Logarithmic spirals, defined as isogonal trajectories of pencils of lines, and their spherical counterparts, the spherical loxodromes, show up at different occasions in geometry.
1. Logarithmic spirals are trajectories of spiral transformations. The orbits of points under discrete stretch rotations generate a spiral grid. Recently, K. Myrianthis proved that the Voronoi cells of the grid points are mutually similar hexagons (or quadrangles) with a circumcircle. Stereographic projection yields spherical spiral polyhedra with hexagonal faces. They can also be interpreted as helical surfaces in the Cayley-Klein model of hyperbolic geometry.
2. Logarithmic spirals play a role in kinematics: When a logarithmic spiral rolls on a line, its asymptotic point traces a straight line.
Hence, wheels with the shape of a logarithmic spiral can be used for a stair climbing robot. Two congruent logarithmic spirals can roll on each other while their asymptotic points remain fixed. A composition of two such rollings results in a two-parametric motion, which allows a second decomposition of this kind (R. Bricard).
3. Spirals are also important in the geometry of gearing. When involute spur gears are to be generated by virtue of the principle of Camus, the auxiliary curves must be logarithmic spirals. The same is valid on the sphere for involute bevel gearing. Study's principle of transference, applied to this kind of gearing, could pave the way to a spatial counterpart of involute gears with skew axes. However, the result of this process is still open.
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Chemin du Musée 3
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