface for defining layout problems makes use of rectangle relations. The user can select individual relations, rectangular portions, or rows or columns of the matrix by pointing and dragging. The requirements can also be specified by the intersection, union and difference of a basic set of predefined relations such as inside, non-overlap and adjacent. Taking the inverse of a relation, and rotating a matrix such that the horizontal and vertical relations are interchanged are operations that transform a single matrix. We have been able to derive all the spatial relations of interest from a basic set using these operations. 6 Conclusion We tried to define the design requirements of layout problems. There are topological, dimensional and orientation; unary, binary and global constraints. In order to be useful, a layout system should enable the designer to express a wide range of functional and stylistic requirements and preferences, and to restrict the solution set from an exponential number of solutions to a manageable size. As the number of solutions is reduced, a system that can use constraints intelligently can search more efficiently and find the solutions in less time. The layout design process proposed consists of two searches; the search for the set of constraints and the search for configurations. This requires two way interaction between designer and program that is graphical and intuitive. The representation of design requirements independent of solution methods is useful for comparing solution methods and the set of solutions found by a program. The solutions are not given but are constructed by a program based on the representations that it uses. Therefore the set of significantly different solutions to a problem differ among programs. Similarly, the requirements formulated for different programs may mean different things even though the names of the spatial relations used may be the same. 7 Bibliography [ALL 83] Allen, J.F Maintaining Knowledge about Temporal Intervals, Communications of the ACM 26, pp

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