Sinuous Meander Patterns in Natural Coordinates
Inhaltsverzeichnis
Reference
David Chappell: Sinuous Meander Patterns in Natural Coordinates. In: Bridges 2012. Pages 183–190
DOI
Abstract
Natural (or intrinsic) coordinate systems parameterize curves based on their inherent properties such as arc length and tangential angle, independent of external reference frames. They provide a convenient means of representing many organic, flowing curves such as the meandering of streams and ocean currents. However, even simple functions written in natural coordinates can produce surprisingly complex spatial patterns that are difficult to predict from the original generating functions. This paper explores multi-frequency, sine-generated patterns in which the tangential angle of the curve is related to the curve’s arc length through a series of sine functions. The resulting designs exhibit repeating forms that can vary in subtle or dramatic ways along the curve depending on the choice of parameter values. The richness of the “pattern space” of this equation suggests that it and other simple natural equations might provide fertile ground for generating geometric, organic and even whimsical patterns.
Extended Abstract
Bibtex
Used References
[1] C. Bier, Symmetry and Symmetry-Breaking An Approach to Understanding Beauty, Bridges 2005 Symposium Proceedings, pp. 219-226, 2005.
[2] J. A. Staab, Katherine Westphal and Wearable Art, Textile Society of America 2004 Symposium Proceedings, Ed. C. Bier, pp. 227-233 (CD-ROM), OmniPress, 2005.
[3] “Fort Yukon, Alaska.” Map, Google Maps, Web, 7 January 2012.
[4] W. Whewell, Of the Intrinsic Equation of a Curve, and its Application, Transactions of the Cambridge Philosophical Society, 8, pp. 659-671, 1849.
[5] L. B. Leopold and W. B. Langbein, River Meanders, Scientific American, 214, pp. 60-70, June 1966.
[6] B. Hayes, Up a Lazy River, American Scientist, 94, pp. 490-4, 2006.
[7] P. Gailiunas, Meanders, Bridges 2005 Symposium Proceedings, pp 25-31, 2005.
[8] N. Movshovitz-Hadar and A. Shmukler, River Meandering and a Mathematical Model of this Phenomenon, Physica Plus, physicaplus.org.il, issue 7, 2006.
[9] C.G. Fraser, Mathematical technique and physical conception in Euler’s investigation of the Elastica, Centaurus, 34, pp. 211-246, 1991.
[10] R. L. Levien, From Spiral to Spline: Optimal Techniques in Interactive Curve Design, Ph.D. thesis, University of California, Berkeley, 2009.
Links
Full Text
http://archive.bridgesmathart.org/2012/bridges2012-183.pdf
