also a pair of curves may not have a unique intersection. Hence by using curves in parallel coordinates instead of lines, the point line duality is lost together with all the other properties of projective geometry, and the known nice higher-dimensional patterns corresponding to (hyper)planes, curves, several smooth (hyper)surfaces, proximities, convexity and recently non-orientability. The goal is to map n-dimensional relations into 2D patterns. Hence, parallel coordinates is not a point-to-point mapping but rather a nD subset to 2D subset mapping, there is no loss of information. Note: even a point in nD is not mapped into a point in 2D, but to a polygonal line—a subset of 2D. Statistical considerations When used for statistical data visualisation there are three important considerations: the order, the rotation, and the scaling of the axes. The order of the axes is critical for finding features, and in typical data analysis many reorderings will need to be tried. Some authors have come up with ordering heuristics which may create illuminating orderings. The rotation of the axes is a translation in the parallel coordinates and if the lines intersected outside the parallel axes it can be translated between them by rotations. The simplest example of this is rotating the axis by 180 degrees. Scaling is necessary because the plot is based on interpolation (linear combination) of consecutive pairs of variables. Therefore, the variables must be in common scale, and there are many scaling methods to be considered as part of data preparation process that can reveal more informative views. A smooth parallel coordinate plot is achieved with splines. In the smooth plot, every observation is mapped into a parametric line (or curve), which is smooth, continuous on the axes, and orthogonal to each parallel axis. This design emphasizes the quantization level for each data attribute. Reading Inselberg () made a full review of how to visually read out parallel coords' relational patterns. When most lines between two parallel axis are somewhat parallel to each other, it suggests a positive relationship between these two dimensions. When lines cross in a kind of superposition of X-shapes, it's a negative relationship. When lines cross randomly or are parallel, it shows there is no particular relationship. Limitations In parallel coordinates, each axis can have at most two neighboring axes (one on the left, and one on the right). For a d-dimensional data set, at most d-1 relationships can be