The Minkowski Metric

The k-Minkowski metric between two points p=(x1,y1) and q=(x2,y2) can be given as

where k >= 1.

For k=2, this becomes,

which is the Euclidean distance between the two points.

For k=∞, this becomes

The figure below shows the locus of points around a mesh node by taking d^k (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 .

If L denotes the set of points in the line drawing, then we can give a more formal definition of a curve point using the Minkowski metric.

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.