On the Efﬁcient Graph Representation of Collinear Relation in the Shape

Grammars

Kamila Kotulska and Leszek Kotulski

AGH University of Science and Technology, Mickiewicza 30, 30-059 Krakow, Poland

Keywords:

Shape Grammars, Graph Representation, Computational Model(s), Grammar, Computer-aided Conceptual

Design.

Abstract:

Shape grammars are a powerful, generative approach to description, interpretation and evaluation of many

designs. However, their practical implementation has problems related to computational and spatial efﬁciency.

Because of that, successful examples are restricted to small graphs or those with reduced numbers of rules and

shapes. While executing a project inspired by Antonio Gaudi’s designs, we found those limitations critical and

initiated a series of research tasks to improve of the efﬁciency of their implementation. The most important

task consists in developing an efﬁcient graph representation of the collinear segments. The proposed solution,

based on a classical application of shape grammars – Stiny’s Chinese lattice design – has been compared with

two most popular existing representations.

1 INTRODUCTION

Shape Grammars have been introduced in 1971 by

Stiny (Stiny and Gips, 1971) and further developed

by himself. Stiny deﬁned them as a “set of rules of

transformation applied recursively to an initial form,

generating new forms” (Stiny, 1980). Since then, they

have been used as a powerful generative approach to

description, interpretation and evaluation of many de-

signs. For over 45 years, the notation of the formalism

has been signiﬁcantly changed and developed. De-

spite that, shape grammars were used constantly as a

rule-based system for describing and generating de-

signs (Knight, 1999), (Stiny, 2006).

Shape Grammars are used for creating and un-

derstanding designs directly, by performing compu-

tations with shapes. They have been applied to tackle

a variety of design tasks from analysis to synthesis.

Basing on a classiﬁcation by Terry Knight and George

Stiny (Knight, 2015), which distinguishes seven areas

of application of shape grammars, the following rep-

resentatives can be pointed out in each area.

Painting is an area where shape grammars are used

to analyse transformations of the style of a painting.

Terry Knight (Knight, 1989) has developed her re-

search on transformations of De Stijl Art: The Paint-

ings of Georges Vantongerloo and Fritz Glarner. She

used the formalism to describe stylistic changes in de-

sign.

Shape grammars can be also used for product de-

sign. Jay P. McCormack, Jonathan Cagan and Craig

M. Vogel (Jay P McCormack, 2004) used them to

explore brand identity and showed that they might

have an important role in studio work, engineering

and marketing.

Terry Knight and George Stiny also mentioned the

area of craft. In his work, Rizal Muslimin (Mus-

limin, 2010) restructured weaving performance in ar-

chitecture. He has analysed the tacit knowledge of

traditional weavers through perceptual study and con-

verted it into explicit rules in computational design.

Mechanical design is another area where shape

grammar-based methods have been developed. Agar-

wal, Cagan and Stiny (Agarwal et al., 2000) have

shown that they are capable of generating coupled

forms, by means of so-called function shape gram-

mars. They have achieved that by satisfying the mini-

mal required functionality and then modifying the de-

vice to obtain the desired speciﬁcations.

Stiny and Mitchell (Stiny and Mitchell, 1980) also

used shape grammars for landscape design. They

used parametric shape grammars to design Mughal

gardens.

Jos Pinto Duarte (Duarte, 2005) researched shape

grammars in the area of architecture. He has de-

scribed an interactive computer system for design of

customised mass housing. The role of shape gram-

mars in such a system is to systematise the design

Kotulska K. and Kotulski L.

On the Efﬁcient Graph Representation of Collinear Relation in the Shape Grammars.

DOI: 10.5220/0006503800600066

In Proceedings of the International Conference on Computer-Human Interaction Research and Applications (CHIRA 2017), pages 60-66

ISBN: 978-989-758-267-7

rules.

One of the most developed area of shape grammar

application is urban design. Jos Nuno Beiro (Beirao,

2012) deﬁned all the aspects of using shape gram-

mars in this ﬁeld and successfully applied his theses

in practice. Although he mentioned that rules had be

hidden from the designer, he gave the designer some

freedom by using discernible names for the moves

and their parameters.

Analysis of the mentioned examples leads us to

the conclusion that Shape Grammars can be success-

fully applied in many ﬁelds. However, the graph

structures, which formally represent problems, are

based on small graphs or have limited numbers of

rules.

During our work on the project inspired by Anto-

nio Gaudi’s designs we have noticed a serious prob-

lem with graph processing efﬁciency. We work on

different types of structures, which are described by

large numbers of nodes. The project requires rep-

resentation of thousands or even millions of edges,

nodes and shapes which are not represented by lines.

Even the most advanced among the exiting meth-

ods are incapable to process such vast collections

data. Real-life application of shape grammars de-

mands their efﬁcient implementation for shapes con-

sisting of thousands of elements.

Translation of shapes into graph grammars re-

quires many problems to be solved, including:

• Complexity reduction of designation of the left-

hand side of transformation rules: In a general

case, it is an NP-complete problem, just like sub-

graph isomorphism. But with some restrictions on

the application rule form and the generated graph

structure, it can be solved in a polynomial time

(cf. (Flasinski, 1990)).

• Introduction of a mechanism of parallel applica-

tion of the transformation rules with implicit syn-

chronization; cf. (Kotulski and Sedziwy, 2010).

However, the most basic and yet still open prob-

lem is proper representation of the shape elements.

The shape graph presented in Figure 1 consists of

two overlapping squares (basic elements). In this ex-

ample, the line segment (a, b) has been split into three

collinear segments (the other line segments also have

been split). The representation of relations among

them and segment (a,b) has a strong inﬂuence on the

implementation of the shape grammar.

This problem has been outlined by Grasl and

Economou (Grasl and Economou, 2011), the authors

of one of the best shape graph interpreters called

GRAPE. The efﬁciency of the line segment represen-

tation, with each line consisting of several segments,

will be considered here. The representation proposed

Figure 1: Collinearity problem in Shape Grammars.

in the following part of the paper has been compared

with two others – Grasl (Grasl and Economou, 2011)

and Kelles et al. (Keles et al., 2010) – based on the

classical Stiny shape grammar application for Chi-

nese ice-ray lattice design (Stiny, 1977).

2 COLLINEAR SEGMENTS

IMPLEMENTATION

One of the most important problems associated with

graph representation of shapes is the representation

of collinear shapes in a way that allows their use it

for the analogical relations for Bezir curve elements.

The maximum line segments are a set of lines created

by combining all collinear line segments that touch

or overlap. The most straightforward approach is to

map points to the nodes and segments to the edges. In

such a representation, if a line is divided into several

segments by a crossing line, it is difﬁcult to designate

the other collinear segments. Note that two or more

connected collinear segments can appear in the trans-

formation rule as one segment.

To overcome this problem, Keles et al. (Keles

et al., 2010) join all node pairs in such lines. Unfor-

tunately, this approach results in creation of too many

edges. For n points along the line, n ·(n − 1)/2 edges

have to be created. Thus, even for a small grid of

lines (over a dozen intersecting lines), like the one

in Stiny’s ice-ray (Stiny, 1977), a large number of

edges is generated: 1653 edges for Chengtu Szee-

hwan, 1825 AD and 1392 edges for Hanchow, Szech-

wan 1875 AD. The mentioned examples of Chinese

lattice design creates a good background for evalua-

tion of the efﬁciency of the considered methods.

Grasl and Economou analyse 7 other representa-

Figure 2: Keles’s long line representation.

Figure 3: GRAPE long line representation.

tions and ﬁnally introduce the representation in which

both points and segment lines are represented as

nodes (points as a black nodes and lines as a white

nodes). The collinear 5-segment line in both ap-

proaches is presented in Figure 3 and Figure 2. This

representation is efﬁcient for collinear lines, but we

have to analyse it in the entire context of the generated

shapes. Thus, an intermediate solution has to be con-

sidered and two such approaches are presented here.

3 CHINESE LATTICE DESIGN

Stine (Stiny, 1977) analyses several examples of Chi-

nese lattice design to show the expressiveness and ca-

pabilities of shape grammars. In this paper we will

Figure 4: Chinese Lattice design - Chengtu, Szechwan 1800

AD.

consider the Chengtu Szeehwan, from 1800 AD (see

Figure 4) and from 1825 AD (see Figure 5), as well

as Hanchow, Szechwan from 1875 AD (see Figure 6).

Table 1 shows number of lines consisting from n

collinear nodes in the designs from 1800, 1825 and

1875 years.

Table 1: Number of collinear nodes in the design.

NCN 1800 AD 1825 AD 1875 AD

2 262 42 0

3 0 84 24

4 0 0 4

5 0 0 6

7 0 0 4

9 0 0 3

11 0 0 2

13 0 8 5

15 0 7 0

17 0 0 4

Figure 5: Chinese Lattice design - Chengtu, Szechwan 1825

AD.

Figure 6: Chinese Lattice design - Hanchow, Szechwan

1875 AD.

4 SHAPE GRAPH

REPRESENTATION

During deﬁnition of a shape grammar, we have to de-

ﬁne what kinds of the basic elements can be used as

building blocks for shapes generated by the grammar.

In the Virtual Gaudi project we assume that we will

describe the shape as Bezir curves (linear, quadratic,

cubic or higher order) connected by common nodes.

Such a curve will be denoted by S

(x,y), where x

and y are the nodes representing end points, and A

a set of other (n − 2) attributes deﬁning the Bezier

curve of the order n. For two curves, we denote

(x,y) ⊂ S

(w,z) if all points of the ﬁrst curve be-

long to the second one. For a given n and a set of

attributes A

, we will say that two shapes, are co-n-

bezier shapes (denoted ↔) as if the following condi-

tions are met: for any nodes x,y, u,w,v,z:

• S

(x,y) ↔ S

(u,z) if S

(x,y) ⊂ S

(u,z)

• ↔ relation is closed under symmetry i.e.

(x,y) ↔ S

(u,z) ⇒ S

(u,z) ↔ S

(x,y)

• ↔ relation is closed under transitivity i.e.

(x,y) ↔ S

(u,w) and S

(u,w) ↔ S

(v,z)

⇒ S

(x,y) ↔ S

(v,z)

The S

(x,y) that does not contain any curves ↔ re-

lated with them will be called minimal and its end

nodes will be connected with direct edges.

The shapes grammars use hybrid notation:

• for the presentation layer, the mentioned basic

shapes are used to express the designed item,

• for efﬁcient generation of this item, a more com-

plex graph-based representation (based on graph

transformation rules) is used.

Here, we compare the efﬁciency of several graph

representations according to the number of source

items (nodes and edges). There is a necessity to re-

member the information about the generated shapes;

the number of nodes and edges in the graph are the

basic parameters while considering the computational

complexity of graph-based algorithms.

Let us note that in S

(x,y) representation of the

Bezier curves we separate the notation of end nodes

and A attributes. Thus, in the graph representation we

will represent only these end nodes, while the A at-

tributes will be remembered either as attributes of the

edge representing shapes (in case of Keles’ approach)

or as attributes of the node representing shapes (ac-

cording to Grasl). The advantage of such a represen-

tation is that it is the same for all orders of Bezier

curves. For the simplicity of presentation, we will

illustrate the mentioned representation based on the

co-1-bezier shape called line segments (shortened to

S(x,y)) and collinear shape relation.

As already mentioned, two opposite graph repre-

sentations of the collinearity problem have been con-

sidered in literature. The ﬁrst one has been pro-

posed by Keles (Keles et al., 2010). He mentioned

that shape nodes are represented as graph nodes and

shapes are represented as edges (A attributes are

also remembered in edges). The representation of

collinear segments deﬁend using 5 nodes is illustrated

in Figure 2. We will have 10 collinear segments and

the same number of edges. A more compact represen-

tation suggested by Grasl and Economu (Grasl and

Economou, 2011) introduces white nodes for repre-

sentation of the shape edges. The representation of

collinear segments deﬁned by 5 nodes is illustrated

in Figure 3. The A attributes will be remembered in

white nodes. Two segments are collinear if their end

nodes are connected with the same white node.

The advantage of the Grasl representation is not

obvious in the case of more complex shapes, such as

the one presented in Figure 7.

Figure 7: Complex shape.

Keles’s representation of the shapes presented in

Figure 7 is presented in Figure 8.

Grasl’s representation of the same shape presented

in Figure 9 is not that simple. The shape is now rep-

resented using 18 nodes and 28 edges, while in Ke-

les’s approach, the same shape is represented using

10 nodes and 16 edges.

4.1 Virtual Gaudi Project

The Virtual Gaudi Project, developed in our depart-

ment, is inspired by the art of Antonio Gaudi. The

general idea is to formally describe the existing works

of Antonio Gaudi, analyse them using various AI sys-

tems and ﬁnally start to generate a new design in the

Figure 8: Keles’s two-square representation.

Figure 9: Grasl’s two square representation.

“Gaudi style”. It is obvious that composed artworks

consist of millions of details. Because of that, efﬁ-

cient graph representation is a principal requirement

for this idea. In contrast to human perception, which

prefers one consistent representation of the problem,

the graph grammar formal notation has no problem

in describing the same concept in two or more ways,

depending on some parameters e.g. the size of the rep-

resented elements. This fact is the basis for the pro-

posed solution.

We Will Merge Both Presented Representations.

For the two (alternatively three) collinear segments,

we will use direct inline edges (as in Keles’s solution)

and if there are more segments, Grasl’s representation

will be used.

We analyse these four approaches on the Chinese

lattice designs.

In Chengtu, Szechwan 1800 AD design we have

167 (black) nodes and 262 direct (dashed) edges. The

(white) nodes, additional edges and the overall num-

ber of the given method of representation are pre-

sented in Table 2.

Table 2: Comparision of representation for 1800 AD.

represen- Extra Extra Together

tation nodes edges

Keles’2 0 0 429 100,00%

Grasl’s 262 524 1215 238,22%

VG-2 0 0 429 100,00%

VG-3 0 0 429 100,00%

In Chengtu, Szechwan 1825 AD design we have

237 (black) nodes and 262 direct (dashed) edges. The

(white) nodes, additional edges and the overall num-

ber of the given method of representation are pre-

sented in Table 3.

Table 3: Comparision of representation for 1825 AD.

represen- Extra Extra Together

tation nodes edges

Keles’2 0 1249 1890 100,00%

Grasl’s 141 545 1327 70,21%

VG-2 15 293 949 50,21%

VG-3 15 293 949 50,21%

In Hanchow, Szechwan 1875 AD design we have

165 (black) nodes and 276 direct (dashed) edges. The

(white) nodes, additional edges and the overall num-

ber of the given method of representation are pre-

sented in Table 4.

Table 4: Comparision of representation for 1875 AD.

represen- Extra Extra Together

tation nodes edges

Keles’2 0 1116 1557 100,00%

Grasl’s 52 328 821 52,73%

VG-2 28 280 749 48,11%

VG-3 24 276 741 47,59%

5 APPLICATION OF VG-X

NOTATION TO SHAPE

TRANSFORMATION RULES

The analysed examples show that the introduced VG-

2 and VG-3 representations have the same or bet-

ter expressiveness in comparison with Keles’s ap-

proach; their advantage grows when longer collinear

Figure 10: Online conversion of Keles’s to Grasl’s repre-

sentations - inline edge.

segments appear in the shape. They are always bet-

ter than Grasl’s approach. The VG-3 representation

seems to be slightly better, but the implementation of

splitting one of the collinear segments becomes more

complex. Thus, we will sketch the solution for VG-2.

As mentioned above, there is no problem for graph

transformation rules to use the both representations of

collinearity in the same system. The left side of the

transformation rule lhs will use the Grasl’s collinear-

ity representation. While searching the subgraph of

the entire graph G that is isomorphic to lhs, we will

convert:

• Keles’s in-line edge (black line) and the (only)

node connected with the in-line endnodes with

direct edges (dashed one) — into four nodes in

Grasl’s representation (see Figure 10),

• direct edges that are not considered in the previ-

ous in-line relation — into three nodes in Grasl’s

representation (see Figure 11).

The opposite transformation will be performed af-

ter the application of the transformation rule – all the

white nodes that participate in this rule and do not

point to at least three black nodes are converted to the

Grasl’s notation.

In the real life the ﬁnal project consist of many

different styles and shapes. we can present such a sit-

uation by summing up Chengtu, Szechwan 1800 AD,

1825 AD and Hanchow, Szechwan 1875 AD. That

proves that introduced VG-2 and VG-3 representa-

tions are the most efﬁcient ones (see table 5).

Figure 11: Online conversion of Keles’s to Grasl’s repre-

sentations - direct edge.

Table 5: Comparision of representation for the sum.

represen- Extra Extra Together

tation nodes edges

Keles’2 0 2365 3876 100,00%

Grasl’s 455 1397 3363 86,73%

VG-2 43 573 2127 54,88%

VG-3 39 569 2119 54,67%

6 CONCLUSIONS

The more compact graph representation of the shapes

generated by shape grammars is very important, as

the efﬁciency of graph transformation algorithms de-

pends on the graph size (number of nodes and edges).

In this context, efﬁcient representation of the collinear

segments is proposed. The deﬁnition of transfor-

mation rule application, extended by the conversion

mechanisms, allows us to combine two different rep-

resentations of the collinear segments. It should also

be noted that the mentioned approach can be extended

to representation of co-n-bezier shapes (deﬁned in

section 4), which is important in real-world appli-

cation of shape grammars. For example, in Antoni

Gaudi’s Sagrada Familia, line nearly do not appear.

ACKNOWLEDGEMENTS

This work has been partially supported by AGH UST

research project 11.11.120.859

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