An Algorithm to Find the Shortest Path through Obstacles of Arbitrary Shapes and Positions in 2D

An algorithm is described to find the shortest route through a field of obstacles of arbitrary shapes and positions. It has the appreciable advantage of not having to find mathematical formulas to represent the obstacles: it works directly with a digital image of the terrain and is implemented solely with standard graphical functions. Key to this algorithm is the definition of digraphs, the edges of which are built with obstacle bitangents and border enveloping convex arcs that incorporate the fundamental features of shortest paths. These graphs have a remarkably lower cardinality than those that have been proposed before to solve this problem; their edges are a concatenation of sequences of what individual edges and nodes are in formerly defined graphs. Furthermore, a thorough analysis of the topology of the terrain yields a procedure to eliminate the edges that have no possibility of being part of the shortest path. The A* graph optimization algorithm is adapted to deal with this type of graph. A new quite general theorem is proved, which applies to all graphs in which the triangle inequality holds, which allows the discarding of one of the normal steps of the A* algorithm. The effectiveness of the algorithm is demonstrated by calculating the shortest path for real complex terrains of areas between 25 km2 and 900 km2. In all cases, the required calculation time is less than 0.6 s on a Core i7-10750H CPU @ 2.60 GHz laptop computer.