the construction of curved shapes

7/31/2019 The Construction of Curved Shapes

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1 Introduction

Shape grammars are a formal production system whereby languages of designs are

generated according to shape replacement rules (Stiny, 1980). Ever since their concep-tion, efforts have been made to take advantage of the generative properties afforded by

the shape grammar formalism by developing shape grammar interpreters, a discussion

of which is presented by Gips (1999). These are computational systems intended to

automate the application of shape rules, and thereby generate networks of designs

within a design space. However, the development of such systems is not trivial because

the problem of general subshape detection is yet to be solved. Indeed, shape grammar

interpreters typically are concerned with generating shapes within limited algebras of

design (Stiny, 1991), in which shapes are composed of a limited range of geometric

elements, such as straight lines. In this paper, algorithms are presented that enable thedevelopment of shape grammar interpreters in which designs are generated within

algebras of shapes composed of parametric curve segments. These algorithms provide

a significant advantage over previous approaches to implementing shape grammars on

curved shapes, such as that presented by McCormack and Cagan (2003), since they do

not rely on degenerate representations of shapes in order to afford subshape detection.

Instead the algorithms employ established results from differential geometry in order

to enable an intrinsic comparison of shapes composed of parametric curve segments.

This ensures that shapes are compared according to their embedding properties, defined

under Euclidean transformations.

The construction of curved shapes

Iestyn JowersSchool of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, England;e-mail: [email protected]

Christopher EarlFaculty of Mathematics, Computing and Technology, The Open University, Milton KeynesMK7 6AA, England; e-mail: [email protected]

Received 6 August 2008; published online 19 October 2009

Environment and Planning B: Planning and Design 2010, volume 37, pages 42 ^ 58

Abstract. Application of a shape grammar involves the repetitive task of matching and replacingsubshapes of a design under transformation, and as such is well suited for computer implementation.As a result, ever since the conception of the shape grammar formalism, efforts have been made todevelop computer programs that automate shape grammar applications. Much of this effort hasbeen directed towards the problem of subshape detection, which involves recognising subshapesembedded in a design. Solutions to this problem have been presented for shapes composed ofrectilinear geometric elements, such as straight lines, and algorithms based on these solutions

have been implemented in a variety of shape grammar interpreters. However, there has beenless research concerning the solution of the subshape detection problem for shapes composedof nonrectilinear geometric elements, such as curve segments. In this paper a method of intrinsicmatching is presented, which enables comparison of the embedding properties of parametric curves.This method has been employed in order to develop shape algorithms which can be implementedin shape grammar interpreters for shapes composed of parametric curve segments, arranged intwo-dimensional or three-dimensional space.

doi:10.1068/b35093


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2 Background

2.1 Shape grammars

A shape grammar consists of an initial shape and a set of rules of the form a 3 b,where a and b are both shapes, as illustrated in figure 1. In this context, a shape isdefined visually, as a finite arrangement of geometric elements, such as lines or curves,

each with a definite boundary and limited but nonzero extent. A rule is applicable

to a shape g if some similarity transformation (typically a Euclidean transformation)

of the shape a on the left-hand side of the rule is a subshape of g, denoted a 4 g,

where 4 is the subshape relation.

Application of the rule removes the transformed instance of the subshape a and

replaces it with a similarly transformed instance of the shape b on the right-hand side

of the rule. For example, the shape rule in figure 1 removes a lens and replaces it with

a translated lens, as indicated by the coordinate axes. Repeated application of shape

rules to an initial shape leads to the generation of a sequence of designs, as illustrated

in figure 2.

Shape grammars differ from many other generative systems of design, such asevolutionary algorithms (Lee et al, 2005), in that the structure of a design is not fixed

but changes dynamically throughout a generative process. As a result, application of a

shape rule is not restricted to the geometric elements initially used to define a design;

instead they are applicable to any subshapes that can be perceived to be embedded

in the design. Indeed, within the shape grammar formalism the structure of a design

is defined retrospectively according to shape rule applications (Stiny, 1994). A conse-

quence of this is that shape grammars often generate designs that incorporate some

unexpected results that follow from the recognition and manipulation of new interpre-

tations of shapes. For example, the design sequence in figure 2 begins with an initialshape [figure 2(a)], and ends with a shape that is a rotation of this initial shape

[figure 2(c)], but this sequence is generated via application of a rule that merely

translates a subshape of the initial shape (figure 1). This unexpected result occurs

because after a single application of the shape rule additional instances of the lens

in the left-hand side of the rule have emerged [figure 2(b)]. Further applications of

the shape rule can now be used to recognise and manipulate these emergent shapes,

and restructure the design so that it appears to be rotated.

Shape grammars have been found to be a powerful generative mechanism and have

been employed in a range of design disciplines, from architecture (see, for example,Koning and Eizenberg, 1981) to engineering (see, for example, Brown and Cagan, 1997).

Figure 1. An example shape rule.

(a) (b) (c)

Figure 2. Example design sequence: (a) initial shape; (b) result of one rule application; (c) resultof two rule applications.

The construction of curved shapes 43


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However, few of these applications are computationally implemented and instead it is

common for rules to be applied and designs generated as a paper-based exercise. This

is because the task of developing a shape grammar interpreter is not trivial, requiring asolution to the subshape detection problem.

2.2 Shape grammar interpreters

Typically, a shape grammar interpreter requires mechanisms to enable the recognition

of embedded subshapes within a design under a similarity transformation, along

with mechanisms that enable the recognised subshapes to be removed and replaced.

Implementation of these mechanisms is difficult because shapes can generally be

represented in an uncountable number of ways and the task of determining whether

or not two representations refer to the same shape can be computationally expensive, if

not impossible. Instead, implementation can be simplified through the use of a canonicalrepresentation of shapes, typically the maximal representation. Under this representation,

shapes are composed of the smallest possible set of geometric elements. For example,

a shape composed of straight lines is reduced to its maximum representation by merging

any colinear lines that overlap or share any endpoints, so that the shape is composed of the

smallest possible set of lines; these are the maximal lines of the shape. The first algorithms

to provide the required mechanisms for shape grammar implementation were presented

by Krishnamurti (1980; 1981). These algorithms enable implementation of shape grammars

on shapes composed of straight lines arranged in a plane and have been implemented

in a variety of shape grammar interpreters, such as those by Krishnamurti and Giraud(1986), Chase (1989), and Tapia (1999). They have also been shown to be extendable

to include shapes composed of lines, planes, or solids, arranged in two-dimensional

or three-dimensional Euclidean space (Krishnamurti, 1992).

In the algorithms, maximal geometric elements are represented by a descriptor and

a boundary, where the descriptor defines the abstract object of infinite extent in which a

geometric element is embedded (for example, an infinite line) and the boundary

defines the location of a geometric element with respect to its descriptor, for exam-

ple, endpoints of a line. In this way, shapes are represented by lexicographically

ordered lists where maximal lines are partitioned according to their descriptorsinto colinear sets and these sets are themselves ordered according to boundaries.

Binary shape operations, such as shape difference, are applicable to two lines only

if they share the same descriptors; that is, a line can only be added to or subtracted

from other colinear lines. As a result, shape operations are implemented efficiently

by comparing the ordered lists that represent the shapes. However, under Euclidean

transformations, a straight-line segment is self-similar and can be embedded in

any other straight line in an infinite number of ways. Consequently, matching of

embedded subshapes is facilitated by the definition of distinct points, such as the

points of intersection between two lines, and these points are used to determinethe spatial relations and embedding properties of shapes.

An obvious limitation of this approach is that it can generate only designs com-

posed of rectilinear geometric elements, such as lines or planes. Extension to include

nonlinear elements such as curves or surfaces is not straightforward due to the differ-

ences in embedding properties. However, Chau et al (2004) have adapted the approach

in order to develop a shape grammar interpreter for shapes composed of straight lines

and circular arcs arranged in three-dimensional space. This is possible because

circular arcs share many of the self-similarity properties apparent in straight lines.

Similarly, McCormack and Cagan (2003) have employed aspects of the algorithmsin order to develop a shape grammar interpreter for shapes composed of curves. In

particular, shapes composed of freeform curve segments are associated with distinct

44 I Jowers, C Earl


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construct designs, and intrinsic comparison of descriptors provides valuable insight

with respect to the embedding properties of curve segments.

Given a space curve C, the natural parameters of C, denoted s, are defined by thearc length. This arc length is not uniquely defined since it is measured from a specific

point of the curve, in a specific direction. As a result, if s1 and s2 are both natural

parameters ofC then s2 s1 c, for some constant c. The curvature and torsion ofCare defined as functions of this arc length and are denoted k(s) and t(s), respectively.

These functions are intrinsic properties of C since they are independent of the natural

parameter used to represent the curve. That is, the value of the curvature and torsion

at any point of C is the same regardless of the direction in which the arc length is

measured and the point it is measured from. The curvature function k(s) is a measure

of a curve's departure from linearity; that is, it is a measure of how much the curveturns. Consequently, it follows that a straight line has zero curvature, and conversely

any curve with zero curvature is a straight line. Similarly, the torsion function t(s) is

a measure of a curve's departure from planarity; that is, it is a measure of how much

the curve twists. Consequently, it follows that a planar curve has zero torsion, and

conversely any curve with zero torsion is a planar curve.

Formally, the fundamental theorem of space curves states that two space curves C1and C2 can be mapped onto each other under a Euclidean motion if and only if,

for some constant c,

k1 s k2 s c , (1)

t1 s t2 s c . (2)

These equations hold for curves that are parameterised according to natural param-

eters, defined by arc length. However, Farouki and Sakkalis (1991) have shown that

real curves, other than straight lines, cannot be parameterised according to rational

functions of a natural parameter. This means that the descriptors of parametric curve

segments commonly used by designers in computational systems of design, such as

B-splines or NURBS, cannot be readily parameterised according to natural param-eters. As such, they cannot be easily compared according to equations (1) and (2).

Fortunately, the principles of differential geometry hold for any regular curves irres-

pective of the parameters used to define them (Do Cormo, 1976), and the equations can

be restated for curves defined according to arbitrary parameters. Accordingly, given

two space curves C1 and C2 , parameterised according to arbitrary parameters t

and u, respectively, the two curves can be mapped onto each other under a Euclidean

motion if and only if, for some continuous reparameterisation function u u(t),

k1 t k2ut

, (3)

t1 t t2ut

. (4)

Equations (3) and (4) are suitable for comparing the embedding properties of

parametric curves under Euclidean motion: that is, translation and rotation. How-

ever, in shape grammar implementations, it is customary to consider the embedding

properties of shapes under Euclidean transformations: that is, Euclidean motion

augmented by reflection and isotropic scale. As a result, it is desirable to expand

these equations in order to enable comparison of curves under the Euclidean trans-

formations, and to do so, it is necessary to consider how the curvature and torsionfunctions of a curve are modified under reflection and isotropic scale.

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Given a space curve C, defined parametrically according to the function x(t) [x(t),y(t), z(t)], the curvature ofC is

k jx H x HHj

jx Hj3

,

and the torsion function ofC is

t jx Hx HHx HHHj

jx H x HHj2

,

where jx Hx HHx HHHj denotes a scalar triple product x H(x HH x HHH), and the derivatives x H,x

HH, and x HHH are functions of arbitrary parameter t (Do Carmo, 1976). IfC is modified

according to a scaling factor l then the derivativesx

H

,x

HH

, andx

HHH

are scaledequivalently:

xt 3 lxt A x Ht 3 lx Ht , and x HHt 3 lx HHt .

As a result, it can be shown that the curvature and torsion function of C are scaled

according to a factor of l1

. Similarly, it can be shown that, ifC is modified according

to a reflection and if C is a planar curve, then the curvature function is negated;

otherwise ifC is nonplanar the torsion function is negated.

These results, when combined with equations (3) and (4), mean that given two

space curves C1 and C2 , parameterised according to arbitrary parameters t and u,respectively, the two curves can be mapped onto each other under a Euclidean

transformation if and only if

k1 t l1k2

ut

, (5)

t1 t l1t2

ut

, (6)

for some constant l( T 0) and some continuous reparameterisation function u u(t).These equations provide a means of determining whether or not two space curves can

be mapped onto each other under a Euclidean transformation. If the space curves actas descriptors for curve segments in a shape, then the equations reveal whether or not

the segments lie on infinite curves that can be mapped onto each other, and whether or

not the curve segments can be embedded in each other. In the remainder of this paper,

equations (5) and (6) form the basis of shape algorithms that enable implementation of

shape grammars on shapes composed of parametric curve segments.

4 Shape algorithms for curved shapes

Implementation of a shape grammar involves repeated application of shape rules in the

form a 3 b. Such a rule is applicable to a shape g if there is a similarity transformationT that will match the shape a on the left-hand side of the rule with a subshape of g.This condition is tested according to the subshape relation, denoted by 4, and if it is

satisfied then a is said to be a subshape of g, denoted a 4 g. A shape rule is applied to

g by removing the subshape that matches the transformed shape T(a) and replacing it

with a similarly transformed shape T( b), where b is the shape on the right-hand side of

the rule. This shape replacement involves applying a shape subtraction operation,

followed by a shape addition operation, and results in the shape [g T(a)] T( b).As a result, the algorithms necessary to implement a shape grammar interpreter include

algorithms for the subshape relation and the shape operations of addition and subtrac-tion. The algorithms introduced here are for implementing shape grammars on shapes

composed of parametric curve segments that are arranged in two- or three-dimensional

Euclidean space, with shape operations applied under Euclidean transformation. It is



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assumed that the curve segments are sufficiently smooth that their curvature and torsion

functions are continuous.

The algorithms will be illustrated by considering application of the shape rulein figure 1 to the curved shape S in figure 3 [which is also the shape in figure 2(b)].

S is composed of nine curve segments, S1 ^S9 , and shape operations will be applied

to Sby considering the intrinsic properties of the descriptors of these composite curves

according to equations (5) and (6).

4.1 Maximal representation of curved shapes

As discussed above, when implementing shape operations it is beneficial to consider

a canonical representation of shapes, typically the maximal representation. This

approach was successfully employed by Krishnamurti (1980; 1981) in shape algorithms

for shapes composed of straight lines, and is also used here in shape algorithms forcurved shapes. Given a shape composed of parametric curve segments, such as the

shape S in figure 3, its maximal representation results from merging any curve

segments that are on the same infinite curve and have any points in common. For

example, consider the curve segments S1 and S3 in figure 3. Let S1 and S3 be para-

meterised according to arbitrary parameters t and u by the vectors x1 (t) and x2 (u) on the

intervals t0 4 t 4 t1 and u0 4 u 4 u1 , respectively. If S1 and S3 are on the same

infinite curve and have any points in common, then they can be merged to form a

single-curve segment. The infinite curves on which S1 and S3 lie are defined by their

descriptors, which are compared according to equations (5) and (6). This intrinsic com-parison, along with a comparison of the spatial relation between the infinite curves, will

reveal whether or not S1 and S3 are embedded in the same infinite curve. If there exists a

scaling factor l( T 0) and a continuous function u u(t) such that the intrinsic proper-ties of the infinite curves defined by the descriptors of S1 and S3 satisfy equations (5)

and (6) then S1 and S3 are on infinite curves which can be mapped onto each other

under Euclidean transformations. In addition, if x1 (t) x2 [u(t)] then S1 and S3 are oninfinite curves that have the same spatial position and indeed are the same. In figure 3,

S1 and S3 are on the same infinite curve, as is S5 , as illustrated in figure 4.

Comparison of the endpoints of S1 and S3 will determine whether or notthey have any points in common and consequently whether or not they can be

merged to form a single curve. If S1 and S3 overlap and have one or more points

S8 S1

S9

S7 S2S6 S3

S5 S4

Figure 3. A curved shape S.

S8 S1

S9S7 S2

S6 S3

S5 S4

Figure 4. Description ofS1 , S3 , and S5 .

48 I Jowers, C Earl


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in common, as illustrated in figures 5(a) ^ 5(d), then they can be merged to form

a single curve. Alternatively, if S3 is completely embedded in S1 , as illustrated in

figures 5(e) and 5(f), then S3 can simply be removed from the shape without causing

any visual alterations. Similarly, if S1 is completely embedded in S3 , as illustrated

in figures 5(g) and 5(h), then S1 can simply be removed from the shape without

causing any visual alterations. For any other arrangements of endpoints, S1 and S3are completely disjoint and are both maximal with respect to each other.

In this example, S1 and S3 share an endpoint and as a result can be merged to

form a single curve, say S13. Similarly, S13 and S5 share an endpoint and can bemerged to form a single maximal curve, say Sa. Comparison of all the curves in Swith

each other reveals that S2 , S7 , and S9 can be merged to form a single maximal curve,

say Sb, and that S4 , S6 , and S8 can be merged to form a single maximal curve, say Sc,

resulting in a maximal representation of Sas illustrated in figure 6.

This procedure for discovering the maximal representation of a curved shape is

summarised in algorithm 1. The input for the algorithm is a shape, represented by an

array of N curves. Every curve segment in the shape is compared with every other

curve segment according to the intrinsic properties of the infinite curves defined by

their descriptors. If, for a pair of infinite curves, there exists a constant l ( T 0) and acontinuous function u u(t) that satisfy equations (5) and (6) then they can bemapped onto each other under a Euclidean transformation. In such a case the spatial

positions of the two infinite curves are compared to determine if they are the same.

If they are the same then the endpoints of the curve segments associated with the

descriptors are compared to determine if the curve segments can be merged to form

a single maximal curve segment. Once all the curve segments in the shape have been

compared with each other the algorithm outputs a shape represented by an array of N

maximal curves. Note that the number, N, of curves that are input into the algorithm

need not necessarily equal the number, N, of curves that are output from the algo-rithm. Curves are merged and deleted as the algorithm progresses and as a result N,

the number of curves in the shape, is constantly being updated.

u0 u1 t0 t0

t0 t0 u0 u1

t0 t1 t1 t0 u0 u1 u1

u u t t

u0

(a) (b) (c) (d)

(e) (f ) (g) (h)

t1u1

t1u0

u1t1

u0t1

Figure 5. Overlapping curve segments, where (a) t0 4 u0 4 t1 4 u1 ; (b) t0 4 u1 4 t1 4 u0 ;(c) u0 4 t0 4 u1 4 t1 ; (d) u1 4 t0 4 u0 4 t1 ; (e) t0 4 u0 , u1 4 t1 ; (f ) t1 4 u0 , u1 4 t0 ;

(g) u0 4 t0 , t1 4 u1 ; (h) u1 4 t0 , t1 4 u0 .

Sa

Sb

Sc

Figure 6. Maximal representation ofS.



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4.2 Intrinsic subshape detection

The first stage in applying a shape rule, a 3 b, to a shape g is determining whether ornot the shape a, on the left-hand side of the rule, is a subshape ofg. This is determined by

comparing the shapes according to the subshape relation. For example, consider the

shape rule in figure 7 where a lens is translated along the length of its central axis,

as indicated by the coordinate axis. If this rule is applicable to the shape S in figure 6

the shape a, on the left-hand side of the rule, can be embedded in Sunder a Euclidean

transformation. Here, the shape a, is composed of two curve segments a1 and a2 , and if

a transformation of a is a subshape ofS, then transformations of the curve segments a1and a2 must be subshapes of the curve segments Si in S. Accordingly, comparing the

descriptors of a1 and a2 with descriptors of the maximal curves that compose the shapeS, according to equations (5) and (6), will reveal whether or not a can be embedded in S.

Let Sa and a1 be parameterised according to the arbitrary parameters t and u by

the vectors x1 (t) and x2 (u) on the intervals t0 4 t 4 t1 and u0 4 u 4 u1 , respectively.

If the infinite curves defined by the descriptors of Sa and a1 can be mapped onto each

other under a Euclidean transformation, such that the endpoints of a1 are within the

Algorithm 1. Make maximal.

Data: A shape represented by an array ofN curvesResult: A shape represented by an array ofNmaximal curves

ifNb 1 thenfor i2 0 to N 2 do

for j2 0 to N 1 dofirstCurve designArrayi;secondCurve designArrayj;if l exists then

if u (t) exists thenif firstCurvet secondCurveut then

if firstCurve and secondCurve overlap thenthirdCurve firstCurve secondCurve;Add thirdCurve to designArray;Delete firstCurve from designArray;Delete secondCurve from designArray;Update N;

else if firstCurve is embedded in secondCurve thenDelete firstCurve from designArray;Update N;

else if secondCurve is embedded in firstCurve thenDelete secondCurve from designArray;

Update N;end

endend

endend

endend

a1 b1

a2 b2

Figure 7. A shape rule for a 3 b.

50 I Jowers, C Earl


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endpoints of Sa , then a1 can be embedded in Sa . Now, the infinite curves defined by

the descriptors of Sa and a1 can be mapped onto each other under a Euclidean

transformation if there exists a scaling factor l( T 0) and a continuous functionu u(t) such that the intrinsic properties of the infinite curves satisfy equations (5)and (6). Also, comparison of the endpoints of the two curves, as illustrated in figure 5,

will reveal whether or not a1 can be embedded in Sa . In this example, a1 can be

embedded in Sa under a Euclidean transformation, and as a result a1 is a subshape

of Sa . Comparison of all the curve segments that compose a with all the curve seg-

ments that compose Swill determine the transformations that embed each a curve in

each S curve. If all curves in a can be embedded in curves in S under the same

transformation then a is a subshape ofS. Indeed, in this example, there are, disregard-

ing reflective symmetry, three transformations that embed a in S, as illustrated infigure 8.

This procedure for applying the subshape relation to curved shapes is summarised

in algorithms 2 and 3. Algorithm 2 compares all the curves in a with all the curves in

S in order to calculate the transformations that embed the a curves in S. The input

for the algorithm is a shape grammar that is composed of a shape, called design,

represented by an array of N curve segments and an array of L shape rules. In each

shape rule there is a shape a that is represented by an array of M curve segments.

Sa Sa Sa

Sb Sb Sb

Sc Sc Sc

Figure 8. Subshape detection

Algorithm 2. Calculate transformations.

Data: A shape grammar consisting of L shape rules, each rule containing a shape a (alpha)represented by an array ofM curves, and a shape design represented by an array ofN curvesResult: An array of transformations transArray associated with each curve in alpha

for ruleCount 0 to L 1 dorule ruleArrayruleCount;M number ofalpha curves in rule;for alphaCount 0 to M 1 do

alphaCurve ruleXalphaArrayalphaCount;

fordesignCount

0 toN 1 dodesignCurve designArraydesignCount;

if l exists thenif u (t) exists then

if alphaCurve can be embedded in designCurve thenCalculate transformation between alphaCurve and designCurve;transformation.matchWith designCount;Add transformation to alphaCurve.transArray;

endend

end

endend

end



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Algorithm 3. Compare transformations.

Data: A shape grammar consisting ofL shape rules, each rule containing a shape a (alpha) defined by an arrayofMcurvesResult: A Boolean value indicating whether or not all curve segments in a shape alpha are associated with thesame transformation

if nextRule TRUE thenruleMatch ruleMatch 1mod L;nextRule FALSE;

endfor ruleCount ruleMatch to L 1 do

rule ruleArrayruleCount;ifMb 1 do

firstCurve alphaArray0;T1 number of transformations associated with firstCurve;for firstTransCount matchPosition to T1 1 do

firstTrans firstCurveXtransArrayfirstTransCount;for alphacount 1 to M 1 do

alphaCurve alphaArrayalphaCount;Talpha number of transformations associated with alphaCurve;for alphaTransCount 0 to Talpha 1 do

alphaTrans alphaCurveXtransArrayalphaTransCount;if firstTrans alphaTrans then

firstCurveXtransMatchIndex firstTransCount;alphaCurveXtransMatchIndex alphaTransCount;

foundMatch TRUE;ruleMatch ruleCount;if alphaCount ` M 1 then

alphaTransCount Talpha;else

matchPosition matchPosition 1mod T1;if matchPosition 0 then

nextRule TRUE;endReturn foundMatch;

endelse if alphaTransCount Talpha 1 then

alphaCount M;foundMatch FALSE;end

endend

endelse ifM 1 then

alphaCurve alphaArray0;Talpha number of transformations associated with alphaCurve;alphaCurveXtransMatchIndex matchPosition;foundMatch TRUE;ruleMatch ruleCount;matchPosition

matchPosition

1

mod Talpha;if matchPosition 0 then

nextRule TRUE;endReturnfoundMatch;

endmatchPosition 0;

endif ruleMatch b 0 then

foundMatch TRUE;nextRule TRUE;

endreturn foundMatch;

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The algorithm systematically compares the curves that compose the a shape in each

shape rule with the curves that compose the shape design in order to determine whether

or not transformations exist that embed the a curves in the design curves. If such atransformation exists it is calculated and then associated with a variable matchWith that

captures the index of the design curve in which the a curve is embedded. In order to

make shape matching a more efficient process this algorithm is applied only once per

rule application and the transformations are stored in transformation arrays that are

associated with the a curves to which they apply.

Algorithm 3 scrolls through the rules in the grammar and compares the trans-

formations associated with the a curves in order to determine whether or not the shape

a as a whole can be embedded in the shape design. The input for the algorithm is an

array ofL shape rules each containing a shape a represented by an array ofMcurves.For each rule in turn the algorithm compares the transformation associated with the

first curve in the array of a curves with the transformations associated with the other a

curves. If all of the curves in an a shape are associated with the same transformation

then that a is a subshape of the shape design and the algorithm terminates.

The algorithm can be applied repeatedly in order to explore different match and

rule possibilities and the variables nextRule, ruleMatch, and matchPosition are global

variables that are in place in order to record the current state of the comparison. When

the algorithm is applied next the comparison continues from the current state accord-

ing to these global variables. If a shape a is found such that all its composite curvesare all associated with the same transformation then the shape is a subshape of design.

The variable transMatchIndex records the index of the transformation in the arrays

associated with the a curves. When the algorithm terminates it outputs the Boolean

value of foundMatch which is TRUE if a match is found and FALSE otherwise.

4.3 Shape replacement

If a shape g is found to have a subshape that matches a shape a on the left-hand side of

shape rule under a Euclidean transformation T, then that rule can be applied to g by

replacing the shape T(a) with the shape T(b). For example, application of the subshape

relation has revealed that the shape a, on the left-hand side of the rule in figure 7, is asubshape of the shape S in figure 6, under the Euclidean transformation T. The rule

can be applied to Sby first applying the shape difference operation in order to remove

the transformed shape T(a) from S, followed by the shape union operation in order to

add the similarly transformed shape T(b).

The shape difference operation is applied by comparing the curve segments that

compose the shapes. For example, let Sa and a1 be parameterised according to the

arbitrary parameters t and u by the vectors x1 (t) and x2 (u) on the intervals t0 4 t 4 t1and u0 4 u 4 u1 , respectively. It is already known from the subshape algorithm

(algorithm 2) that the descriptors ofSa and a1 can be mapped onto each other underEuclidean transformation and that a1 can be embedded in Sa under the transforma-

tion T. The shape Sa T(a1 ) results from applying the shape difference operation tothe curve segments Sa and T(a1 ). This shape will be composed of one curve segment if

Sa and T(a1 ) have endpoints in common, otherwise it will be composed of two curve

segments, as illustrated in figure 9.

In this example, Sa and T(a1 ) share an endpoint, as illustrated in figure 8, and as

a result the shape Sa T(a1 ) is composed of a single curve. Similarly, Sb T(a2 ) is ashape composed of a single curve, and the shape S T(a), that results from applying

the shape difference operation to S and T(a) is illustrated in figure 10.The shape union operation is applied by simply adding the curve segments

that compose the shape T(b), to the array of curves that represents the shape. In this



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example the curves T( b1 ) and T(b2 ) are added to the shape S T(a), resulting in theshape [S T(a)] T(b), as illustrated in figure 11.

This procedure for shape replacement according to a shape rule a 3 b, issummarised in algorithms 4 and 5. Algorithm 4 transforms the shapes a and b accord-

ing to the Euclidean transformation T. This transformation was determined according

to the subshape relation and embeds the shape a in the shape design. The input for

the algorithm is a shape rule, composed of two shapes a and b that are represented

x1 (t) x2 (u)

u0 u1 u1 u0 u1 u0 u0 u1t0 t1 t1 t0 t0 t1 t1 t0

(a) (b) (c) (d)

Figure 9. Shape difference between curve segments, where (a) t0 ` t1 and u0 ` u1 ; (b) t0 b t1 andu0 b u1 ; (c) t0 ` t1 and u0 b u1 ; (d) t0 b t1 and u0 ` u1 .

Sa T(a1 )

Sb T(a2 )

Sc

Figure 10. Subshape removal.

T( b2 )

T( b1 )

Figure 11. Result of rule application.

Algorithm 4. Shape transformation.

Data: A shape a (alpha) represented by an array ofMcurves, and a shape b (beta) represented

by an array ofP curvesResult: A shape Talpha represented by an array ofM curves, a shape Tbeta represented by anarray ofP curves

for alphaCount 0 to M 1 doalphaCurve alphaArrayalphaCount;transMatch CurveXtransArrayalphaCurveXtransMatchIndex;TalphaCurve transMatchaplhaCurve;Add TalphaCurve to TalphaArray;

endfor betaCount 0 to P 1 do

betaCurve betaArraybetaCount;TbetaCurve transMatchbetaCurve;Add TbetaCurve to TbetaArray;

end

54 I Jowers, C Earl


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by an array of M and P curves, respectively. Each curve is considered in turn

and transformed according to the transformation T. The output of the algorithm is the

transformed shapes T(a) and T( b).

Algorithm 5 replaces the subshape T(a), in the shape design, with the shape T( b).The input for the algorithm is a shape T(a) represented by an array of M curves,

a shape design represented by an array ofNcurves, and a shape T( b) represented by an

array of P curves. The algorithm deletes the shape T(a) from the shape design by

considering each design curve in turn. The variable matchWith associated with each

T(a) curve is the index of a design curve in which it can be embedded. The shape

difference of a design curve and a T(a) curve results in two curves, as illustrated in

figure 9. If the two curves have any endpoints in common then one of these curves will

be of zero length. When all the design curves have been considered then the shape

design will have been replaced with the shape design T(a). The algorithm then addsall the curves in the array that represents the shape T( b) to the array that represents

the shape design, and outputs the shape [design T(a)] T( b).

Algorithm 5. Shape replacement.

Data: A shape T(a) (Talpha) represented by an array ofM curves, a shape design representedby an array ofN curves, and a shape T(b) (Tbeta) represented by an array ofP curvesResult: A shape, design Talpha Tbeta

for designcount 0 to N 1 dodesignCurve designArraydesignCount;Add designCurve to minusalphaArray;for alphaCount 0 to M 1 do

TalphaCurve TalphaArrayalphaCount;transMatch TalphaCurveXtransArraytransMatchIndex;if transMatchXmatchWith designcount then

count number of curves in minusalphaArray;for i 0 to count 1 dodelete FALSE;designCurve minusalphaArrayi;if t0 ` t1 & u0 ` u1 OR t0 b t1 & u0 b u1 then

firstCurve curve defined between t0 and u0;secondCurve curve defined between t1 and u1;Add firstCurve and secondCurve to minusalphaarray;delete TRUE;

else if t0 ` t1 & u0 b u1 OR t0 b t1 & u0 ` u1 thenfirstCurve curve defined between t0 and u1;

secondCurve curve defined between t1 and u0;Add firstCurve and secondCurve to minusalphaarray;delete TRUE;

endif delete TRUE then

Delete designCurve from minusalphaArray;end

endend

endenddesignArray minusalphaArray;for betaCount 0 to P 1 do

TbetaCurve TbetaArraybetaCount;Add TbetaCurve to designArray;

end



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The five shape algorithms presented here provide the mechanisms necessary to

implement a shape grammar interpreter for shapes composed of parametric curves in

two- or three-dimensional space. Within such a system shape grammars would producesequences of designs via repeated application of shape rules, according to sequential

use of the algorithms, as follows:

Algorithm 1: calculate maximal representation of shape S.

Algorithms 2 and 3: apply subshape relation to determine applicability of shape rule

a 3 b.Algorithms 4 and 5: remove the transformation of a from Sand replace with a transfor-

mation of b.

If the subshape relation produces a negative result then the shape a cannot be

embedded in the shape S and the shape rule cannot be applied. However, whilethe subshape relation continues to produce positive results the shape grammar can

continue to produce new designs.

5 Discussion

The shape grammar formalism is based on a visual representation of shape rather than

a formal representation, and shape grammar interpreters afford shape manipulations

that reflect this. It is this representational aspect of shape grammars that gives rise to

the distinction between a geometric element, such as a curve, and the abstract object

defined by its descriptor. Geometric elements are visual and they have a definiteboundary and limited but nonzero extent. On the other hand, a descriptor defines an

abstract object given by a formal mathematical representation, with no boundary and

possibly infinite extent. This distinction can lead to some interesting consequences with

respect to implementing shape grammars on curved shapes. For example, consider the

curve segment illustrated in figure 12 which is represented according to two distinct

descriptors, and as a result is a segment of two distinct infinite curves.

Visually, there is no problem with this situation, and perceptually the curve

segment could be represented according to either, or both, descriptors. Indeed, it is

entirely feasible for this situation to arise when considering the types of piecewiseparametric curves commonly used by designers in computational systems of design,

such as B-Splines or NURBS. However, this raises complications with respect to

shape grammar implementation. As discussed above, implementation of the mecha-

nisms necessary for a shape grammar interpreter is simplified if shapes can be

reduced to a canonical representation. In the algorithms presented in this paper,

and in the algorithms presented by Krishnamurti (1980; 1981), a canonical represen-

tation of a shape is given according to maximal geometric elements, and shape

operations are applied through consideration of each of these elements along with

their descriptors. Jowers et al (2004) warn that, if a canonical representation of ashape is required, then the situation illustrated in figure 12 should be avoided,

since shapes composed of curve segments associated with multiple descriptors would

Figure 12. A curve segment represented according to two descriptors.

56 I Jowers, C Earl


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have multiple maximal representations. As such, piecewise curves, such as B-Splines

or NURBS, should not be considered as single curves; instead, they could be treated as

shapes composed of multiple geometric elements, each defined according to a differentdescriptor. The formal representation of shapes supports this approach. Indeed, con-

sideration of a mathematical representation of curves, as defined by descriptors,

suggests that the situation illustrated in figure 12 could not arise practically. This is

because, mathematically, if regular curves are equal for a finite extent of their length

then they are equal for their whole length.

The distinction between visual and formal representations of shapes is not so

apparent for shapes composed of straight lines. This is because if two lines are

associated with different descriptors then it is usually visually obvious. This difference

between curved and rectilinear shapes results from differences in embedding propertiesof geometric elements. Further differences can also be noted. For example, difficulties

arise when comparing the embedding properties of rectilinear geometric elements, such

as straight lines, because in an intrinsic sense there is no distinction between different

segmentsin the words of Blake ``a line is a line in its minutest subdivisions'' (1974,

page 878). As a result, a line can be embedded in another line in an uncountable

number of ways. That is, the embedding relations between different line segments is

infinite and intrinsic comparison, such as that outlined in the algorithms presented

here, is not possible. Similarly, intrinsic matching is not possible for shapes com-

posed of geometric elements with constant intrinsic properties, such as circular arcs.For such shapes, it is necessary to use alternative methods for determining the

embedding properties. For example, Krishnamurti's algorithms rely on the definition

of distinct points in order to determine the spatial relations and embedding proper-

ties of shapes composed of straight lines. On the other hand, intrinsic matching is

possible for shapes composed of geometric elements with varying intrinsic properties.

This is because the varying intrinsic properties result in limited embedding relations

between geometric elements. These embedding relations are defined by a scalar value l

and the continuous function u u(t), and allow for intrinsic comparison of curve

segments according to the equationsk1 t l

1k2

ut

,

t1 t l1

t2ut

.

A general method for determining l and u u(t), or indeed of proving theirexistence, has not been investigated here. Indeed, l and u u(t) are both dependenton the particular parametric curves used to compose a shape. An example in which

the algorithms are used to implement shape grammars on shapes composed of Be zier

curves will be presented in a paper to follow. Such an implementation will increase

the scope of shape grammars applications to design disciplines where rectilinear shapesare not sufficient, such as product design.

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