几类特殊曲面曲线的法曲率测地曲率和测地挠率的计算

Calculations of normal curvature, geodesic curvature and geodesic torsion of some special types of curves on surface

特殊曲面曲线是曲面论中的一个重要研究对象,坐标曲线、渐近曲线、曲率线和测地线是曲面上常见的几类特殊曲面曲线,而法曲率、测地曲率和测地挠率又是曲面曲线的三个重要的数字特征。本文首先推导出法曲率、测地曲率和测地挠率的性质;其次,列举了三者之间最常见的几种关系;最后,给出上述几类特殊曲面曲线的法曲率、测地曲率和测地挠率的计算公式。

Special curve on surface is an important research object in surface theory. Coordinate curves, asymptotic line, curvature line and geodesic line are some special curves which are common on surface, while normal curvature, geodesic curvature and geodesic torsion are the three important numerical traits. This paper first derives the nature of normal curvature, geodesic curvature and geodesic torsion, then lists some common relations among the three, and lastly provides the calculation formulas of normal curvature, geodesic curvature and geodesic torsion of the special types of normal curvature.