MEANDER CURVE Curve studied by Luna Léopold and Walter Langbein in 1966 (cf. this article), that they named sine-generated curve. This curve is also found under the name patterned curve, in this study by Rovenski. Cf. also this article from AmericanScientist.

 Intrinsic equation 2: . Intrinsic equation 1: . Cartesian parametrization: ( ); complex parametrization: . Transcendental curve. Remark: the curve parametrized by is similar to the previous one.

The meander curves are the curves such that the angle formed by the tangent and a fixed direction is a sinusoidal function of the curvilinear abscissa. Such a curve was considered by Luna Léopold because he had noticed in an experiment that a compass needle on a boat sailing at constant speed down a river with meanders oscillated in a sinusoidal way over time.
In view of the intrinsic equation 1 above, the meander curves also are the curves for which the curvature varies as a sinusoidal function of the curvilinear abscissa.

 The meander curve is also the curve described by a point M such that if the angle formed by a line (D) and the tangent at M is proportional to the curvilinear abscissa, then the distance between the projection H of the centre of curvature I on (D) and M is a constant.  Here is the evolution of the curve according to the values of the maximal tangential angle : When is less than , the curve has a sinusoidal shape; above, .   It is when that the curve looks the most like meanders.   Look when ,  more precisely when  When , the curve looks like a lemniscate            REMARK: the values of for which the curve is bounded are defined by (which implies ).

 When goes to infinity, the meander curve (in red) "goes" to the clothoid: (in blue), here when = 10:   Conversely, for small values of , the meander curve is very close to the elastic curve (and this is explained on the corresponding page); in fact, the interpretation of the meanders as the curve minimising the curvature leads to modelling meanders rather by the elastic curve, but the comparison between the two curves shows that Luna Léopold had indeed conducted a good experimental observation.

Compare also with the curve of constant gyration and the curves with sinusoidal radius.  The meanders of the Seine.

© Robert FERRÉOL 2017