We propose a new approach for the accurate numerical computation of high-order derivatives along geodesic curves on surfaces. The method is based on the observation that for geodesics the Darboux frame and the Frenet–Serret frame are locally equal up to a constant rotation around the tangent. It computes derivatives of arbitrary order from the result of the numerical method employed for computing the geodesic. Since it does not rely on finite difference approximations, no additional discretization errors are introduced. Applications of the method include motion planning of autonomous vehicles and geometric modeling with developable surfaces.

中文翻译：

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光滑表面上测地线路径的精确高阶导数

我们提出了一种新方法，用于精确数值计算沿表面测地曲线的高阶导数。该方法基于以下观察：对于测地线，Darboux 框架和 Frenet-Serret 框架在局部等于围绕切线的恒定旋转。它根据用于计算测地线的数值方法的结果计算任意阶的导数。由于它不依赖于有限差分近似，因此不会引入额外的离散化误差。该方法的应用包括自动驾驶汽车的运动规划和具有可展曲面的几何建模。