The line defined by the vector and the central point of the torus is known as the axis of the torus. With this torus construction, we can solve the problem provided the initial and final positions are sufficiently far apart. If a torus of this type is constructed around both the initial and final positions, there will be a line through each axis that is also tangent to each torus. In space there are four possible paths, just as there are four possible paths in the two-dimensional case. One of these paths always represents the shortest and most efficient path between the initial and final positions.
Many problems remain to be solved.
In the case of positions that are relatively close together, the simple analogue of moving from circles to torii does not appear to work. Various proposed solutions have been examined, but the robot is always required to make a sharper turn than its structure allows. The presence of obstacles must also be considered, including the object being inspected, since it may lie across the direct path considered by this algorithm. A more involved problem is to find a path that uses the least energy rather than just finding the shortest physical path. This path would be significant if it required less energy to move the robot in one direction than in another.