MANNHEIM CURVE

 Notion studied by Mannheim in 1859, name given by Wölffing in 1899. Amédée Mannheim (1831-1906): French mathematician and artillery captain.

 If the intrinsic equation 1 of the rolling curve is , then the Cartesian equation of the Mannheim curve is: , the axis Ox being the rolling axis. If the intrinsic equation 2 of the rolling curve is , then the Cartesian parametrization of the Mannheim curve is: .

The Mannheim curve associated to a curve is the locus of the centre of curvature at the contact point of this curve rolling without slipping on a line.

Examples:

 initial curve Mannheim curve circle line alysoid (including the catenary) parabola cycloid circle cycloidal curve ellipse pseudo-cycloidal curve hyperbola logarithmic spiral line Cornu spiral rectangular hyperbola curve with sinusoidal radius sinusoid involute of a circle parabola catenary of equal resistance catenary Ribaucour curve of index k Ribaucour curve of index k  1 pseudo-spiral of index n curve See an application of the Mannheim curves for the wheel-road couples.

A possible generalisation is for the curve to roll on any curve instead of a straight line, for example on a circle. In this latter case, the locus of the centre of curvature can be called polar Mannheim curve. If the intrinsic equation 1 of the rolling curve is , then the polar equation of the polar Mannheim curve is: , where a is the radius of the circle.

NOTE: in the literature, another notion can be referred to as "Mannheim curve". It consists in the curves the radius of curvature of which is proportional to the signed distance to a fixed point.
This notion is tackled on the page dedicated to the Norwich spiral.

© Robert FERRÉOL  2017