Analysis on the Manhattan geometry - ISISS M. Casagrande (Pieve di Soligo)

Since ancient times, the grid street map has been widely used in the urban planning of cities. Examples include ancient Giza, Babylon, Rome, as well as modern cities like Manhattan, Barcelona, and Lyon. In these cities, the streets run at right angles to each other, forming a grid. The frequent intersections and orthogonal geometry facilitate movement, orientation, and wayfinding. In this article, we address some problems associated with such a grid street map. The first chapter deals with the shortest path problem. When all segments of the grid are of
uniform length, there are multiple shortest paths between two points, all with the same total length. The challenge here is to count the number of these shortest paths. When the segments have different lengths, the paths will generally also have different lengths, and the problem becomes finding the shortest path.
In the second chapter we model the grid as a metric space using Manhattan distance, considering only the intersections as points of interest. After developing some classical geometric notions in this setting, we apply them to address some urban planning issues. In particular, we introduce a new definition for the straight line and provide two algorithms to compute the resulting point-line distance. We then study conics and their properties in this geometry, applying them to solve some planning issues. With another geometric locus, the axis of two points, we address the problem of dividing the city between hospitals and determining the minimum number of hospitals needed to cover the city to ensure rapid medical aid.
Finally, in the last chapter we attempt to generalize the space by considering all points on the grid as points of interest.