Differential Geometry and Geometric Structures
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Student's Work: Symmetric rolling motion

Twin crank mechanism with counter-rotating cranks

For the displayed mechanism the polodes are congruent hyperbolas. The fixed hyperbola coloured in blue has the focal points A0, B0 and the centre 00. The moving hyperbola coloured in green has the focal points A, B and the centre 0.

The locus of all points, which are instantaneously at an inflection point of their path, is the so-called inflection circle (orange), which touches the pole tangent (red) in P. The antipode of P is known as inflection pole W.

The intersection between the affine normal (red dashed line) in P with respect to the fixed polode and the inflection circle results in the Ball point U. This is the only point, which is instantaneously at an undulation point of its path.

The locus of all points which are instantaneously at a vertex of their path is called cubic of stationary curvature (red). This cubic contains the instantaneous pole P, the Ball point U and the focal points of the moving polode, which are the Burmester points of the constrained motion.

Due to reasons of symmetry, the cubic of stationary curvature splits up into a circle and a straight line if P is at a vertex of the hyperbola.

Created by Christoph Teufel (2014) using Cinderella.

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Hans Havlicek
Friedrich Manhart
Georg Nawratil


Institute of Discrete Mathematics and Geometry

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Last modified on June 6th, 2014.