One‐parameter families of Legendre curves and plane line congruences
One‐parameter families of Legendre curves and plane line congruences
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DOI：
10.1002/mana.201900327
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发表时间：
2022-07
影响因子：
1
通讯作者：
Y. Kabata;Masatomo Takahashi
中科院分区：
文献类型：
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作者：
Y. Kabata;Masatomo Takahashi
欧几里得平面上的曲线族自然包含奇异曲线，经典微分几何的框架不能很好地工作。引入了欧几里得平面上的单参数勒让德曲线族、全等等价和曲率的概念。特别地，单参数的勒让德曲线族可以包含奇异曲线，并且由曲率到同余决定。我们还给出了勒让德曲线的单参数族的性质。作为应用，我们给出了勒让德曲线的单参数族与勒让德曲面之间的关系。此外，我们还研究了平面直线同余（平面上直线的单参数族）作为勒让德曲线的单参数族的曲率。
Families of curves in the Euclidean plane naturally contain singular curves, where the frame of classical differential geometry does not work well. We introduce the notions of one‐parameter family of Legendre curves in the Euclidean plane, congruent equivalence and curvature. Especially, a one‐parameter family of Legendre curves can contain singular curves, and is determined by the curvature up to congruence. We also give properties of one‐parameter families of Legendre curves. As applications, we give a relation between one‐parameter families of Legendre curves and Legendre surfaces. Moreover, we study plane line congruences (one‐parameter families of lines in plane) in terms of the curvatures as one‐parameter families of Legendre curves.