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No but parallel lines have a constant distance between them
parallel lines - they are parallel when the distance between them remains constant
No.
If the distance between the lines is constant then they are parallel.
The answer depends on the grid. On the taxicab grid, which was studied by Minkowski, the distance is the sum of the vertical and horizontal distances between a and b. See, for example, https://en.wikipedia.org/wiki/Taxicab_geometryIf, a and b have horizontal and vertical coordinates which are, respectively, (xa, ya) and (xb, yb) then the grid distance is abs(xa - ya) + abs(xb - yb).