有关抛物面上测地线的研究

Research on Geodesics on Parabolic Surfaces

摘要: 从方程的角度求解曲面上的测地线，是我们求测地线的常用方法。但是这种方法在求一些具体问题时计算量很大，同时也不利于人们对于曲面上测地线的直观想象。本文从测地曲率几何意义的角度出发，求出了椭圆抛物面、旋转抛物面和双曲抛物面上经过原点的所有平面曲线类型的测地线，求解过程简单，有助于人们直观理解，并利用数学软件Mathematica画出了测地线的图像。

Abstract: Solving geodesics on surfaces from the perspective of equations is a common method we use to find geodesics. However, this method is very computationally intensive when solving some specific problems, and it is also not conducive to people’s intuitive imagination of geodesics on surfaces. In this paper, from the perspective of the geometric meaning of geodesic curvature, the geodesic of all plane curve types passing through the origin on elliptic paraboloid, rotating paraboloid and hyperbolic paraboloid are obtained, and the solution process is simple and helpful for people to intuitively understand, and use the mathematical software Mathematica to draw the geodesic image.