THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART
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 spira...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SORGING
2020-08-01
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| Series: | Journal of Industrial Design and Engineering Graphics |
| Subjects: | |
| Online Access: | http://www.sorging.ro/jideg/index.php/jideg/article/view/27 |
| _version_ | 1797716142476754944 |
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| author | Hellmuth Stachel Giorgio Figliolini Jorge Angeles |
| author_facet | Hellmuth Stachel Giorgio Figliolini Jorge Angeles |
| author_sort | Hellmuth Stachel |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-12T08:18:08Z |
| format | Article |
| id | doaj.art-bb30270c6181498989e9c9e0f75bda67 |
| institution | Directory Open Access Journal |
| issn | 1843-3766 2344-4681 |
| language | English |
| last_indexed | 2024-03-12T08:18:08Z |
| publishDate | 2020-08-01 |
| publisher | SORGING |
| record_format | Article |
| series | Journal of Industrial Design and Engineering Graphics |
| spelling | doaj.art-bb30270c6181498989e9c9e0f75bda672023-09-02T18:45:38ZengSORGINGJournal of Industrial Design and Engineering Graphics1843-37662344-46812020-08-01141THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPARTHellmuth Stachel0Giorgio Figliolini1Jorge Angeles2Vienna University of TechnologyUniversity of Cassino and Southern LazioMcGill UniversityLogarithmic 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.http://www.sorging.ro/jideg/index.php/jideg/article/view/27logarithmic spiralinvolute spur gearstwo-parametric motionspherical loxodromeinvolute bevel gearshyperbolic screws |
| spellingShingle | Hellmuth Stachel Giorgio Figliolini Jorge Angeles THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART Journal of Industrial Design and Engineering Graphics logarithmic spiral involute spur gears two-parametric motion spherical loxodrome involute bevel gears hyperbolic screws |
| title | THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART |
| title_full | THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART |
| title_fullStr | THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART |
| title_full_unstemmed | THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART |
| title_short | THE LOGARITHMIC SPIRAL AND ITS SPHERICAL COUNTERPART |
| title_sort | logarithmic spiral and its spherical counterpart |
| topic | logarithmic spiral involute spur gears two-parametric motion spherical loxodrome involute bevel gears hyperbolic screws |
| url | http://www.sorging.ro/jideg/index.php/jideg/article/view/27 |
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