Geodesic-like curves on parametric surfaces

Finding the geodesic on a surface is an important task in many fields. As the traditional discrete algorithms or numerical approach can only find a list of discrete points, this study aims to propose a new, elegant and accurate method for approximating geodesics on a surface. The proposed geodesic-like curve is a mathematical curve that approaches to the geodesic on surface when the order of the curve reaches to infinity. The geodesic-like curve is also compared to some other well-known methods in this study.