Tangential Symmetries of Planar
Curves and Space Curves

Bernd Wegner

Fachbereich Mathematik, TU Berlin
Straße des 17. Juni 135,
D - 10623 Berlin

        1. Introduction

Mathematically symmetries occur for many objects and they have a quite general notion. They are automorphisms of a mathematical structure leaving invariant characteristic properties or quantities defined for this structure. For example, in Euclidean geometry symmetric figures mostly admit a non-trivial isometric self-map of the ambient Euclidean space, which also maps the figure onto itself. Hence this map, restricted to the figure, obviously preserves all the metric properties of that figure. But also combinatorial symmetries are considered where the self-map of the figure only preserves the combinatorial structure, while metric relations may change after having applied the map. Lots of different types of such symmetries are known, and all of them distinguish the shape of such a figure from the shape in the general case in a way which more or less immediately can be noticed by looking at that figure.

In the case of planar curves reflectional or rotational symmetries are considered as remarkable properties. Also self-similarities like in the case of spirals are of interest, and they are used to characterize certain spirals by admitting a big family of self-similarities. These properties have consequences for utilizing certain curves in the applications of geometry. Spatial generalizations of these notions are obvious, and in space the possibilities of possessing symmetries are even richer for curves. Compare a helix with a circle, for example. The aim of this short note is to explain the impact of so-called tangential symmetries on the shape of curves in the Euclidean plane and in Euclidean 3-space. As symmetry notions they seem to be quite weak, but nevertheless they have visible consequences for the shape of these curves.