Centre symmetry sets of families of plane curves. (English) Zbl 1336.57041
From the text: The centre symmetry set (CSS) of a hypersurface \(M\) in \(\mathbb{R}^{k+1}\) is the envelope of (infinite) straight lines joining pairs of points of \(M\) with parallel tangent hyperplanes or “parallel tangent chords” as we shall call them.
We study centre symmetry sets and equidistants for a \(1\)-parameter family of plane curves where, for a special member of the family, there exist two inflexions with parallel tangents. Some results can be obtained by reducing a generating family to normal forms, but others require direct calculation from the generating family.
The article is organized as follows. In section 2 we study supercaustics in more detail than is needed for our main application. In section 3, we study the CSS of a family of curves \(\gamma_\varepsilon\), parametrized by \(\varepsilon\), which contains a member \(\gamma_0\) with parallel but distinct tangents at inflexion points. In particular, we show that the union of the CSS for all small \(\varepsilon\) – the “big CSS” – is a cuspidal edge surface, but with the function \(\varepsilon\), whose level sets are the separate CSS, being very degenerate. In section 4, we study families of equidistants associated with a fixed \(\gamma_\varepsilon\) and close to certain special values of \(\lambda\) . In section 5, we show how, in some situations, it is possible to reduce the generating family to a normal form. These allow us to recognize the big CSS and the evolution of the momentary CSS as \(\varepsilon\) changes, but unfortunately not the momentary CSS in the parallel inflexional tangents case. We also identify the “big equidistant” and evolution as \(\varepsilon\) changes of the momentary equidistants for a fixed value of \(\lambda\) away from the special values.
We study centre symmetry sets and equidistants for a \(1\)-parameter family of plane curves where, for a special member of the family, there exist two inflexions with parallel tangents. Some results can be obtained by reducing a generating family to normal forms, but others require direct calculation from the generating family.
The article is organized as follows. In section 2 we study supercaustics in more detail than is needed for our main application. In section 3, we study the CSS of a family of curves \(\gamma_\varepsilon\), parametrized by \(\varepsilon\), which contains a member \(\gamma_0\) with parallel but distinct tangents at inflexion points. In particular, we show that the union of the CSS for all small \(\varepsilon\) – the “big CSS” – is a cuspidal edge surface, but with the function \(\varepsilon\), whose level sets are the separate CSS, being very degenerate. In section 4, we study families of equidistants associated with a fixed \(\gamma_\varepsilon\) and close to certain special values of \(\lambda\) . In section 5, we show how, in some situations, it is possible to reduce the generating family to a normal form. These allow us to recognize the big CSS and the evolution of the momentary CSS as \(\varepsilon\) changes, but unfortunately not the momentary CSS in the parallel inflexional tangents case. We also identify the “big equidistant” and evolution as \(\varepsilon\) changes of the momentary equidistants for a fixed value of \(\lambda\) away from the special values.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 57R45 | Singularities of differentiable mappings in differential topology |
| 53A04 | Curves in Euclidean and related spaces |
| 53A15 | Affine differential geometry |
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