Sine-generated curves belong to a class of intrinsic functions which describe a curve by specifying its “direction angle”. The curve is determined by ω, the maximum angle which the curve makes with the horizontal, and the fact that the direction angle changes in a sinusoidal fashion along the path. Sine-generated curves are shown to be excellent approximations to the path of minimal average curvature, and expressions for radius of curvature and curve sinuosity are derived.
References
[1]
Leopold, L.B. and Langbein, W.B. (1966) River Meanders. Scientific American, 214, 60-70.
http://dx.doi.org/10.1038/scientificamerican0666-60
[2]
Langbein, W.B. and Leopold, L.B. (1966) River Meanders: Theory of Minimum Variance. US Geological Survey Professional Paper 422-H.
[3]
Adam, J. (2006) Mathematics in Nature: Modeling Patterns in the Natural World. Princeton University Press, Princeton.
[4]
Movshovitz-Hadar, N. and Shmukler, A. (2006) River Meandering and a Mathematical Model of this Phenomenon. Physica Plus, 7, 1-23.
[5]
Von Schelling, H. (1951) Most Frequent Particle Paths in a Plane. Transactions American Geophysical Union, 32, 222-226.
http://dx.doi.org/10.1029/TR032i002p00222
[6]
Arfken, G.B., Weber, H.J. and Harris, F.E. (2013) Mathematical Methods for Physicists. Elsevier Press, Oxford.
[7]
Nahin, P.J. (2007) When Least Is Best. Princeton University Press, Princeton.
