The point-line duality (Inselberg, 1985) is the most fundamental concept for parallel coordinates, as it provides the basis for mapping patterns from Cartesian to parallel coordinates and vice versa. The duality consists of two parts:

points in Cartesian coordinates are mapped to lines in parallel coordinates and

points in parallel coordinates are represented by lines in Cartesian coordinates.

The first part of the duality is straight-forward: mapping a point from Cartesian to parallel coordinates is a matter of locating the vertical position of the point-coordinates on the respective axes in parallel coordinates. The line that connects these two points is the line (in parallel coordinates) that represents a point (in Cartesian coordinates). To see how this works, add points to the Cartesian coordinate system on the left by clicking on the Canvas and see their dual lines appear in the parallel coordinate system on the right:

References

Inselberg, A. (1985). The Plane with Parallel Coordinates. The Visual Computer, 1(4), 69–91. doi:10.1007/BF01898350