Conference Paper

Description of Surfaces in Parallel Coordinates by Linked Planar Regions

DOI: 10.1007/978-3-540-73843-5_12 Conference: Mathematics of Surfaces XII, 12th IMA International Conference, Sheffield, UK, September 4-6, 2007, Proceedings
Source: DBLP


An overview of the methodology covers the representation (i.e. visualization) of multidimensional lines, planes, flats, hyperplanes,
and curves. Starting with the visualization of hypercubes of arbitrary dimension the representation of smooth surfaces is
developed in terms of linked planar regions. The representation of developable, ruled, non-orientable, convex and non-convex surfaces in ℝ3 with generalizations to ℝ
are presented enabling efficient visual detection of surface properties. The parallel coordinates methodology has been applied
to collision avoidance algorithms for air traffic control (3 USA patents), computer vision (1 USA patent), data mining (1
USA patent), optimization and elsewhere.

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Available from: Alfred Inselberg, Dec 03, 2015
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    ABSTRACT: This story is about multidimensional visualization with Parallel Coordinates (abbr. ||-coords) which transform the search for relations in multivariate datasets into a 2-dimensional pattern recognition problem. The discovery process is illustrated by finding: ground features from remote sensing satellite data, and useful rules for gold, foreign currencies and stockmarket trading from a nancial dataset. A complex dataset with two categories is classified using a geometric classification algorithm based on ||-coords. The minimal set of variables required to state the rule is found and ordered by their predictive value. A visual model of a real country's economy is constructed showing how multivariate relations can be modeled by means of hypersurfaces. Interior points corresponding to feasible economic policies are constructed interactively finding that two sectors unknowingly compete for the same group of workers. An overview of modern ||-coords provides foundational understanding. The representation of an M-dimensional hypersurface is obtained from its (M−1) subsets which are constructed recursively from its points. There are examples of surfaces where convexity can be seen in any dimension as well as non-orientability(as in the Möbius strip), and features like folds, crevices, bumps which are hidden or distorted in other types of displays. This is a prelude of what is on the way: recursive multidimensional interactivity for uncovering the secrets in massive datasets.
    No preview · Chapter · Dec 2008
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    ABSTRACT: This is one book that can genuinely be said to be straight from the horse's mouth. Written by the originator of the technique, it examines parallel coordinates as the leading methodology for multidimensional visualization. Starting from geometric foundations, this is the first systematic and rigorous exposition of the methodology's mathematical and algorithmic components. It covers, among many others, the visualization of multidimensional lines, minimum distances, planes, hyperplanes, and clusters of "near" planes. The last chapter explains in a non-technical way the methodology's application to visual and automatic data mining. The principles of the latter, along with guidelines, strategies and algorithms are illustrated in detail on real high-dimensional datasets. © Springer Science+Business Media, LLC 2009. All rights reserved.
    Full-text · Chapter · Jan 2009
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    ABSTRACT: With parallel coordinates multivariate relations and multidimensional problems can be visualized [7], [9], [8]. After an overview providing foundational understanding, we focus on some exciting recent results [6]. Hypersurfaces in Ndimensions are represented by their normal vectors, which are mapped into (N – 1) points in ℝ2, forming (N – 1) planar regions. In turn the shape and interior of these regions reveal key properties of the hypersurface. Convexity, various nonconvexities and even non-orientability (as for the Möbius strip) can be detected and “viewed” in high dimensions from just one orientation making this surface representation preferable even for some applications in 3-dimensions. Examples of data exploration & classification and Decision Support are illustrated at the end. The parallel coordinatesmethodology has been applied to collision avoidance algorithms for air traffic control (3 USA patents), computer vision (USA patent), data mining (USA patent) for data exploration, automatic classification, optimization, process control and elsewhere.
    No preview · Chapter · Oct 2009