Curves in the Minkowski plane and their contact with pseudo-circles. (English) Zbl 1267.53011
The authors study evolutes, caustics, and parallel curves of a smooth, regular curve \(\gamma\) in the pseudo-Euclidean plane (Minkowski plane). A point on \(\gamma\) is called “space-like”, “time-like, “light-like” if the tangent in this point is space-like, time-like, light-like, respectively. The behavior of caustics at light-like points of \(\gamma\) are discussed. For instance, \(\gamma\) and the caustic are tangent to each other in such a point and lie locally in distinct halfplanes determined by the common tangent. Another result is that the caustic of an oval curve \(\gamma\) lies in the exterior of \(\gamma\). A parallel curve to \(\gamma\) can be defined only at non-light-like points of \(\gamma\). The behavior of the Minkowski parallel curves at an ordinary vertex of \(\gamma\) is compared to the behavior of their Euclidean counterparts at such a point.
Reviewer: Anton Gfrerrer (Graz)
MSC:
| 53A35 | Non-Euclidean differential geometry |
| 58K05 | Critical points of functions and mappings on manifolds |
| 53D12 | Lagrangian submanifolds; Maslov index |
Keywords:
caustic; evolute; Minkowski plane; pseudo-Euclidean plane; parallel curves; singularities; ovalsReferences:
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