球面和平面曲线的旋转极小标架刻画
Characterization of Spherical and Plane Curves Using Rotation Minimizing Frames
摘要：
在这项工作中，我们研究了平面和球面曲线在欧几里德和洛伦兹-闵可夫斯基3-空间，采用旋转极小化（RM）框架。通过方便地写出球面上曲线的曲率和挠率，给出了如何求球面曲线的主法线与RM向量场之间的夹角。之后，我们将平面曲线和球面曲线刻画为其位置向量位于由其单位切线和RM向量场所张成的移动平面上的曲线。最后，作为一个应用，我们刻画了Bertrand曲线和斜螺旋线的曲线，其所谓的自然伴侣是球面和一般的螺旋线，分别。
英文摘要：
In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. Later, we characterize plane and spherical curves as curves whose position vector lies, up to a translation, on a moving plane spanned by their unit tangent and an RM vector field. Finally, as an application, we characterize Bertrand curves and slant helices as curves whose so-called natural mates are spherical and general helices, respectively.
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