The Minkowski Metric
The k-Minkowski metric between two points p=(x1,y1) and q=(x2,y2) can be given as
where k >= 1.
which is the Euclidean distance between the two points.
The figure below shows the locus of points around a mesh node by taking dk (p,q)= for k=1,2 and ∞. For k=∞, the locus is a square of side T. For k=2, it is a circle of radius and for k=1, it is a rhombus of diagonal T and side .
Definition: For a line drawing L and a mesh gap T, a mesh node is a curve point, according to the k-Minkowski metric quantization, if and only if the distance from the node to the closest point in the set L is less than .
If k=∞, the quantization scheme becomes the square quantization and for k=2, it becomes the circular quantization scheme. For k=1, a new quantization scheme arises called the rhombic quantization. Convex quantizations can be referred to as all schemes where the region checked around the mesh node is convex.