Modification of Manhattan Geometry In a gridworld, the usual metric used is the Manhattan Metric. From a given coordinate, there are only four possible directions: up, down, left, right. 
I want to modify the metric by giving four more possible directions (totaling eight): usual 4 directions, northwest, northeast, southwest, southeast. 
I want to ask if there is a name to this kind of modification. This is usually found in Artificial intelligence and machine learning.  

If we are allowed to take steps in any of the eight directions you describe, the number of steps it takes to get from $(x_1,y_1)$ to $(x_2,y_2)$ is the maximum of $|x_1-x_2|$ and $|y_1-y_2|$. This is often called the $L^\infty$ metric.
