Isoperimetrically optimal polygons in the triangular grid with Jordan-type neighbourhood on the boundary

摘要

The digital spaces have some properties that are not present in the Euclidean space. A digitized circle do not necessarily have the smallest (digital arc length) perimeter of all objects having a given area. In digital geometry various measures of perimeter and area lead to various definitions of digital circles using the digital version(s) of the isoperimetric inequality. Usually the square grid is used as digital plane with either the cityblock or the chessboard neighbourhood relation. In this paper the triangular grid is also used with two types of neighbourhood relation that play importance in Jordan curves. We search for those (digital) objects that have optimal measures and therefore they can be considered as digital circles by our definition. We show that special, (almost) regular hexagons are Pareto optimal, i.e. they fulfil both versions of the isoperimetric inequality: they have maximal area among objects having perimeter at most a given length; and they have minimal perimeter among objects enclosing at most a certain area. The optimal objects can be build in a similar way as the Wang-spiral for the square grid.