Design Computing and Cognition DCC’12. J.S. Gero (ed), pp. xx-yy. © Springer 2012

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Representing 3D Shape Grammars in a Generative Product Design System

Jia Cui and Ming-Xi Tang The Hong Kong Polytechnic University, Hong Kong

A new representation of 3D shape, referred to as Dynamic Shape Representation, is introduced in this paper. This representation is aimed at better supporting the generative application of shape grammars in product design. Inspired from real design process at a cognitive level, shape transforming actions in product design instead of geometric and topologic features of the product form are represented and manipulated in our system. This Dynamic Shape Representation offers better support to generative design of conceptual product forms with flexible shape crea-tions. Two product design examples are presented in this paper in order to evalu-ate the feasibility of this new representation for 3D shape grammar applications in design.

Introduction

Shape Grammars (SG) have been developed for more than three decades since its first introduction by Stiny and Gips [1] in 1972. Its outstanding visual description, recursive usage and shape vague tolerance (emergence) play an important role in design formalization, stylization and simulation. Shape Grammars can build a strong connection between a design domain, which is traditionally recognized as a human dominated area, and comput-er technology, which is the modern logic science, for the development of advanced Computer Aided Design (CAD). Designers are more sensitive about the shapes deriving from semantics, feelings and emotions, than ge-ometric and topologic representations of shape which are the fundamental elements used in shape grammars. There is a need to develop a better inte-gration between shape grammar and conceptual design in order to utilize the mature symbol computation developed in computer sciences.

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A generative product design system can scatter one existing conception into a huge solution space for more alternatives. Working with a shape grammar representation, a generative product design system can enhance the usability of the analyses collected by shape grammars to generate novel design candidates. As early as 1981, Knight[2] proposed two basic ways to use the generative product design method, i.e., changing the partial rules and changing the conditions on the shape parameters. After that, he intro-duced redundant rules [3], and Liew illustrated equivalence rules [4], and Trescak introduced a data-sensitive and unordered production rules [5] en-larging the generative functions in shape grammar applications.

In this paper, we introduce a Dynamic Shape Representation (DSR) which can provide better support not only to design simulation and analy-sis, but also to design creation at a conceptual level. By using the DSR, several logic clues can be found for generative production of 3D conceptu-al designs. A shape grammar in product design using the DSR can record the design process in terms of design actions, and support emergent shapes in a more flexible way than traditional approach which is based mostly on the shapes.

Significance

Designers are those with visual sensitive insights and emotional feelings towards forms, who focus on the shapes, or the ways of transformation. In a design domain, all the design methodologies and design philosophies serve the purpose of finding novel design solutions or new creations. Simply speaking, the status of a design in terms of shape is the most inter-esting issue for a designer to start with. However, for design researchers, the process of generating a good design is more essential than finding the design solution itself. In this context, the design process means the way in which the shapes are transformed from an original concept to a final solu-tion. The dynamic nature of this design process holds more useful infor-mation than the shape itself on the design evolution. This dynamic nature needs to be understood by researchers and modeled in computer based de-sign support systems. When CAD researchers appreciate a design, they will consciously or unconsciously try to find the reasons in terms of how to generate (transform) it from the shapes with which they are familiar.

In the development of shape grammars, the shapes are the focuses. A shape set is one of the four key factors in the definition of Shape Gram-mars[6]. The main inference body of shape grammars, i.e., the rule, is formed based on shape replacement. How to represent a shape can directly

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influence the effectiveness of shape grammar applications. Popular repre-sentations of geometric shapes include: B-Rep (Boundary Representation), CSG (Constructive Solid Geometry), Variation Geometry and Feature Representations. Briefly, B-Rep approach focuses on the boundary infor-mation such as faces, edges and vertices. The CSG approach uses a set of primitives and a set of Boolean operations (the Set operation[7]). Variation modeling allows designers to use formulas to model design components, and feature based approach builds shapes by feature primitives extracted from the design samples.

The current shape representations mostly describe the geometric and topological characteristics of shapes in a static way. In this paper, we use a dynamic viewpoint to approach the shapes, which can help CAD research-ers to find clearer clues for generative production. Adopting the CSG method, we chose the basic primitives with a rule-based approach to de-scribe the generating process, named Dynamic Shape Representation (DSR). The DSR can support more flexible representations through con-trolling the variables embedded in the initial shapes or changing the primi-tive shapes without changing the rules. In the following sections, the de-tails of the definition and the application of the DSR are presented.

Method

Understanding Design Feature

A design cannot merely be explained by logical information, as there are some intangible factors which are more significant than tangible aspects. Some of these intangible factors cannot be fully understood by the re-searchers in AI and Computer Sciences. Therefore, there is an assumption which says that before designating a tool to support design, one should learn from designers first. When developing new tools or computational methods supporting design activities, it is necessary to try to follow the thinking of designers instead of the information process which is more convenient only for computer representation and manipulation.

At conceptual design stage of the design process, there are three essen-tial features governing the complexity of the matter. These are dynamic feature, environment sensitive feature, and multilateral feature. Addressing these features in design research is useful for pushing the design technolo-gy to an advanced level.

Dynamic feature - From an original design conception to the final de-sign solution, there are always dynamic changes until the design process terminates either by a designer’s decision, satisfaction of the objectives, or

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some other reasons. In the design process, a designer’s thinking process develops in highly activating manner. This process may inspire the design-er because of a tiny change in the information and the knowledge that are both evolving. In a shape grammar system, it is therefore difficult to enlist all the potential shapes in advance since many of these shapes may only appear in a designer’s mind as the design process goes on. It is also diffi-cult to model and trace the whole evolving process, as it is most likely to be influenced by some intangible factors, such as the feelings, emotions, experiences, subjectivity, and even chances. A common factor is that the changing will not stop until a final design solution is confirmed. There is therefore a need for a representation scheme capable of not only describing the tangible design shapes in static terms such as geometric or topological features, but also in a dynamic way by the rules of grammatical mecha-nism.

Environment sensitive feature – At conceptual design stage, a designer may change parts of the design not because the parts are unsatisfactory, but because they are unsuitable for the entire design. In this situation, a family of rules instead of a single one can describe the shapes and its variations better in terms of supporting the design thinking rather than simply creat-ing the design results. This is because some better results may be created unexpectedly by the ever changing of design thinking of the designers.

Multilateral feature – A designer’s thinking has unlimited boundary and scope. Even designers themselves sometimes cannot speak clearly how an idea is emerged and why it is so emerged. The same design may require different imaginations of different designers with different backgrounds. With no technical restrictions, if we ask several designers to create one same shape, there will be much more than one way for them to accomplish it. In other words, there are diverse ways which can build the same result. Nevertheless, some of these diverse ways will be better than others in dif-ferent design contexts.

Dynamic Shape Representation

The shape grammar mechanism has powerful analyzing abilities to keep the styles and characters of the design of architecture, product or art appli-cations. Most applications of shape grammar are concerned with simulat-ing some specific designs, and then generatively generating large amounts of similar candidates for designer’s selections. In the rule base of a shape grammar system, every single rule represents one replacement of shape/sub-shape in a shape set. Rule replacement only focuses on the par-tial transformation of a shape, not the entire effect. When we restrict the rule application conditions, the possibilities of generating novel solutions

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will be reduced. When we relax the limit on rule application conditions through a generative product system, large useless productions will be cre-ated which will decrease the effectiveness of the system.

In this paper, we emphasize the three features of design generation pro-cess discussed above and introduce a rule-based shape description on three dimensions, referred to as Dynamic Shape Representation (DSR), with the aim of increasing the efficiency of shape grammar with generative capabil-ity.

The CGA shape[8, 9] (Computer Graphics Architecture), a novel shape grammar for the procedural modeling of CG architecture, produces build-ing shells with high visual quality and geometric details. Complex model-ing can be generated by the operations of several primitives including cyl-inder, rectangle, sphere and so on. Similar with the CGA shape, we use some fundamental primitives as the elements for basic rules, named Ele-mental Rule (ER). In this approach, a shape is composed of several ERs which can be applied in a sequential way. Each single ER is the same as the rule in a general shape grammar process, which describes a partial transformation of the object shape. The formal definition is as follows: Definition 1: Elemental Rule

An Elemental Rule is a finite set of shape rules of the form αàβ, where αand βare the shapes being included in the permitted primitive shape set with an orientation, called ER(i).

Normally, the primitive shape set contains the basics which are easily expressed in a computer system, such as cube, sphere etc. For the consid-eration of feasible implementation in computers, any shapes that can be generated by definite Boolean operations do not belong to the primitive shape set. Definition 2: DSR Initial Shape

A DSR initial shape is a collection of definite primitive shapes by union operation. Definition 3: DSR Shape

A DSR Shape is a finite set of shapes, which are generated by using a family of Elemental Rules in a specific order. A DSR Shape can be formal-ly represented by the family of ERs with the application sequence, {ER(0), ER(1), … ER(i)}.

A simple example is shown in Figure 1. Definition 4: DSR Shape Grammar

A DSR Shape Grammar has four components: (1) S is a finite set of DSR Shapes; (2) L is a finite set of symbols;

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(3) R is a set of rules of ΑàΒ, whereΑis a labeled shape in S, Βis a labeled shape in S+ (S+ ⊇ S); and

(4) I is a labeled DSR Initial Shape.

Figure 1a shows a DSR Initial Shape, which is composed of a union of eight cubes. Figure 1b shows a DSR Shape, following a sequential applica-tion of ER(1), ER(2), ER(3) and ER(4), formally represented as {ER(1), ER(2), ER(3), ER(4)}. Figure 1c is the default direction of all the shapes in the working environment. For 3D shapes, one elemental rule can be divid-ed into several concrete elemental rules based on different directions. For example, ER(1)-ER(4) shown in Figure 1 actually belong to one rule. With different directions in a three-dimensional space, this 3D rule can be divid-ed into 8 different ERs.

From this example, we can know that a DSR shape is not similar to the current status-based shape description. DSR shape is a family of ERs without limiting the size, construction and color of the object shape.

Based on a cognitive theory of computer design introduced by Ramscar, et al [10], in a computer system, all the proposed geometric systems can be a closed-world system aiming at expressing one category or more general real-world objects. Any ‘building’ in the system is represented within ge-

(a) (b)

ER(1) ER(2)

ER(3) ER(4)

(c)

Fig1. Example of Dynamic Shape Representation

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ometrically numeric information. Unless we can generate all the real-world objects by the system, we cannot claim that the system is completely close to the real world. Therefore, it is difficult to prove any ‘closed-world’ in a computer that can strictly correspond to the real world. However, a DSR shape is not a closed system of description. With the DSR, only the trans-forming actions are recorded in the form of Elemental Rules. For the shape set S in the DSR shape grammar, there is no restriction on the configura-tion of initial shapes. Theoretically speaking, any types of shape can be the shapes belonging to S that is operated by the DSR shape grammar. The DSR shape is not focused on static shape description, but the design gener-ating process, which can describe the sequences of design operations for creating a conceptual form or a detailed form.

The DSR shape is an open description. We create a DSR shape on the basis of current classical modeling technique, i.e., CSG shapes. For differ-ent composition of primitives, the application of DSR rules can generate various novel shapes.

The definition of the DSR shape is based on the DSR initial shape. In practical usage, the DSR shapes are not only generated from DSR initial shapes. The Elemental Rules can be applied to any types of initial shapes. So we need to define the details of Elemental Rule application and primi-tive matching in ERs. Definition 5: Primitive Matching.

If Vpi ≥ Vec(pi) * TS, then the primitive(i) is found, named ‘Match’, else primitive(i) is missed, named ‘MisMatch’, where Vpi is the volume of primitive(i), Vec(pi) is the volume of external cube of primitive(i), TS is a threshold ∈(0, 1].

Normally, we can use a cube as the primitive of DSR shape. During DSR shape applications, if the volume of object primitive is less than the required value, then the object primitive will be recognized as missing.

As the DSR is based on the combination of several primitives, it is therefore a shape description but not a design description. There is no need to consider too much of the sub-shape detection problem. Primitive Matching plays the same role as sub-shape detection comparing with the static shape representation approaches. Definition 6: Application of Elemental Rule.

For ER(i) αàβ, if object primitive matches αsuccessfully, then ob-ject primitive will be replaced byβ, else the application of αwill be ig-nored and deleted.

The primitives are already the fundamental elements of the DSR shape. If a primitive is less than the standard requirement, decided by Vec(pi) * TS, then we consider that it will not work for the final DSR shape. This

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primitive will be dismissed. Using the same DSR shape in Figure 1, the three DSR Initial Shape in Figure 2 can generate the same DSR shape after using ER(1)-ER(4).

Considering the practical effect, Definition 5 and Definition 6 will not reduce the function of DSR shape for minute shape transforming. If we de-fine a DSR initial shape with a high granularity (such as 10*10*10), or even higher, then tiny changes can be captured by ERs with efficiency.

Experiment Results

DSR shape can support generative methods to increase the effectiveness in creating design alternatives. In this section, we present two 3D experi-ments to show the usage of DSR shape in grammatical applications.

Case 1: Generative Power of DSR

There is one 5*5*5 cube combination as the DSR Initial Shape. We sup-port three basic rules, Delete Rule, Slice-edge Rule, and Slice-corner Rule. In the 3D coordinate system, the orientation will increase the number of Elemental Rules. So, in a DSR shape application, the orientation factor needs be considered. l Delete Rule. To empty the object primitive.

(a) (b) (c)

(d)

Fig2. DSR shape application

(a) (b) (c) are three different initial shapes, (d) is the DSR shape after applying ER(1)-ER(4) shown in Figure 1, TS = 0.3

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l Slice-edge Rule. To slice the object by one edge of the object

primitive

l Slice-corner Rule. To delete one corner of the object primitive

Concerning the different orientations, there is one ER1 generated from

Delete Rule. There are ER2 to ER13, generated from Slice-edge Rule, as one cube has 12 edges. There are ER14 to ER21, generated from Slice-corner Rule, as one cube has 8 vertices.

The DSR Initial Shape is shown in Figure 3.

After sequentially using Elemental Rules (ER1 – ER21), there are two types of DSR shape shown in Figure 4. Figure 4a and Figure 4b are the balanced shapes, while Figure 4c and Figure 4d are the imbalanced shapes.

Fig3. The DSR Initial Shape for Experiment 1

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There are 45 sequential ERs to generate the balanced DSR shape1. There are 85 sequential ERs to generate the imbalanced DSR shape2. As the Elemental Rule has orientation, when we change the orientation, the results will be different. The primitives of DSR Initial Shape belong to a collection with the inside relationships of each other. Each primitive will only be replaced once by ER application or none replacement. Therefore, when we change the ERs’ application sequence along x-dimension, y-dimension or z-dimension respectively, the replacements will be changed, which will transform the entire results of the DSR shape.

For DSR shape1, we use the first method to change the orientation of the ERs. In Figure 5, there are many new shapes generated based on DSR shape1. Some of those are not interesting enough. However, there are some common characters which are kept. The Slice-corner rule is used to the eight corner areas and the Slice-edge rule is used to the mid part of the body. Because only the external changes of 3D shape can be observed, there are some replacements inside which cannot be shown in Figure 5. From this experiment, we can see that when the orientation is changed, the DSR shape will be transformed. Clearly, the orientation of ER is important to the aesthetic value of the result. The massy feeling of Figure 5 is due to the random orientation selection. In our future research, how to change the orientation in a regular way will be tackled.

(a) (b)

(c) (d)

Fig4. The DSR shape for Experiment 1

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For DSR shape2, we use the second method to generatively create some new shapes. After changing the application sequence along three axial di-rections, some of the results are shown in Figure 6.

Obviously, there are two acceptable ways to generatively produce new shapes based on DSR shape. Furthermore, the DSR approach can increase the novelties of shape generation deriving from the existing solution. DSR shape can also provide more clues to improving the generative production

Fig6. Generative generation of DSR shape2

Fig5. Generative generation of DSR shape1

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method with complex 3D shapes. This is the reason why we want to record the shape generating process, in addition to the geometric information.

Case 2: Application in Shape Grammar

Using DSR shape in shape grammar can also create more alternatives while keeping the design intention. In this section, we illustrate the genera-tion of a tea table design with the DSR shape grammar.

We can create five rules expressed by DSR shape. The ‘cave_surface’ rule is used to generate the tabletop. The ‘chop_width’ rule is used to curve the side of table top. The ‘framework’ rule is used to generate the main structure of the table. The ‘side_leg’ rule is used to cave the shape of table leg. The ‘leg_even’ rule is used to even the table leg.

As the DSR shape records generating actions, not only the geometric ac-tion, we do not need to consider the final result of the tea table when we create the rule. During the rule creating process, the thing that we need to be concerned with is how to transform DSR Initial shape with design in-tention. In this application, all the DSR Initial shapes are based on cube.

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In this application, the initial shape is a cuboid of 10*5*5, shown in Figure 8a, and the standard tea table generation is shown in Figure 8b. With the shape rules generated in the form of DSR shape, we can genera-tively transform the 5 rules to create more new options. There are some new tea table solutions shown in Figure 9.

(a)

(b)

(c)

(d)

(e)

Fig7. DSR Shape rules for tea table generation

(a) ‘cave_surface’ rule; (b) ‘framework’ rule (c) ‘side_leg’ rule; (d) ‘leg_even’ rule (e) ‘chop_width’ rule

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Fig9. Generatively Production of Tea Table

(a)

(b) Fig8. Initial Shape and Tea Table Generation

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Discussions

A single Elemental Rule (ER) in our Dynamic Shape Representation (DSR) represents the shape transformation aiming at modeling a specific design intention. Using a family of ERs to represent and transform a shape is considered a new approach to supporting the dynamic nature of design process.

In DSR, a group of ERs focuses the transforming actions on the initial shape. Different initial shapes using the same ERs will generate variable effects. The essential idea of DSR is to record shape changes in the design process. This will provide more flexible options for the rule developers. Therefore, the DSR shape grammar can be used in design simulation, as well as in the creation of novel design candidates by the designers. Our ul-timate objective is to allow designers to create their own shape grammar rules reflecting their styles or preferences.

For a DSR shape, each primitive is one member of the whole shape set, which means that there are certain relationships among them. When the ERs are created by the rule developers the influence of a single transfor-mation on the entire design is considered. In other words, the group of ERs has some inherent connections, not just the simple union of each single one. In our current research, we have used the sequential order to control these inherent relationships among ERs. Different from parameter-driven CAD technology, the requirement for a DSR shape is only on the final de-sign effect. The ERs are applied with sequences and we do not limit the order of these sequences. Therefore, for the same DSR shape, the represen-tation may be non-unique. This can allow designers create their desired shapes in a free environment. However, the non-uniqueness in the repre-sentation is also a potential problem for design computation, which is be-ing studied further in our research.

Excepting for advanced generative power, another merit of DSR is the support for emergent shapes. Since we only use ERs to transform a shape, applying the same rules to a different initial shape can create some unex-pected design effects. As shown in Figure 10, when we used the DSR rules of standard tea table shown in Figure 8a to the initial shapes of sphere, cyl-inder, and prism, three different effects were generated.

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There are some unsatisfactory characters of our current DSR shape, too. When we use the DSR shape rules to an irregular initial shape, such as

sphere, similar to the one shown in Figure 10d, the rule application mech-anism which is based on shape replacement will create some edges and corners that are unsuitable for maintaining smooth surfaces. How to make the DSR shape rule created based on orthogonal primitives more suitable for curvilinear surface shapes such as sphere, cylinder or other irregular shapes, is a problem for our further investigation.

Because the DSR shape is not only aimed at design simulation, but also design creation, a real-time visualization support will be necessary. A 3D visual shape interpreter is required for further development of our DSR shape grammar system.

DSR shape focuses on the shape interpretation, not the design descrip-tion. The ERs can describe one shape well for the effective support to the applications of shape grammar. For practical applications, however, the DSR shape should be working with a structured shape assembly system in order to support more complex product design tasks.

(a) (b) (c)

(d)

Fig10. Tea Table generated from different Initial Shape

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Conclusions

In this paper, a new representation of 3D shape different from traditional geometric shape description is presented. We propose a Dynamic Shape Representation (DSR) as the basis for reasoning with 3D shapes in a gen-erative system supporting product design. This representation focuses on design actions which can be formulated as shape grammar rules that are applied to a design session in the orders determined by the designers. As such, the DSR supports shape transforming actions, suitable for generative product design with the flexible usage of different kinds of initial shape. Several examples are provided to validate this representation, which have shown promising results that can be further explored. Our current work fo-cuses on how to extend this representation to have more rules that simulate the intention of the designers. Our ultimate objective is for developing generative 3D shape grammars with which designers’ knowledge and styles can be captured, while computational efficiency in the process of design exploration and evaluation is achieved through the DSR by allow-ing designers to create their own rules. At the current stage of the devel-opment of this research project, we are exploring the exploration power of the DSR in complex transformation actions. Our next step will be examin-ing the rules, initial shapes and application orders of rules in order to cre-ate the designs that represent the styles of designers, or the types of prod-ucts.

Acknowledgements

This research is sponsored by the Hong Kong UGC research grant, the general research fund (project number BQ-20L).

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