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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.
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© Robert FERRÉOL 2017