Gate points in continuous location between regions with different ℓp norms

In this paper, a set of notable points is given from the ℓp-distance between a point P and a straight line r. The problem of the geodesic path between two points of R2 separated by a straight line r with different norms in each half-plane is then studied from the notable points of these points. This geodesic path provides a point in r, called the “Gate point”, which is studied by varying ℓp- and ℓq-distances, and the slope m of the straight line r. The analytical expression of the Gate point is given in some particular cases.