Higher Dimensions
Adding more dimensions in parallel coordinates (often abbreviated ||-coords or PCs) involves adding more axes. The value of parallel coordinates is that certain geometrical properties in high dimensions transform into easily seen 2D patterns. For example, a set of points on a line in n-space transforms to a set of polylines (or curves) in parallel coordinates all intersecting at n − 1 points. For n = 2 this yields a point-line duality pointing out why the mathematical foundations of parallel coordinates are developed in the Projective rather than Euclidean space. Also known are the patterns corresponding to (hyper)planes, curves, several smooth (hyper)surfaces, proximities, convexity and recently non-orientability. It is worth mentioning that since the process maps a k-dimensional data onto a lower 2D space, some loss of information is expected. The loss of information can be measured using Parseval's identity (or energy norm).
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