A generic geometric transformation that unifies a wide range of natural and abstract shapes
Inhaltsverzeichnis
Reference
Gielis, J.: A generic geometric transformation that unifies a wide range of natural and abstract shapes. American Journal of Botany 90, 333–338 (2003)
DOI
http://dx.doi.org/10.3732/ajb.90.3.333
Abstract
o study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man-made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non- Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.
Extended Abstract
Bibtex
Used References
BARR, A. H. 1981. Superquadrics and angle preserving transformations. IEEE Computer Graphics Applications 1: 11–23.
BECKER, T. R. 2000. Taxonomy, evolutionary history and distribution of the middle to late Fammenian Wocklumeria (Ammonoida, Clymeniida). Mit- teilungen Museum für Naturkunde Berlin, Geow. Reihe 3: 27–75.
BLANC, C., AND C. SCHLICK. 1996. Ratioquadrics: an alternative method for superquadrics. Visual Computer 12(8): 420–428.
BOOKSTEIN, F. L. 1996. Biometrics, biomathematics and the morphometric synthesis. Bulletin of Mathematical Biology 58: 313–365.
D’ARCY THOMPSON, W. 1917. On growth and form. Cambridge University Press, Cambridge, UK.
DECARLO, D., AND D. METAXAS. 1998. Shape evolution with structural and topological changes using blending. IEEE Transactions on Pattern Rec- ognition and Machine Intelligence, 20: 1186–1205.
DUMAIS, J., AND L. G. HARRISON. 2000. Whorl morphogenesis in the dasy- cladalean algae: the pattern formation viewpoint. Philosophical Trans- actions of the Royal Society of London B 355: 281–305.
GIELIS, J. 1999. Methods and devices for synthesizing and analyzing patterns using a novel mathematical operator. USPTO patent application NЊ 60/ 133,279.
GIELIS, J. 2001. De uitvinding van de Cirkel. Geniaal, Antwerp, Belgium. GREEN, P. B. 1999. Expression of pattern in plants: combining molecular and calculus-based biophysical paradigms. American Journal of Botany 86: 1059–1076.
GRENANDER, U. 1993. General pattern theory. Oxford University Press, Ox- ford, UK.
GRIDGEMAN, N. T. 1970. Lame ́ ovals. Mathematical Gazette 54: 31. HANSON, A. J. 1988. Hyperquadrics: solid deformable shapes with complex polyhedral bounds. Computer Vision, Graphics and Image Processing 44: 191–210.
HERSH, R. 1998. What is mathematics, really? Vintage, London, UK. JACKLIC, A., A. LEONARDIS, AND F. SOLINA. 2000. Segmentation and recov- ery of superquadrics. Kluwer Academic, Dordrecht, Netherlands.
JEAN, R. V. 1994. Phyllotaxis: a systemic study in plant morphogenesis. Cam- bridge University Press, Cambridge, UK.
JENSEN, R. J. 1990. Detecting shape variation in oak leaf morphology: a comparison of rotational-fit methods. American Journal of Botany 77: 1279–1293.
KINCAID, D. T., AND R. B. SCHNEIDER. 1983. Quantification of leaf shape with a microcomputer and Fourier transform. Canadian Journal of Bot- any 61: 2333–2342.
KUHL, F. P., AND D. R. GIARDINA. 1982. Elliptic Fourier features of a closed contour. Computer Graphics and Image Processing 18: 236–258.
LIESE, W. 1998. The anatomy of bamboo culms. International Network for Bamboo and Rattan Technical Report 18, New Delhi, India.
LORIA, G. 1910. Spezielle algebraische und transzendente ebene Kurven. B.G. Teubner, Leipzig-Berlin.
LOWE, C. J., AND G. A. WRAY. 1997. Radical alterations in the roles of homeobox genes during echinoderm evolution. Nature 389: 718–721. MCGOWAN, D. J. 1889. Square bamboo. Nature 33: 560.
MCLELLAN, T. 1993. The roles of heterochrony and heteroblasty in the di- versification of leaf shapes in Begonia dreigei (Begoniaceae). American Journal of Botany 80: 796–804.
MEINHARDT, H. 1998. The algorithmic beauty of sea shells. Springer Verlag, Berlin, Germany.
MOLINA-FREANER, F., C. TINOCO-OJANGUREN, AND K. J. NIKLAS. 1998. Stem biomechanics of three columnar cacti from the Sonoran desert. American Journal of Botany 85: 1082–1090.
PRUSINKIEWICZ, P. 1998. Modeling of spatial structure and development of plant: a review. Scientia Horticulturae 74: 113–149.
PRUSINKIEWICZ, P., AND A. LINDENMAYER. 1989. The algorithmic beauty of plants. Springer Verlag, Berlin, Germany.
ROHLF, F. J. 1996. Morphometric spaces, shape components and the effects of linear transformations. In L. F. Marcus et al. [eds.], Advances in mor- phometrics, 284: 117–129. Plenum, New York, New York, USA.
ROUND, F. E., R. M. CRAWFORD, AND D. G. MANN. 1991. Diatoms. Cam- bridge University Press, Cambridge, U.K.
TAKENOUCHI, Y. 1931. Systematisch-vergleichende Morphologie und Ana- tomie der Vegetationsorgane der japanischen Bambus-arten. Mem. Fac. Sci. Agr., Taihoku Imperial Univ., III: 1–60.
VAN OYSTAEYEN, F., J. GIELIS, AND R. CEULEMANS. 1996. Mathematical aspects of plant modeling. Scripta Botanica Belgica 13: 7–27.
WAINWRIGHT, S. A. 1988. Axis and circumference. Harvard University Press, Cambridge, Massachusetts, USA.
WEYL, H. 1952. Symmetry. Princeton University Press, Princeton, New Jer- sey, USA.
ZHOU, L., AND C. KAMBHAMETTU. 1999. Extending superquadrics with ex- ponent functions. In: Proceedings 1999 IEEE Computer Society Confer- ence on Computer Vision and Pattern Recognition II: 73–78.
Links
Full Text
http://www.amjbot.org/content/90/3/333.full.pdf+html
