THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART

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

References

Airy, G.B.: On the forms of the teeth of wheels. Transactions of the Cambridge Philosophical Society, 1825, Vol. II, Part I., pp. 277–286.

Bennett, J.A.: George Biddell Airy and horology. Annals of Science 37, (3), 269–285 (1980).

Bool, F.H., Kist, J.R., Locher, J.L., Wierda, F.: M.C. Escher: His Life and Complete Graphic Work. Harry N. Abrams, Inc., New York 1992.

Bricard, R.: Sur les mouvements plans à deux paramètres doublement décomposables. Bulletin Société de la Mathématique de France, Comptes Rendus des Séances, 41, 27–29 (1913).

Bricard, R.: Sur une hexaédre particulier. Nouvelles annales de mathématiques, Série 4, 13, 24–29 (1913).

Bricard, R.: Sur les mouvements doublement décom-posables. Comptes Rendus Hebdomadaires des Séan-ces de l’académie des sciences 158, 110–112 (1914).

Bricard, R.: Sur un mouvement plan à deux paramè-tres, doublement décomposables. Nouvelles annales de mathématiques, Série 6, 2, 105–109 (1927).

Bricard, R.: Leçons des Cinématique, Tome 2. Gau-thier-Villars et Cie, Paris 1927.

Figliolini, G., Angeles, J.: Algorithms for involute and octoidal bevel-gear generation. ASME J. of Me-chanical Design 127/4, 664–672 (2005).

Figliolini, G., Rea, P., Angeles, J.: Synthesis of the pitch surface of non-circular skew-gears. Proc. of the ASME 2010 IDETC-CIE Conference, Aug. 2010, Montreal/Quebec, Canada, Paper: DETC2010-28902.

Figliolini, G., Stachel, H., Angeles, J., A new look at the Ball-Disteli diagram and its relevance to spatial gearing. Mechanism and Machine Theory 42/10, 1362–1375 (2007).

Figliolini, G., Stachel, H., Angeles, J.: The role of the orthogonal helicoid in the generation of the tooth flanks of involute-gear pairs with skew axes. ASME J. of Mechanisms and Robotics 7/1, 011003, 9 pages, (2015).

Figliolini, G., Stachel, H., Angeles, J., Kinematic Properties of Planar and Spherical Logarithmic Spi-rals: Applications to the Synthesis of Involute Tooth Profiles. Mechanism and Machine Theory, https:// doi.org/10.1016/j.mechmachtheory.2019.02.010

Koenigs, G.: Sur les mouvements doublement dé-composables et sur les surfaces qui sont le lieu de deux familles des courbes égales. Comptes Rendus Hebdomadaires des Séances de l’académie des sciences 157, 988–991 (1913).

Krys, V., Mostyn, V., Kot, T.: The synthesis and testing of a shaped wheel for stairs climbing robot. Appl. Mech. Mater. 555, 178–185 (2014).

Marcotte, J., Salomone, M.: Loxodromic Spirals in M.C. Escher’s Sphere Surface. J. Humanist. Math. 4/2, 25–46 (2014).

Mostyn, V., Krys, V., Kot, T., Bobovsky, Z., Novak, P.: The synthesis of a segmented stair-climbing wheel. Int. J. Adv. Robot. Syst. 15/1, 1–11 (2018).

Reuleaux, F., The Kinematics of Machinery, Dover Publications Inc., New York 1963, pp. 155–156.

Stachel, H.: A Family of Multiply Decomposable Euclidean Motions. In I. Bárány, K. Böröczky (eds.): Intuitive Geometry. Bolyai Society Mathematical Studies 6, János Bolyai Mathematical Society, Buda-pest 1997, pp. 413–427.

R. Willis: Principles of mechanism. 2nd ed., Long-mans, Green and Co., London 1870.

W. Wunderlich: Ebene Kinematik. Bibliographi-sches Institut, Mannheim, Wien, Zürich 1970.

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Published

2020-08-17

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Section

Applied Geometry and Graphics