THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART

Hellmuth Stachel; Giorgio Figliolini; Jorge Angeles

Authors

Hellmuth Stachel Ph.D. Vienna University of Technology
Giorgio Figliolini Ph.D. University of Cassino and Southern Lazio
Jorge Angeles Ph.D. McGill University

Keywords:

logarithmic spiral, involute spur gears, two-parametric motion, spherical loxodrome, involute bevel gears, hyperbolic screws, 3-web

Abstract

Logarithmic spirals are isogonal trajectories of pencils of lines. From a series of geometric consequences, we pick out a few which are relevant for 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. When involute spur gears are to be generated by virtue of the principle of Camus, the auxiliary pitch curves must be logarithmic spirals. Two congruent logarithmic spirals can roll on each other while their asymptotic points remain fixed. A composition of two such rollings gives a two-parametric motion which allows a second decomposition of this kind. Some of these properties hold similarly for the spherical counterparts, the spherical loxodromes. For example, when in spherical geometry a loxodrome rolls on a circle, both asymptotic points trace circular involutes. Therefore, spherical loxodromes are auxiliary pitch curves for involute bevel gearing. On the other hand, spherical loxodromes can also be seen as helical curves in the projective model of hyperbolic geometry, where the sphere serves as a Clifford surface. This paves the way for remarkable arrangements of loxodromes on a sphere, e.g., a 3-web.

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Author Biographies

Hellmuth Stachel Ph.D., Vienna University of Technology

Professor emeritus Institute of Discrete Mathematics and Geometry

Giorgio Figliolini Ph.D., University of Cassino and Southern Lazio

Professor Dept. of Civil and Mechanical Engineering

Jorge Angeles Ph.D., McGill University

Professor Department of Mechanical Engineering & CIM

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