Geodesic Path Planning using Exponential Map with Injectivity Radius

Geodesic path planning is crucial in applications such as robotics, computer graphics, and autonomous navigation, focusing on finding the shortest path between two points on a curved surface while accounting for intrinsic geometry. Traditional methods, including energy function minimization, heat flow, and curvature-based techniques, often face local minima and computational inefficiencies, particularly in irregular environments. This paper presents a novel method that integrates the exponential map with the injectivity radius, ensuring globally optimal paths. Our approach avoids local minima, guarantees the shortest path, and provides real-time computational efficiency. Simulations show that our method outperforms existing techniques in optimality, local minima avoidance, and computation time.