2. Tangential symmetries in the plane


For closed smooth curves in the plane three types of tangential symmetries are of interest. These symmetries always consist of a group of smooth self-maps on the domain S1 for the parametrization c: S1 E2 of the curve. The maps are assumed to assign points with parallel tangents to each other, and may be subject to additional conditions.

The general case without restrictive conditions has been studied by J. Shaer (unpublished) and F.J. Craveiro de Carvalho and S.A. Robertson [CR1]. Some classification results are obtained for open curves by the latter while J. Shaer provided a complete description of the shape of closed curves having a non-trivial group of tangent preserving self-maps. Mathematically spoken, the asignment of their unit tangents factorizes through a multiple covering map of the unit sphere for these curves, after having identified tangent vectors with opposite directions. For example in the locally convex case, the group Z2 characterizes strictly convex ovals. Centrally symmetric curves may serve as examples for the nonconvex case. More general groups lead to curves with a finite number of loops, directed to the same "interior" side of the curve in the locally convex case and possessing symmetries concerning the shape of "interior" and "exterior" loops and bumps in the general case. This shape can be visualized very easily, and it obviously determines a fairly special structure for these curves.

If the tangential symmetry is assumed to preserve the normal lines in addition, then we get what has been introduced by H. Farran and S.A. Robertson [FR] as exterior self-parallelism in the general context of immersions. For closed curves this reduces to the notion of rosettes of constant width. In particular, these curves are locally strictly convex, only the cyclic group of order two can appear as a non-trivial group of self-parallelisms, and the rosettes can be generated in a very special way: A rigid line segment could be moved along the curve, connecting point and image point for the only non-trivial tangential symmetry, such that this segment always will be a normal to the curve at its endpoints. It is easy to imagine the shape of such curves, though they are more general than the classical rosettes, where a rotational symmetry can be observed. In this general case the rotational symmetry only refers to the loop structure. Details on rosettes of constant width can be found in the paper of W. Cieslak and W. Mozgawa [CM] and in [W1].

Finally, assuming as a stronger requirement, that any normal line of the curve can intersect the curve as a normal line only, we arrive at the notion of transnormality. This has been introduced by S.A. Robertson [R1] for the more general situation of immersions into Euclidean spaces, but in the simple case of a closed curve in the plane it leads immediately to convex curves of constant width bounding planar convex domains of constant width. This characterization is a classical result. There is a vast amount of literature on ovals of constant width. For, example a comprehensive survey on classical results already could be found in the book of T. Bonnesen and W. Fenchel [BF]. Later surveys are given in handbooks on convexity. These curves may be considered as the trivial case of rosettes of constant width where no loops occur. Hence the kinematic interpretation is simple. They are frequently used in applied geometry. Most famous is their application to the construction of the cylinder and the piston for the Wankel engine.


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