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Shape decompositions and their algebras

Published online by Cambridge University Press:  02 November 2005

DJORDJE KRSTIC
DJORDJE KRSTIC
Affiliation:
Alcatel, Calabasas, California 91302, USA
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Abstract

Shapes play an important role in many human activities, but are rarely seen in their natural form as raw and unanalyzed. Rather, shapes come analyzed, structured in terms of their certain parts, forming shape decompositions. Different kinds of shape decompositions are developed, the most interesting among which are the decompositions that could be used as shape approximations. Two kinds of such decompositions, discrete and bounded, are examined in greater detail. Computations with shapes conducted in the framework of shape grammars and related shape algebras have been standard for over 3 decades. Similar computations are possible with analyzed shapes or shape decompositions. Different algebras to compute with shape decompositions are developed and compared to the shape algebras. The measure of their agreement determines how well the shapes are approximated by their decompositions.

Keywords

Shape AlgebrasShape ApproximationsShape DecompositionsShape Decompositions AlgebrasShape Grammars
Type
Research Article
Information
AI EDAM , Volume 19 , Issue 4 , November 2005 , pp. 261 - 276
Copyright
© 2005 Cambridge University Press

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References

REFERENCES

Agrawal, M. & Cagan, J. (1998). A blend of different tastes: the language of coffee makers. Environment and Planning B: Planning and Design 25, 205–226.CrossRefGoogle Scholar
Birkhoff, G. (1993). Lattice Theory. Providence, RH: American Mathematical Society.
Bleicher, M.N., Schneider, H., & Wilson, R.L. (1973). Permanence of identities on algebras. Algebra Universalis 3, 72–93.CrossRefGoogle Scholar
Carlson, C., Woodbury, R., & McKelvey, R. (1991). An introduction to structure and structure grammars. Environment and Planning B: Planning and Design 18, 417–426.CrossRefGoogle Scholar
Chase, S.C. (1989). Shapes and shape grammars: from mathematical model to computer implementation. Environment and Planning B: Planning and Design 16, 215–242.CrossRefGoogle Scholar
Earl, C. (1997). Shape boundaries. Environment and Planning B: Planning and Design 24, 668–687.CrossRefGoogle Scholar
Flemming, U. (1987). The role of shape grammars in analysis and creation of designs. In Computability of Designs (Kaley, Y.E., Ed.), pp. 245–272. New York: Wiley.
Gratzer, G. (1979). Universal Algebra. Providence, RI: American Mathematical Society.
Gratzer, G. & Whitney, S. (1978). Infinitary varieties of structures closed under the formation of complex structures [Abstract]. Notices of the American Mathematical Society 25, A224.Google Scholar
Guatam, N. (1957). The validity of equations of complex algebras. Archives Mathematic Logik Grundlagenforch 3, 117–124.CrossRefGoogle Scholar
Knight, T. (1988). Comparing designs. Environment and Planning B: Planning and Design 7, 73–110.CrossRefGoogle Scholar
Knight, T. (2003a). Computing with emergence. Environment and Planning B: Planning and Design 30, 125–155.Google Scholar
Knight, T. (2003b). Computing with ambiguity. Environment and Planning B: Planning and Design 30, 165–180.Google Scholar
Krishnamurti, R. (1982). SGI: A Grammar Interpreter [Research Report]. Centre for Configurational Studies, The Open University, Milton Keynes.
Krishnamurti, R. Giraud, C. (1986). Towards a shape editor: the implementation of a shape generation system. Environment and Planning B: Planning and Design 13, 391–403.CrossRefGoogle Scholar
Krstic, D. (1996). Decompositions of shapes. PhD Thesis. University of California, Los Angeles.
Krstic, D. (1999). Constructing algebras of design. Environment and Planning B: Planning and Design 26, 45–57.CrossRefGoogle Scholar
Krstic, D. (2001). Algebras and grammars for shapes and their boundaries. Environment and Planning B: Planning and Design 28, 151–162.CrossRefGoogle Scholar
Krstic, D. (2004). Computing with analyzed shapes. In Design Computing and Cognition '04 (Gero, J.S., Ed.), pp. 397–416. Dordrecht: Kluwer Academic.CrossRef
Piazzalunga, U. & Fitzhorn, P.I. (1998). Note on three-dimensional shape grammar interpreter. Environment and Planning B: Planning and Design 25, 11–33.CrossRefGoogle Scholar
Shafaat, A. (1974). On varieties closed under construction of power algebras. Bulletin Australian Mathematical Society 11, 213–218.CrossRefGoogle Scholar
Stiny, G. (1980). Introduction to shape and shape grammars. Environment and Planning B: Planning and Design 7, 243–251.CrossRefGoogle Scholar
Stiny, G. (1982). Spatial relations and grammars. Environment and Planning B: Planning and Design 9, 113–114.CrossRefGoogle Scholar
Stiny, G. (1990). What is a design. Environment and Planning B: Planning and Design 17, 97–103.CrossRefGoogle Scholar
Stiny, G. (1991). The algebras of design. Research in Engineering Design 2, 171–181.CrossRefGoogle Scholar
Stiny, G. (1992). Weights. Environment and Planning B: Planning and Design 19, 413–430.CrossRefGoogle Scholar
Stiny, G. (1994). Shape rules: closure, continuity, and emergence. Environment and Planning B: Planning and Design 21, S49–S78.Google Scholar
Stiny, G. (2001). How to calculate with shapes. In Formal Engineering Design Synthesis (Antonson & E.K., Cagan, J., Eds.), pp. 24–60. Cambridge: Cambridge University Press.CrossRef
Tapia, M.A. (1999). A visual implementation of a shape grammar system. Environment and Planning B: Planning and Design 26, 59–73.CrossRefGoogle Scholar
Vickers, S. (1989). Topology Via Logic. New York: Cambridge University Press.