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Logic based design modeling with shape algebras
ELSEVIER
Automation
in Construction
6 (1997) 31 l-322
Logic based design modeling with shape algebras Scott C. Chase
*
National Instituteof Standardsand Technology, Manufacturing Engineering Laboratory, Gaithersburg, MD 20899, USA
Abstract A new method of describing designs by combining the paradigms of shape algebras and predicate logic representations is presented. Representing shapes and spatial relations in logic provides a natural, intuitive method of developing complete computer systems for reasoning about designs. The advantages of shape algebra formalisms over more traditional representations of geometric objects are discussed. The method employed involves the definition of a large set of high level design relations from a small set of simple structures and spatial relations. Examples in architecture and geographic information systems are illustrated. 0 1997 Published by Elsevier Science B.V.
1. Background 1.1. Reductionist
separate parts which, in various combinations, make up the whole: us. holistic approaches
to design
Evolutionary models supporting dynamic schema modification are a necessary requirement in the development of future CAD systems [ll. Current systems development does not appear to support such
models, in that they are constructed using a bottom-up ‘kit-of-parts’ approach, beginning with the design of low level data structures and operations. While this method is generally -used to facilitate efficient object manipulation, it also forces the designer or user into a specific manner of representing and manipulating those objects. What this does is fix the model’s structure at the beginning of design. This can be viewed as a reductionist philosophy of design, in which the design is perceived as a composition of
* Corresponding author. As of January Design Computing, Uniwrsity of Sydney, Australia, E-mail: [email protected]
1998, Key Centre of Sydney, NSW 2006,
It is a natural human tendency to separate a whole into its parts, to categorize and classify, to draw boundaries between parts, and to define classes on the basis of rigidly defined boundaries. Boundaries so defined may be useful for some purposes, but they may badly confuse the accomplishment of other purposes [2]. By fixing the structure of a data model at the beginning of the modeling process, the possibility of other desirable forms in the future may be precluded. It is extremely difficult, if not impossible, to anticipate all possible ways in which one might wish to view or classify parts of a model, due to the generally unmanageable amount of information required. This can be seen as one of the causes for the failure of early CAD systems for buildings in the 1970’s and early ’80s which generally required the predetermination of all types of information of interest, and for this information to be stored in a single model [3].
0926-5805/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SO926-5805(97)00048-4
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Holistic approaches, which work from the whole rather than the parts, don’t restrict the user to a specific structure. This enables the emergence of features and properties unanticipated at the beginning of design. Such a ‘top-down’ approach to modeling can enable a more flexible design development path, i.e. from the abstract to the specific. Shape algebras [4] support both holistic and reductionist approaches. By considering shapes as finite sets of elements which can carry fixed properties, a ‘kit-of-parts’ reductionist view is supported. The shape elements themselves are defined in such a manner as to enable the emergence of features which are not apparent in the initial formulation of a shape, thus supporting a holistic view. In addition, the generality of their representations, their reliance upon a minimum of structure, and their use in combination can provide the semantic richness needed in design. The utility of these algebras has been demonstrated with their use in shape grammars [51, production systems generating languages of designs. In Section 2 we demonstrate how emergence is supported in these representations. 1.2. Shape grammars Shape grammars have been shown to be a powerful means of design generation and stylistic analysis. Grammars have been developed which encapsulate styles of designs in areas such as fine arts [6], architecture [7] and landscape design [8]. Recent work includes the development of grammars which generate new languages of designs [9]. Although shape grammars have a formal mathematical foundation, their use continues primarily in paper and pencil exercises. In other words, shapes and spatial relations are described by drawing, thus limiting their use primarily to non-parametric shapes. Consequently, most attempts at representing parametric shapes and grammars have consisted of natural language descriptions of constraints and rule application. The dearth of computer implementations can be seen as due to problems of computational complexity. Because of this, computer implementations of shape grammars (and indeed, design systems in general) tend to have many restrictions and simplifications of the formal representations in order to reduce or eliminate computational problems. This
generally has the effect of limiting practical application [lo- 131.
their
use in
1.3. Logic based modeling systems The development of computer systems for design reasoning can be facilitated by representing shapes and spatial relations in first order predicate logic. This provides a natural, intuitive method of specifying the symbolic constraints inherent in definitions of parametric shapes and spatial relations. The use of logic as a specification and programming tool has become widespread over the past two decades, initially with the Prolog language and continuing with constraint logic programming languages [ 141. Among the advantages over more traditional procedural programming methods is the ability to describe the knowledge to be encapsulated in a model without the need to prescribe data manipulation procedures [15]. Such an approach facilitates top-down development, from the abstract to the specific. This is possible because the symbolic abstractions of logic formulations enable one to denote entire classes of data structures and procedures while ignoring their details. This can be a more natural method of development than having to deal with often unintuitive formulations. Use of a logic based paradigm is not new to design; general surveys may be found in Refs. [16,17]. Shape grarnmar and geometric reasoning systems based in logic include Refs. [10,18-20,121. However, such systems suffer from significant weaknesses: those of the shape grammar implementations are in their limited utility due to restrictions on rule application in order to reduce computational complexity; those of other logic based systems are in their representations of shape, which-with few exceptions-cannot support emergence. I .4. Approach Any data modeling system consists of both data representations as well as the means to manipulate that data. This research focuses on representational issues. The approach taken is of modeling designs using spatial relations based upon shape algebraic representations. First principles of geometry, topology and logic are used to develop a formal model of
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shape and spatial relations. Logic specifications extend the formalisms of shape algebras, making possible more precise, generalized, parametric definitions of shape and spatial relations. These relations can be used to describe designs in more ways than simply geometrical composition: they have the potential to represent behavioral, psychological and cultural issues. The model’s value and its advantages over more traditional models of shape is shown through a number of examples utilizing generalized spatial relations in typical spaltial reasoning problems. The remainder of the paper is organized as follows. Section 2 offers an introduction to the shape algebraic representations used and illustrates how they support feature emergence. Section 3 shows how shapes and spatial relations can be formally defined in logic. Sec:tion 4 offers examples from the domains of architectural plans and geographic information systems. Section 5 discusses the problems inherent in a computer implementation of the model, and Section 6 offers a summary and conclusions.
2. Shape Algebras Shape grammars are capable of generating a large variety of designs from a small set of production rules. This is possible because the properties of shape algebras enable emergence, allowing unanticipated forms to be found within existing shapes. In Section 2.1 we demonstrate how these properties are manifested in algebras of lines. 2.1, An algebra of lines Shapes are defined as finite sets of geometric elements (also called basic elements). The elements typically of interest to designers are points, lines, two dimensional regions and three dimensional solids. Each element type is manipulated in its own algebra. We focus here on algebras of lines and shapes composed of lines. A line has finite, nonzero length. It may be decomposed into its parts, which consist of other lines embedded within it. There are an infinite number of lines which may be part of a given line. We define the part-of relation I to describe this situa-
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tion, e.g., I, I I, means that line 1, is embedded in or part of line I,. Collinear lines which overlap in any part or share an endpoint may combine via reduction rules for + (sum), - (difference) and . (product) to form new lines [lo]. A shape consists of a set of lines which are maximal, i.e., no line in the shape contains any parts which are part of any other line in the shape. Consequently, no two lines in the shape can be combined to form a single line. As shapes are simply finite sets of maximal lines, the relation I and the operations +, - and . are also defined for shapes by repeated application of the reduction rules for lines (Fig. la). The relation I between shapes is called the subshape relation. It should be noted that this representation is very different from the shape representations typically used in vector-based computer graphics systems (with the possible exception of brep solid models). In such systems shapes are usually represented as sets of lines which cannot be further decomposed, i.e., no parts of lines can be easily recognized (Fig. lb). This difference is significant, as use of the maximal line representation allows the recognition of emergent subshapes, which are defined as subshapes containing at least one maximal line which is not maximal in the original shape (Figs. 2 and 3). Assuming a maximal line representation, there are 22 instances of Euclidean transformations (compositions of translation, scaling, rotation or mirroring) of shape A which are subshapes of S in Fig. 2. If, instead, we assume A and S are sets of non-decomposable line segments corresponding to the maximal lines, there are IU) subsets of transformations r of A in S. In order to identify the same subshapes, A and S have to be represented by different sets of non-decomposable line segments A’ and S’, using a larger number of line segments (Fig. 3). In order to find other subshapes, we never have to modify the maximal line representation but may need to modify the non-decomposable segment representation. For example, the shape B in Fig. 3, represented with maximal lines, is a subshape of S. If, however, it is represented with non-decomposable lines, it is not a subshape of S. Thus, predetermination of desirable subshapes before fixing a shape’s representation is a necessity when using non-decomposable lines. It is this combination of non-predetermination of
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A
0 * A+B
in Construction 6 (1997) 31 l-322 C
B
‘y
q
A-B
cl *
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A-B
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1. in algebras. A, and are. of Note: the symbol * shown here and in other figures represents accompanying text are not considered part of the shape itself.
lines; A, B and C are sets of non-decomposable line segments. a position reference marker for use in comparing two shapes. It and any
structure with a minimal representation producing maximum expression that makes the maximal line representation so appealing. In effect, one does not have to worry about schema modification, as an infinite number of features can be discovered at any time during the design process. With traditional representations, the potential for schema modification is a major concern. A: 4 maximal lines
b
S: S maximal lines
PI
2.2. Algebras of other geometries Algebras supporting emergence in a manner similar to the maximal line algebra can also be defined for geometric objects of other dimensions, e.g., planar regions (Fig. 41, and solids [4]. These algebras can also be extended to support non-geometric attributes of shape, e.g., material properties, function, and cost [21]. AEC applications generally have strong requirements for the simultaneous use of multiple representations [22]. To that end, different algebras can com-
A’: 5
segments
8: a subshape of S (maximal lines) nc4 a subshape of S’ (non-decomposable lines)
r
s:
30 segments 12
Fig. 2. The 22 possible subshapes transformations of A.
of S that can be produced
by
-
I-I-I
*-
*
4
*
’-I
14 --
*
Fig. 3. Line segments needed for a possible representation shapes in Fig. 2 using non-decomposable lines.
of the
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in Construction 6 (1997) 311-322
B
3.1. Algebras of basic elements
b rl * * A+B
A-B
315
B-A
: * Fig. 4. Examples of operations
in an algebra of planar regions.
bine to produce new algebras of shape containing multiple types of elements, and grammars written to manipulate shapes in these compound algebras [21].
3. Logic definitions of shape and spatial relations In this section, we define shape algebras and spatial relations by using the non-graphic symbolism of logic. In doing so, a solid mathematical foundation is established which enables generalized but precise definitions of shape and spatial relations applicable to shapes of any dimension and facilitates reasoning about these shapes in a design context. This also provides an extensible framework for adding new shape algebras and spatial relations to capture additional properties of designs. Our focus is mainly on topological descriptions of shape and spatial relations, with the assumption that an underlying data :structure is implementable for the geometric description of shape. We recognize that the problems of low-level geometric computation have been researched by others and that solutions exist which are adequate to support investigation of the issues here. The amount of research in computational geometry and computer graphics over the past 30 years tends to support this argument. The bulk of the author’s research has been spent in the development of logic specifications for definitions of shape and a large set of spatial relations. Only a few simple relations are illustrated here. Complete, detailed definitions may be found in Ref. [231.
Shapes are composed of finite basic elements which are manipulated in algebras Uij, indicating elements of dimension i in a space of dimension j. For example, U,,, and U,, describe, respectively, points and lines in the plane, U,, describes three dimensional (solid) elements in 3-space. Basic elements in Uij are finite and can be distinguished by a boundary and a descriptor. The boundary partitions space between an element’s interior (finite) and its exterior (infinite). It consists of a set of elements in the algebra of next lowest dimension, e.g., points bound lines, lines bound planes, etc. The descriptor provides additional information (as needed) about an element, including its unique carrier, the infinite element in which it is embedded. Only elements with equal descriptors (considered cohyperplanar) may interact in the operations +, * and the relation I . Collinearity and coplanarity are common examples of cohyperplanarity. 3.2. Spatial relations
Spatial relations useful for modeling designs are easily constructed from the definitions of shape and their operations as described above. Boundary/descriptor representations of shape serve as the basis for spatial relations which apply to multiple element types. In this way, spatial relations are parameterized and can apply to shapes in any dimension. As boundary relations between objects (e.g., adjacency, abutting) are a significant aspect of design, we illustrate two such relations, share-boundary and contiguous.
“” n “22
J
m *
*
Fig. 5. Share-boundary relation. The curves (in U,,) boundary (endpoint); the regions (I&) share a portion boundary lines.
share a of their
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been constructed by progressively defining common spatial relations based upon primitive definitions and previously defined relations [23]. In this way one constructs a network of relations with a hierarchical structure. General spatial relations are used to define more application specific ones, as illustrated in Section 4. Interestingly, a particular definition may be at the whim of the designer, as there may be multiple ways to define a relation from a set of existing ones. How a relation is used in practice can have an influence on its definition.
u22 ml
*
*
Fig. 6. Contiguous relation.
A single definition of the relation share-boundary can be constructed for both linear elements and regions (and indeed, elements of higher dimension). Two basic elements A and B share a boundary if the product of their boundaries is non-empty, i.e., their boundaries overlap (Fig. 5). It should be noted that we use an informal logical notation in order to facilitate reading. Variables (capitalized) are assumed to be universally quantified unless otherwise specified.
4. Applications This section offers a few examples of the paradigms described above: shape algebras and spatial relations defined with logic. These are taken from the domains of architecture and geographic information systems, and demonstrate the relative ease with which the spatial relations and operations described above can be used for generating and reasoning about designs, including the ability to represent emergent features.
Share-boundary ( A,B) @ boundary (A) . boundary (B) # 0 From the definition of share-boundary, the relation contiguous can be defined. Two elements are contiguous if they are discrete (i.e., their product (.> is empty) and share a boundary (Fig. 6).
4. I. Floor plans As floor plans are a common architectural representation, they provide a good application domain for the illustration of spatial relations. Following are examples involving the inference of wall centerlines and enclosed spaces from double line wall represen-
Contiguous ( A,B) @ A . B = 0 & share-boundary (A,B) A large set of spatial relations and operations has
3 a) node types
w-
1
L4
w-
b) wall segments defined by a pair of node types
-___
121eb T
IF= *_3E Fc,
Fig. 7. (a) Wall segment intersection nodes (up to four). (b) Some of the 18 configurations dashed line represents the centerline of the wall segment in question.
of wall segments defined by pairs of nodes. The
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u
*
-_)
v *
a
The following conditions constrain wall thicknesses. These are conditions of architectural context, based upon actual design constraints and the drawing scale. Here, distance is a function whose value is the perpendicular distance between the two parallel lines: min-wall-thickness I distance( 1,)E,> I maxwall-thickness min-wall-thickness 5 distance(Z,,l,) I maxwall-thickness A number of elements and relations can be constructed from the explicit conditions. For example, the parallel relation between 1, and 1, can be inferred from the combination of paraZlel(l,,l,) and cohyperplanar(l,,&). The constructed centerline 1, is parallel to l,, with endpoints pI1 and P,~ being midpoints of the lines ( p1 ,ps} and {p2 ,p3} respectively. I, is constructed in a similar manner. The contiguous relation (which here describes lines which share an endpoint) is used to construct the sequence l,,l,,l,,l,,l, forming a chain of connected line segments: contiguous(l,,l,) contiguous( 1,, lb) contiguous( 1,) 1,) contiguous(l,,Z,) There are a number of additional required and inferred spatial relations which hold in this example; they have been omitted here for the sake of brevity.
_,_____[---------i c--__-_--_
Fig. 8. (a) Shape replacement rule (spatial relations described in Fig. floor plan shape generated from similar to (a), based on the shapes
--I
for wall centerline generation 9 and in text). (b) A possible multiple invocations of rules in Fig. 7b.
tations using algebras q, (points, lines and regions in the plane). 4.1.1. Centerline generation There has been a significant amount of research in wall and location identification from plan drawings. Among tbe techniques developed are line following Ref. [24], and constraint management to construct a structural hierarchy of spatial objects such as walls, doors and rooms [25]. Here, the power of the shape algebras and logic specifications facilitates development of the spatial relations needed for such interpretations without concern for control mechanisms. Fig. 7 illustrates spatial relations representing typical configurations of intersecting walls. A wall segment can be constructed by using two node types as the end conditions of the segment. Although tbe nodes are drawn as walls intersecting as perpendicular segments, this is not a necessary condition for the relations to hold. Fig. 8 illustrates a rule which can generate wall centerlines from wall outlines for case l-3-b in Fig. 7 (T-shaped intersection), and a design which can be generated using this rule and similar ones. Fig. 9 and the formulas following illustrate some of the relations necessary for the construction. We illustrate here a few of the necessary spatial conditions for the T-intersection. The points in Fig. 9 . . are in the algebra U,, and the lmes m U,,. First, there are conditions of parallelism, perpendicularity and collinearity (specified formally in [231) which must be satisfied by certain wall segments: paraZleltt,,l,) perpendicular(l,,Z,) parallel(13,15) cohyperplanar( 1,) 1/&
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4.1.2. Recognition of enclosed spaces A desirable feature of an architectural design interpreter is the inference of (implicit) spaces from (explicit) building elements (walls, floors, etc.). Since architectural spaces are three dimensional, plan drawings cannot always capture their volumetric qualities. However, they do in general serve as adequate notational devices for representing space. The
Fig. 9. Shape (with elements centerlines.
labeled) for the l-3-b case of wall
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Fig. 10. Boundary
in Construction 6 f 1997) 3Il-322
lines (U,,) for wall segment regions.
Fig. 14. Some emergent subspaces interest.
Ir_l--+(
I
Fig. 11. Maximal regions (Q,) resulting segment regions into a shape W.
from reduction
of wall
of 0, most of which are not of
I!.!-
*
*
*I
*
Fig. 15. Rules for ‘virtual wall’ construction.
I--
fl
-l
I I I. __-
I I I I
II I I I I LJ
r----I
---I I
I
t_r__--J
I I I I
I I I CL--_-_-___
*
Fig. 12. Convex hull of the shape W.
Fig. 13. Open space 0 = conuex-hull( W) - W.
Fig. 16. ‘Virtual walls’ added.
Fig. 17. Subspaces constructed
from new set of boundary
lines.
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example here simplifies the problem by treating elements embedded in a wall (such as doors and windows) as discontinuities in the wall (where appropriate). The advantage of the shape algebras over more traditional representations becomes evident here, as generation of the spaces involves no more than the basic shape algebraic operations performed on the wall segments identified in the previous example. Some emergent features are also easily found, a difficult or impossible task with other representations. There are several steps to this process, resulting in a figure-ground of wall vs. open space (Figs. 10-13). First, wall segments defined from the construction in Figs. 7-9 are transformed into individual regions, then reduced to a shape W of maximal regions. The shape W is subtracted from its convex hull (the smallest convex region enclosing the shape), producing a shape 0 representing the open space in a figure-ground relationship with W. 4.1.2.1. Subspaces. The shape 0, as a set of maximal regions representing, continuous spaces, may be useful for analyses such as air flow. However, since we tend to perceive much smaller spaces, a method of determining subspaces of the larger space is desirable. Although plans do not offer a complete representation of three dimensional space, it is possible to generate some spatial decomposition of the larger space 0 from plans alone. There are an infinite number of emergent subspaces of 0, since any subshape of 0 which has non-maximal parts of 0 as maximal elements can be considered emergent. Of course, most of these are not of interest (Fig. 14). One method of :identifying these subspaces is to construct ‘virtual walls’, which are extensions of the
general view between srse-9
axial view
319
bounding wall segments of a space (Figs. 15 and 16). The new ‘virtual walls’ combine with existing wall segments to form boundary elements for emergent subspaces (Fig. 17). The combination of contiguous subspaces produces larger subspaces. This representation of space (as regions in U,,) also provides room perimeter information, through the boundary elements of the spatial region. Perimeter wall lengths and surface areas can easily be computed by forming the product (. > of a region’s boundary elements and the shape consisting of wall edges. 4.1.2.2. Emergent views. The representation described above is also capable of capturing the implicit concept of views between spaces. Fig. 18 illustrates this emergent feature as a parameterized shape consisting of portal jamb lines, ‘virtual wall’ lines at these portals, and a subspace region (shaded). Through parametric variation of the axis of alignment between the two portals, one can identify general vistas between spaces as well as more classical enjlades characteristic of the designs of Palladio [261. 4.2. Geographic information systems Spatial relations are an essential part of any geographic information system (GIS), as maps depict simple spatial relations between points, lines and regions. In fact, most spatial reasoning research originates from research in GIS [27]. Unfortunately, most geographers and urban planners work in practice with a fixed set of features and GIS systems utilize data structures which represent only this closed set. Thus, support of emergence is generally not possible. An example of emergence is shown in Fig. 19, in which the spatial relation accessible is derived from
symmetdcallyalignedview (enfilade)
general view II
Fig. 18. Emergent views between spaces defined by portals.
recognized A
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Other work by the author in the area of GIS has involved a comparison of the shape algebra model with a typical relational database implementation of a geographic information system [23]. This study showed that in a typical GIS relational database system, all information (i.e., features) of interest (roads, rivers, intersections, bridges, and districts) had to be explicitly entered into the relational tables for points, line segments and polygons. Queries using the relational algebra tend to be nonintuitive and difficult to formulate (Fig. 20a). While the actual computations using the shape algebraic formulations described here may be similar to those of the relational representation, they tend to be hidden from the user; the complexity of computation has been moved from the query language to the inference mechanism. The queries prove to be relatively easy to formulate using such intuitive relations as boundary, within and intersection (Fig. 20b), and allow for the possibility of emergent features, impossible in the more traditional relational representation.
A
c-
? E
\
Fig. 19. Accessibility. each other.
A, B, C, D, and E are all accessible
to
the definition of a continuous shape, i.e. one in which all elements are ‘connected’ (Fig. 19). In other systems, accessibility problems are generally solved by the use of graphs which require explicitly constructed nodes and edges. The connectivity relations must be known before constructing the graph. Here, connectivity is inferred from the spatial relations among the various shape elements, and thus can be considered an emergent feature. Thus, points A and E, and regions B, C, and D are all accessible to each other.
Query: Name the intersections
that are entirely within districts
(not on district boundaries).
a) Relational algebra:
A = Kpoint_label(afeature_type=‘intersection) B = ~left_polygon#right_polygon(line) C = ~beginning_pt(B) ” Xending_pt(B) D=A-C E = ffD.point_label,line.left_polygon(D
w
~D.point_label,line.tight_polygon(D F=
~E.point_label,areafeatwe_name(E
D.point_label=beginning_pt
w w
D.point_label=ending_pt
E.polygon=areapolygon_label
line) u line) area)
Answer = Rpoint.feature_name,F.areafeature_name(F
WI F.point_label=oreapoint_label
Point)
b) Shape algebra: Ans
Fig. 20. Comparison algebras.
= { (District,
Znt) I district22( District) & Znt = {II intersectiongz(Z) & within(l, -within( I, bounuizry( District)) } }
of queries on a GIS database.
(a) Relational
algebraic
District)
&
query; (b) the same query using relations
based on shape
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5. Computer implementations By building a descriptive rather than a prescriptive model, we focus on relations between objects, avoiding issues of low level data structures and algorithm specification. Thus, low level data structures can be developed and modified without affecting the model’s general structure. However, in developing a computer implementation, the issues of computational complexity and control must also be considered. Computations using shape grammars involve the subshape recognition problem, i.e., the determination of applicable rule invocations by searching for occulfences of a rule pattern within a design. The number of applicable subshapes can range from none to an infinite number, in general growing combinatorially with the number of maximal elements in a shape. Therefore, a goal is to reduce the set of possible rule applications to a finite, manageable number. Research by Ref. [13] focuses on presenting a manageable number of choices to the designer when computing with shape grammars. Several ways to accomplish this include placing restrictions on the computational mechanism, numerical representation of shapes, and rules and their application. Thus, implementation invariably requires compromises in the areas of model soundness and completeness by restricting the types of queries and data objects permitted. In addition, the generality of some relations may be sacrificed for algorithm efficiency. Despite these potential problems, it is the author’s belief that developing a model using abstract data structures has great potential. Work on a prototype implementation of the model described here is beginning with the use of the programming language CLP(%) [28], which provides a logic programming environment with the addition of numerical constraints. This involves experimentation with and possible extensions to the GENESZS software described in Ref. [19].
6. Summary Here we have described the construction of an abstract model of shape and spatial relations, based
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on first principles of geometry, topology and logic. The features of this model follows. (1) Shapes defined with shape algebras require minimal predetermination of their structure and can support subshape emergence, thus providing advantages over more traditional representations of shape. (2) Shapes and spatial relations are specified using symbolic logic, thus supporting a precise parameterization schema for shapes of any dimension and description. (3) Spatial relations are constructed using a hierarchical approach, beginning with base primitive operations, and building upon them to generate high level relations and operations which are domain specific. This appears to be a natural and intuitive method of development. (4) By specifying relations symbolically, one is able to focus on high level knowledge rather than implementational issues. (5) Logic formulations are amenable to computer implementation, as evidenced by the abundance of programming languages and database systems based on subsets of first order logic. This work demonstrates that by concentrating on the knowledge to be modeled and not directly on implementation, it is possible to develop more powerful design models. The potential for implementing these models with minimal modification of the model semantics appears to be great. It is hoped that future research will adopt this approach (i.e., focus on the power of the formal model), thereby overcoming the traditional bias of favoring implementation at the cost of representation.
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[5] G. Stiny, Introduction to shape and shape grammars, Environ. Planning B 7 (1980) 343-351. [6] J.L. Kirsch, R.A. Kirsch, The structure of paintings: formal grammar and design, Environ. Planning B: Planning and Design 13 (1986) 163-176. [7] U. Flemming, More than the sum of parts: the grammar of Queen Anne houses, Environ. Planning B: Planning and Design 14 (19871 323-350. [8] G. Stiny, W.J. Mitchell, The grammar of paradise: on the generation of Mughul gardens, Environ. Planning B 7 (1980) 209-226. [9] T.W. Knight, Designing with Grammars, in CAAD Futures ‘91, G.N. Schmitt (Ed.), Vieweg, Wiesbaden, 1992, pp. 33-48. [lo] S.C. Chase, Shapes and shape grammars: from mathematical model to computer implementation, Environ. Planning B: Planning and Design 16 (2) (1989) 215-242. [1 1] R. Krishnamurti, The construction of shapes, Environ. Planning B 8 (1981) 5-40. [12] R. Krishnamurti, C. Giraud, Towards a shape editor: the implementation of a shape generation system, Environ. Planning B: Planning and Design 13 (1986) 391-404. [13] M.A. Tapia, From shape to style, shape grammars: issues in representation and computation, presentation and selection, PhD dissertation, University of Toronto, 1996. [14] F. Benhamou, A. Colmerauer, Constraint Logic Programming: Selected Research, MIT Press, Cambridge, MA, 1993. [15] R.A. Kowalski, Logic for Problem Solving, North-Holland, New York, 1979. [16] R.D. Coyne, M.A. Rosenman, A.D. Radford, M. Balachandran, J.S. Gero, Knowledge-based Design Systems, Addison-Wesley, Reading, MA, 1990.
[17] W.J. Mitchell, The Logic of Architecture, MIT Press, Cambridge, MA, 1990. [18] J.C. Damski, J.S. Gero, A logic-based framework for shape representation, Computer-Aided Design 28 (3) (1996) 169181. [19] J. Heisserman, Generative geometric design, IEEE Comput. Graphics Appl. 14 (2) (1994) 37-45. [20] R. Krishnamurti, The arithmetic of maximal planes, Environ. Planning B: Planning and Design 19 (1992) 43 l-464. [21] G. Stiny, Weights, Environ. Planning B: Planning and Design 19 (1992) 413-430. [22] S.C. Chase, The Use of Multiple Representations to Facilitate Design Interpretation, in ARECDAO ‘93, Barcelona, Spain, March 30-April 1, 1993, pp. 205-217. [23] S.C. Chase, Modeling designs with shape algebras and formal logic, PhD dissertation, University of California, Los Angeles, 1996. 1241 A. Koutamanis, Development of a computerized handbook of architectural plans, PhD dissertation, Debt University of Technology, 1990. [25] J. Chemeff, Knowledge based interpretation of architectural drawings, Report, R90-13 (IESL 90-051, Intelligent Engineering Systems Laboratory, Dept. of Civil Engineering, Massachusetts Institute of Technology, 1990. [26] G. Stiny, W.J. Mitchell, The Palladian grammar, Environ. Planning B 5 (1978) 5-18. 1271 A.U. Frank, W. Kuhn, Spatial Information Theory: A Theoretical Basis for GIS, Springer-Verlag, Berlin, 1995. 1281 J. Jaffar, S. Michaylov, P.J. Stuckey, R.H.C. Yap, The CLP(%) language and system, ACM Trans. Progr. Lang. Syst. 14 (3) (1992) 339-395.
Automation
in Construction
6 (1997) 31 l-322
Logic based design modeling with shape algebras Scott C. Chase
*
National Instituteof Standardsand Technology, Manufacturing Engineering Laboratory, Gaithersburg, MD 20899, USA
Abstract A new method of describing designs by combining the paradigms of shape algebras and predicate logic representations is presented. Representing shapes and spatial relations in logic provides a natural, intuitive method of developing complete computer systems for reasoning about designs. The advantages of shape algebra formalisms over more traditional representations of geometric objects are discussed. The method employed involves the definition of a large set of high level design relations from a small set of simple structures and spatial relations. Examples in architecture and geographic information systems are illustrated. 0 1997 Published by Elsevier Science B.V.
1. Background 1.1. Reductionist
separate parts which, in various combinations, make up the whole: us. holistic approaches
to design
Evolutionary models supporting dynamic schema modification are a necessary requirement in the development of future CAD systems [ll. Current systems development does not appear to support such
models, in that they are constructed using a bottom-up ‘kit-of-parts’ approach, beginning with the design of low level data structures and operations. While this method is generally -used to facilitate efficient object manipulation, it also forces the designer or user into a specific manner of representing and manipulating those objects. What this does is fix the model’s structure at the beginning of design. This can be viewed as a reductionist philosophy of design, in which the design is perceived as a composition of
* Corresponding author. As of January Design Computing, Uniwrsity of Sydney, Australia, E-mail: [email protected]
1998, Key Centre of Sydney, NSW 2006,
It is a natural human tendency to separate a whole into its parts, to categorize and classify, to draw boundaries between parts, and to define classes on the basis of rigidly defined boundaries. Boundaries so defined may be useful for some purposes, but they may badly confuse the accomplishment of other purposes [2]. By fixing the structure of a data model at the beginning of the modeling process, the possibility of other desirable forms in the future may be precluded. It is extremely difficult, if not impossible, to anticipate all possible ways in which one might wish to view or classify parts of a model, due to the generally unmanageable amount of information required. This can be seen as one of the causes for the failure of early CAD systems for buildings in the 1970’s and early ’80s which generally required the predetermination of all types of information of interest, and for this information to be stored in a single model [3].
0926-5805/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SO926-5805(97)00048-4
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Holistic approaches, which work from the whole rather than the parts, don’t restrict the user to a specific structure. This enables the emergence of features and properties unanticipated at the beginning of design. Such a ‘top-down’ approach to modeling can enable a more flexible design development path, i.e. from the abstract to the specific. Shape algebras [4] support both holistic and reductionist approaches. By considering shapes as finite sets of elements which can carry fixed properties, a ‘kit-of-parts’ reductionist view is supported. The shape elements themselves are defined in such a manner as to enable the emergence of features which are not apparent in the initial formulation of a shape, thus supporting a holistic view. In addition, the generality of their representations, their reliance upon a minimum of structure, and their use in combination can provide the semantic richness needed in design. The utility of these algebras has been demonstrated with their use in shape grammars [51, production systems generating languages of designs. In Section 2 we demonstrate how emergence is supported in these representations. 1.2. Shape grammars Shape grammars have been shown to be a powerful means of design generation and stylistic analysis. Grammars have been developed which encapsulate styles of designs in areas such as fine arts [6], architecture [7] and landscape design [8]. Recent work includes the development of grammars which generate new languages of designs [9]. Although shape grammars have a formal mathematical foundation, their use continues primarily in paper and pencil exercises. In other words, shapes and spatial relations are described by drawing, thus limiting their use primarily to non-parametric shapes. Consequently, most attempts at representing parametric shapes and grammars have consisted of natural language descriptions of constraints and rule application. The dearth of computer implementations can be seen as due to problems of computational complexity. Because of this, computer implementations of shape grammars (and indeed, design systems in general) tend to have many restrictions and simplifications of the formal representations in order to reduce or eliminate computational problems. This
generally has the effect of limiting practical application [lo- 131.
their
use in
1.3. Logic based modeling systems The development of computer systems for design reasoning can be facilitated by representing shapes and spatial relations in first order predicate logic. This provides a natural, intuitive method of specifying the symbolic constraints inherent in definitions of parametric shapes and spatial relations. The use of logic as a specification and programming tool has become widespread over the past two decades, initially with the Prolog language and continuing with constraint logic programming languages [ 141. Among the advantages over more traditional procedural programming methods is the ability to describe the knowledge to be encapsulated in a model without the need to prescribe data manipulation procedures [15]. Such an approach facilitates top-down development, from the abstract to the specific. This is possible because the symbolic abstractions of logic formulations enable one to denote entire classes of data structures and procedures while ignoring their details. This can be a more natural method of development than having to deal with often unintuitive formulations. Use of a logic based paradigm is not new to design; general surveys may be found in Refs. [16,17]. Shape grarnmar and geometric reasoning systems based in logic include Refs. [10,18-20,121. However, such systems suffer from significant weaknesses: those of the shape grammar implementations are in their limited utility due to restrictions on rule application in order to reduce computational complexity; those of other logic based systems are in their representations of shape, which-with few exceptions-cannot support emergence. I .4. Approach Any data modeling system consists of both data representations as well as the means to manipulate that data. This research focuses on representational issues. The approach taken is of modeling designs using spatial relations based upon shape algebraic representations. First principles of geometry, topology and logic are used to develop a formal model of
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shape and spatial relations. Logic specifications extend the formalisms of shape algebras, making possible more precise, generalized, parametric definitions of shape and spatial relations. These relations can be used to describe designs in more ways than simply geometrical composition: they have the potential to represent behavioral, psychological and cultural issues. The model’s value and its advantages over more traditional models of shape is shown through a number of examples utilizing generalized spatial relations in typical spaltial reasoning problems. The remainder of the paper is organized as follows. Section 2 offers an introduction to the shape algebraic representations used and illustrates how they support feature emergence. Section 3 shows how shapes and spatial relations can be formally defined in logic. Sec:tion 4 offers examples from the domains of architectural plans and geographic information systems. Section 5 discusses the problems inherent in a computer implementation of the model, and Section 6 offers a summary and conclusions.
2. Shape Algebras Shape grammars are capable of generating a large variety of designs from a small set of production rules. This is possible because the properties of shape algebras enable emergence, allowing unanticipated forms to be found within existing shapes. In Section 2.1 we demonstrate how these properties are manifested in algebras of lines. 2.1, An algebra of lines Shapes are defined as finite sets of geometric elements (also called basic elements). The elements typically of interest to designers are points, lines, two dimensional regions and three dimensional solids. Each element type is manipulated in its own algebra. We focus here on algebras of lines and shapes composed of lines. A line has finite, nonzero length. It may be decomposed into its parts, which consist of other lines embedded within it. There are an infinite number of lines which may be part of a given line. We define the part-of relation I to describe this situa-
313
tion, e.g., I, I I, means that line 1, is embedded in or part of line I,. Collinear lines which overlap in any part or share an endpoint may combine via reduction rules for + (sum), - (difference) and . (product) to form new lines [lo]. A shape consists of a set of lines which are maximal, i.e., no line in the shape contains any parts which are part of any other line in the shape. Consequently, no two lines in the shape can be combined to form a single line. As shapes are simply finite sets of maximal lines, the relation I and the operations +, - and . are also defined for shapes by repeated application of the reduction rules for lines (Fig. la). The relation I between shapes is called the subshape relation. It should be noted that this representation is very different from the shape representations typically used in vector-based computer graphics systems (with the possible exception of brep solid models). In such systems shapes are usually represented as sets of lines which cannot be further decomposed, i.e., no parts of lines can be easily recognized (Fig. lb). This difference is significant, as use of the maximal line representation allows the recognition of emergent subshapes, which are defined as subshapes containing at least one maximal line which is not maximal in the original shape (Figs. 2 and 3). Assuming a maximal line representation, there are 22 instances of Euclidean transformations (compositions of translation, scaling, rotation or mirroring) of shape A which are subshapes of S in Fig. 2. If, instead, we assume A and S are sets of non-decomposable line segments corresponding to the maximal lines, there are IU) subsets of transformations r of A in S. In order to identify the same subshapes, A and S have to be represented by different sets of non-decomposable line segments A’ and S’, using a larger number of line segments (Fig. 3). In order to find other subshapes, we never have to modify the maximal line representation but may need to modify the non-decomposable segment representation. For example, the shape B in Fig. 3, represented with maximal lines, is a subshape of S. If, however, it is represented with non-decomposable lines, it is not a subshape of S. Thus, predetermination of desirable subshapes before fixing a shape’s representation is a necessity when using non-decomposable lines. It is this combination of non-predetermination of
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A
0 * A+B
in Construction 6 (1997) 31 l-322 C
B
‘y
q
A-B
cl *
B-A
a)@I *I-I* 1 * r I
AuB
A-B
An
B-A
B
C E AuB? no
0
1. in algebras. A, and are. of Note: the symbol * shown here and in other figures represents accompanying text are not considered part of the shape itself.
lines; A, B and C are sets of non-decomposable line segments. a position reference marker for use in comparing two shapes. It and any
structure with a minimal representation producing maximum expression that makes the maximal line representation so appealing. In effect, one does not have to worry about schema modification, as an infinite number of features can be discovered at any time during the design process. With traditional representations, the potential for schema modification is a major concern. A: 4 maximal lines
b
S: S maximal lines
PI
2.2. Algebras of other geometries Algebras supporting emergence in a manner similar to the maximal line algebra can also be defined for geometric objects of other dimensions, e.g., planar regions (Fig. 41, and solids [4]. These algebras can also be extended to support non-geometric attributes of shape, e.g., material properties, function, and cost [21]. AEC applications generally have strong requirements for the simultaneous use of multiple representations [22]. To that end, different algebras can com-
A’: 5
segments
8: a subshape of S (maximal lines) nc4 a subshape of S’ (non-decomposable lines)
r
s:
30 segments 12
Fig. 2. The 22 possible subshapes transformations of A.
of S that can be produced
by
-
I-I-I
*-
*
4
*
’-I
14 --
*
Fig. 3. Line segments needed for a possible representation shapes in Fig. 2 using non-decomposable lines.
of the
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B
3.1. Algebras of basic elements
b rl * * A+B
A-B
315
B-A
: * Fig. 4. Examples of operations
in an algebra of planar regions.
bine to produce new algebras of shape containing multiple types of elements, and grammars written to manipulate shapes in these compound algebras [21].
3. Logic definitions of shape and spatial relations In this section, we define shape algebras and spatial relations by using the non-graphic symbolism of logic. In doing so, a solid mathematical foundation is established which enables generalized but precise definitions of shape and spatial relations applicable to shapes of any dimension and facilitates reasoning about these shapes in a design context. This also provides an extensible framework for adding new shape algebras and spatial relations to capture additional properties of designs. Our focus is mainly on topological descriptions of shape and spatial relations, with the assumption that an underlying data :structure is implementable for the geometric description of shape. We recognize that the problems of low-level geometric computation have been researched by others and that solutions exist which are adequate to support investigation of the issues here. The amount of research in computational geometry and computer graphics over the past 30 years tends to support this argument. The bulk of the author’s research has been spent in the development of logic specifications for definitions of shape and a large set of spatial relations. Only a few simple relations are illustrated here. Complete, detailed definitions may be found in Ref. [231.
Shapes are composed of finite basic elements which are manipulated in algebras Uij, indicating elements of dimension i in a space of dimension j. For example, U,,, and U,, describe, respectively, points and lines in the plane, U,, describes three dimensional (solid) elements in 3-space. Basic elements in Uij are finite and can be distinguished by a boundary and a descriptor. The boundary partitions space between an element’s interior (finite) and its exterior (infinite). It consists of a set of elements in the algebra of next lowest dimension, e.g., points bound lines, lines bound planes, etc. The descriptor provides additional information (as needed) about an element, including its unique carrier, the infinite element in which it is embedded. Only elements with equal descriptors (considered cohyperplanar) may interact in the operations +, * and the relation I . Collinearity and coplanarity are common examples of cohyperplanarity. 3.2. Spatial relations
Spatial relations useful for modeling designs are easily constructed from the definitions of shape and their operations as described above. Boundary/descriptor representations of shape serve as the basis for spatial relations which apply to multiple element types. In this way, spatial relations are parameterized and can apply to shapes in any dimension. As boundary relations between objects (e.g., adjacency, abutting) are a significant aspect of design, we illustrate two such relations, share-boundary and contiguous.
“” n “22
J
m *
*
Fig. 5. Share-boundary relation. The curves (in U,,) boundary (endpoint); the regions (I&) share a portion boundary lines.
share a of their
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been constructed by progressively defining common spatial relations based upon primitive definitions and previously defined relations [23]. In this way one constructs a network of relations with a hierarchical structure. General spatial relations are used to define more application specific ones, as illustrated in Section 4. Interestingly, a particular definition may be at the whim of the designer, as there may be multiple ways to define a relation from a set of existing ones. How a relation is used in practice can have an influence on its definition.
u22 ml
*
*
Fig. 6. Contiguous relation.
A single definition of the relation share-boundary can be constructed for both linear elements and regions (and indeed, elements of higher dimension). Two basic elements A and B share a boundary if the product of their boundaries is non-empty, i.e., their boundaries overlap (Fig. 5). It should be noted that we use an informal logical notation in order to facilitate reading. Variables (capitalized) are assumed to be universally quantified unless otherwise specified.
4. Applications This section offers a few examples of the paradigms described above: shape algebras and spatial relations defined with logic. These are taken from the domains of architecture and geographic information systems, and demonstrate the relative ease with which the spatial relations and operations described above can be used for generating and reasoning about designs, including the ability to represent emergent features.
Share-boundary ( A,B) @ boundary (A) . boundary (B) # 0 From the definition of share-boundary, the relation contiguous can be defined. Two elements are contiguous if they are discrete (i.e., their product (.> is empty) and share a boundary (Fig. 6).
4. I. Floor plans As floor plans are a common architectural representation, they provide a good application domain for the illustration of spatial relations. Following are examples involving the inference of wall centerlines and enclosed spaces from double line wall represen-
Contiguous ( A,B) @ A . B = 0 & share-boundary (A,B) A large set of spatial relations and operations has
3 a) node types
w-
1
L4
w-
b) wall segments defined by a pair of node types
-___
121eb T
IF= *_3E Fc,
Fig. 7. (a) Wall segment intersection nodes (up to four). (b) Some of the 18 configurations dashed line represents the centerline of the wall segment in question.
of wall segments defined by pairs of nodes. The
XC. Chase/Automation in Construction 6 (1997) 311-322
u
*
-_)
v *
a
The following conditions constrain wall thicknesses. These are conditions of architectural context, based upon actual design constraints and the drawing scale. Here, distance is a function whose value is the perpendicular distance between the two parallel lines: min-wall-thickness I distance( 1,)E,> I maxwall-thickness min-wall-thickness 5 distance(Z,,l,) I maxwall-thickness A number of elements and relations can be constructed from the explicit conditions. For example, the parallel relation between 1, and 1, can be inferred from the combination of paraZlel(l,,l,) and cohyperplanar(l,,&). The constructed centerline 1, is parallel to l,, with endpoints pI1 and P,~ being midpoints of the lines ( p1 ,ps} and {p2 ,p3} respectively. I, is constructed in a similar manner. The contiguous relation (which here describes lines which share an endpoint) is used to construct the sequence l,,l,,l,,l,,l, forming a chain of connected line segments: contiguous(l,,l,) contiguous( 1,, lb) contiguous( 1,) 1,) contiguous(l,,Z,) There are a number of additional required and inferred spatial relations which hold in this example; they have been omitted here for the sake of brevity.
_,_____[---------i c--__-_--_
Fig. 8. (a) Shape replacement rule (spatial relations described in Fig. floor plan shape generated from similar to (a), based on the shapes
--I
for wall centerline generation 9 and in text). (b) A possible multiple invocations of rules in Fig. 7b.
tations using algebras q, (points, lines and regions in the plane). 4.1.1. Centerline generation There has been a significant amount of research in wall and location identification from plan drawings. Among tbe techniques developed are line following Ref. [24], and constraint management to construct a structural hierarchy of spatial objects such as walls, doors and rooms [25]. Here, the power of the shape algebras and logic specifications facilitates development of the spatial relations needed for such interpretations without concern for control mechanisms. Fig. 7 illustrates spatial relations representing typical configurations of intersecting walls. A wall segment can be constructed by using two node types as the end conditions of the segment. Although tbe nodes are drawn as walls intersecting as perpendicular segments, this is not a necessary condition for the relations to hold. Fig. 8 illustrates a rule which can generate wall centerlines from wall outlines for case l-3-b in Fig. 7 (T-shaped intersection), and a design which can be generated using this rule and similar ones. Fig. 9 and the formulas following illustrate some of the relations necessary for the construction. We illustrate here a few of the necessary spatial conditions for the T-intersection. The points in Fig. 9 . . are in the algebra U,, and the lmes m U,,. First, there are conditions of parallelism, perpendicularity and collinearity (specified formally in [231) which must be satisfied by certain wall segments: paraZleltt,,l,) perpendicular(l,,Z,) parallel(13,15) cohyperplanar( 1,) 1/&
317
4.1.2. Recognition of enclosed spaces A desirable feature of an architectural design interpreter is the inference of (implicit) spaces from (explicit) building elements (walls, floors, etc.). Since architectural spaces are three dimensional, plan drawings cannot always capture their volumetric qualities. However, they do in general serve as adequate notational devices for representing space. The
Fig. 9. Shape (with elements centerlines.
labeled) for the l-3-b case of wall
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Fig. 10. Boundary
in Construction 6 f 1997) 3Il-322
lines (U,,) for wall segment regions.
Fig. 14. Some emergent subspaces interest.
Ir_l--+(
I
Fig. 11. Maximal regions (Q,) resulting segment regions into a shape W.
from reduction
of wall
of 0, most of which are not of
I!.!-
*
*
*I
*
Fig. 15. Rules for ‘virtual wall’ construction.
I--
fl
-l
I I I. __-
I I I I
II I I I I LJ
r----I
---I I
I
t_r__--J
I I I I
I I I CL--_-_-___
*
Fig. 12. Convex hull of the shape W.
Fig. 13. Open space 0 = conuex-hull( W) - W.
Fig. 16. ‘Virtual walls’ added.
Fig. 17. Subspaces constructed
from new set of boundary
lines.
SC. Chase/Automation
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example here simplifies the problem by treating elements embedded in a wall (such as doors and windows) as discontinuities in the wall (where appropriate). The advantage of the shape algebras over more traditional representations becomes evident here, as generation of the spaces involves no more than the basic shape algebraic operations performed on the wall segments identified in the previous example. Some emergent features are also easily found, a difficult or impossible task with other representations. There are several steps to this process, resulting in a figure-ground of wall vs. open space (Figs. 10-13). First, wall segments defined from the construction in Figs. 7-9 are transformed into individual regions, then reduced to a shape W of maximal regions. The shape W is subtracted from its convex hull (the smallest convex region enclosing the shape), producing a shape 0 representing the open space in a figure-ground relationship with W. 4.1.2.1. Subspaces. The shape 0, as a set of maximal regions representing, continuous spaces, may be useful for analyses such as air flow. However, since we tend to perceive much smaller spaces, a method of determining subspaces of the larger space is desirable. Although plans do not offer a complete representation of three dimensional space, it is possible to generate some spatial decomposition of the larger space 0 from plans alone. There are an infinite number of emergent subspaces of 0, since any subshape of 0 which has non-maximal parts of 0 as maximal elements can be considered emergent. Of course, most of these are not of interest (Fig. 14). One method of :identifying these subspaces is to construct ‘virtual walls’, which are extensions of the
general view between srse-9
axial view
319
bounding wall segments of a space (Figs. 15 and 16). The new ‘virtual walls’ combine with existing wall segments to form boundary elements for emergent subspaces (Fig. 17). The combination of contiguous subspaces produces larger subspaces. This representation of space (as regions in U,,) also provides room perimeter information, through the boundary elements of the spatial region. Perimeter wall lengths and surface areas can easily be computed by forming the product (. > of a region’s boundary elements and the shape consisting of wall edges. 4.1.2.2. Emergent views. The representation described above is also capable of capturing the implicit concept of views between spaces. Fig. 18 illustrates this emergent feature as a parameterized shape consisting of portal jamb lines, ‘virtual wall’ lines at these portals, and a subspace region (shaded). Through parametric variation of the axis of alignment between the two portals, one can identify general vistas between spaces as well as more classical enjlades characteristic of the designs of Palladio [261. 4.2. Geographic information systems Spatial relations are an essential part of any geographic information system (GIS), as maps depict simple spatial relations between points, lines and regions. In fact, most spatial reasoning research originates from research in GIS [27]. Unfortunately, most geographers and urban planners work in practice with a fixed set of features and GIS systems utilize data structures which represent only this closed set. Thus, support of emergence is generally not possible. An example of emergence is shown in Fig. 19, in which the spatial relation accessible is derived from
symmetdcallyalignedview (enfilade)
general view II
Fig. 18. Emergent views between spaces defined by portals.
recognized A
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Other work by the author in the area of GIS has involved a comparison of the shape algebra model with a typical relational database implementation of a geographic information system [23]. This study showed that in a typical GIS relational database system, all information (i.e., features) of interest (roads, rivers, intersections, bridges, and districts) had to be explicitly entered into the relational tables for points, line segments and polygons. Queries using the relational algebra tend to be nonintuitive and difficult to formulate (Fig. 20a). While the actual computations using the shape algebraic formulations described here may be similar to those of the relational representation, they tend to be hidden from the user; the complexity of computation has been moved from the query language to the inference mechanism. The queries prove to be relatively easy to formulate using such intuitive relations as boundary, within and intersection (Fig. 20b), and allow for the possibility of emergent features, impossible in the more traditional relational representation.
A
c-
? E
\
Fig. 19. Accessibility. each other.
A, B, C, D, and E are all accessible
to
the definition of a continuous shape, i.e. one in which all elements are ‘connected’ (Fig. 19). In other systems, accessibility problems are generally solved by the use of graphs which require explicitly constructed nodes and edges. The connectivity relations must be known before constructing the graph. Here, connectivity is inferred from the spatial relations among the various shape elements, and thus can be considered an emergent feature. Thus, points A and E, and regions B, C, and D are all accessible to each other.
Query: Name the intersections
that are entirely within districts
(not on district boundaries).
a) Relational algebra:
A = Kpoint_label(afeature_type=‘intersection) B = ~left_polygon#right_polygon(line) C = ~beginning_pt(B) ” Xending_pt(B) D=A-C E = ffD.point_label,line.left_polygon(D
w
~D.point_label,line.tight_polygon(D F=
~E.point_label,areafeatwe_name(E
D.point_label=beginning_pt
w w
D.point_label=ending_pt
E.polygon=areapolygon_label
line) u line) area)
Answer = Rpoint.feature_name,F.areafeature_name(F
WI F.point_label=oreapoint_label
Point)
b) Shape algebra: Ans
Fig. 20. Comparison algebras.
= { (District,
Znt) I district22( District) & Znt = {II intersectiongz(Z) & within(l, -within( I, bounuizry( District)) } }
of queries on a GIS database.
(a) Relational
algebraic
District)
&
query; (b) the same query using relations
based on shape
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5. Computer implementations By building a descriptive rather than a prescriptive model, we focus on relations between objects, avoiding issues of low level data structures and algorithm specification. Thus, low level data structures can be developed and modified without affecting the model’s general structure. However, in developing a computer implementation, the issues of computational complexity and control must also be considered. Computations using shape grammars involve the subshape recognition problem, i.e., the determination of applicable rule invocations by searching for occulfences of a rule pattern within a design. The number of applicable subshapes can range from none to an infinite number, in general growing combinatorially with the number of maximal elements in a shape. Therefore, a goal is to reduce the set of possible rule applications to a finite, manageable number. Research by Ref. [13] focuses on presenting a manageable number of choices to the designer when computing with shape grammars. Several ways to accomplish this include placing restrictions on the computational mechanism, numerical representation of shapes, and rules and their application. Thus, implementation invariably requires compromises in the areas of model soundness and completeness by restricting the types of queries and data objects permitted. In addition, the generality of some relations may be sacrificed for algorithm efficiency. Despite these potential problems, it is the author’s belief that developing a model using abstract data structures has great potential. Work on a prototype implementation of the model described here is beginning with the use of the programming language CLP(%) [28], which provides a logic programming environment with the addition of numerical constraints. This involves experimentation with and possible extensions to the GENESZS software described in Ref. [19].
6. Summary Here we have described the construction of an abstract model of shape and spatial relations, based
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on first principles of geometry, topology and logic. The features of this model follows. (1) Shapes defined with shape algebras require minimal predetermination of their structure and can support subshape emergence, thus providing advantages over more traditional representations of shape. (2) Shapes and spatial relations are specified using symbolic logic, thus supporting a precise parameterization schema for shapes of any dimension and description. (3) Spatial relations are constructed using a hierarchical approach, beginning with base primitive operations, and building upon them to generate high level relations and operations which are domain specific. This appears to be a natural and intuitive method of development. (4) By specifying relations symbolically, one is able to focus on high level knowledge rather than implementational issues. (5) Logic formulations are amenable to computer implementation, as evidenced by the abundance of programming languages and database systems based on subsets of first order logic. This work demonstrates that by concentrating on the knowledge to be modeled and not directly on implementation, it is possible to develop more powerful design models. The potential for implementing these models with minimal modification of the model semantics appears to be great. It is hoped that future research will adopt this approach (i.e., focus on the power of the formal model), thereby overcoming the traditional bias of favoring implementation at the cost of representation.
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