Description of Surfaces in Parallel Coordinates by Linked Planar Regions.
ABSTRACT 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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ABSTRACT: With parallel coordinates multivariate relations and multidimensional problems can be visualized , , . After an overview providing foundational understanding, we focus on some exciting recent results . 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.10/2009: pages 123-141;
Description of Surfaces in Parallel Coordinates
by Linked Planar Regions
Chao-Kuei Hung1and Alfred Inselberg2
1CSIE Department, Shu-Te University, Kaoshiung, Taiwan
2School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
visualization) of multidimensional lines, planes, flats, hyperplanes, and
curves. Starting with the visualization of hypercubes of arbitrary dimen-
sion 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 R3with generalizations to RNare
allel coordinates methodology has been applied to collision avoidance al-
gorithms for air traffic control (3 USA patents), computer vision (1 USA
patent), data mining (1 USA patent), optimization and elsewhere.
Do It in Parallel!
Parallel coordinates (abbreviated as ?-coords) transforms N-dimensional objects
into distinct planar patterns enabling our fantastic pattern-recognition, aided
by interactivity, to obtain insight about multivariate problems. Starting with
an overview of the fundamentals the presentation leads to the representation of
surfaces in ?-coords focusing on core ideas and avoiding unessential technicalities
(for details see  and the forthcoming textbook ).
On the plane with xy-Cartesian coordinates a vertical line, labeled ¯ Xi, is
placed at each x = i − 1 for i = 1,2,...,N. These are the axes of the paral-
lel coordinate system for RN. A point C = (c1,c2,...,cN) is mapped into the
polygonal line C whose N-vertices with xy-coords (i − 1,ci) are on the parallel
axes. In C the full lines, not just the segments between the axes, are included
as illustrated in Fig. 1(left) for N = 5. The representation of an object S in
?-coords is denoted by¯S.
2Overview of Parallel Coordinates
A point P = (p1,p2) on the plane is represented by a line on the points (0,p1)
and (d,p2). As illustrated in Fig.1 (right), points on a line
? : x2= mx1+ b,
Duality in the Plane
R. Martin, M. Sabin, J. Winkler (Eds.): Mathematics of Surfaces 2007, LNCS 4647, pp. 177–208, 2007.
c ? Springer-Verlag Berlin Heidelberg 2007
178C.-K. Hung and A. Inselberg
¯? = (
? : x2= mx1+ b
Fig.1. (left) A point C = (c1,...,c5) ∈ R5is represented by a polygonal line¯C.
(right) Parallel coordinates induce a point ↔ line duality in R2.
are represented in ?-coords by lines intersecting at the point
1 − m,
1 − m
where d is the inter-axes distance. In 2-D then ?-coords induce a point ↔ line
duality (i.e. mapping points into lines and vice versa). Dualities properly reside
in P2; the Projective rather than R2the Euclidean plane. Here this is hinted by
the denominator in eq. 2. As m → 1 the point¯l → ∞ in the constant direction
with slope b/d. The full duality exists in P2for lines with m = 1 corresponding
to ideal points i.e. ‘directions’. Note that parallel lines are represented by points
having the same value of x. Using homogeneous coordinates1the mapping is a
? : [m,−1,b] →¯? : (d,b,1 − m)(3)
where the brackets [,] denote line coordinates. One does not need expertise in
projective geometry to work with ?-coords but awareness is advisable to avoid
1Denoted for a planar point by a triple between ().
Description of Surfaces in Parallel Coordinates by Linked Planar Regions179
(k − 1,pk,1)
(k − 1,pk,2)
(j − 1,pj,1)
(i − 1,pi,1)
(j − 1,pj,2)
(i − 1,pi,2)
Fig.2. The 3 points¯?i,j,¯?j,k,¯?i,k are on a line¯L for distinct i,j,k
2.2 Lines in RN
A line ? in Euclidean N-space RNis completely described in terms of N − 1
independent projections on the xixj2-planes given by
?i,j: xi= mi,jxj+ bi,j.
By the line → point mapping, eq. (2), these 2-D lines are represented by the
points (assuming i < j)
where the distance between adjacent axes¯ Xi−1and¯ Xiis one. Here then d = j−i
and the translation is needed since the¯ Xi and y-axes are (i − 1) units apart.
Hence, ? is represented by N − 1 such points where the indexing is essential
specifying the linear relations between the variable pairs xi and xj and also
needed as input to algorithms (e.g. finding the minimum distance between pairs
of lines ). An important consequence of Desargues theorem is that for any
i ?= j ?= k the 3 points¯?ij ,¯?jk ,¯?ik are collinear Fig. 2. In particular for a
line ? ⊂ R3the¯?12,¯?13,¯?23are always on a line¯L. The 3-point-collinearity
property has higher dimensional generalizations.
The¯?i,i+1 (j = i + 1) are the most commonly used representing points.
A polygonal line¯P on the N − 1¯?i,i+1 necessarily represents a point P =
(p1,...,pi−1,pi,...,pN) ∈ ? since the pair of values pi−1,pisimultaneously satisfy
eq. (4) for every pair i = 2,...,N. In Fig.3 several polygonal lines are seen inter-
secting at the¯?i−1,i representing a line ? ⊂ R10. That is, both the line and its
j − i
1 − mi,j
+ (i − 1),
1 − mi,j
180C.-K. Hung and A. Inselberg
Fig.3. Interval on a line ? ⊂ R10, the polygonal lines intersect at 9 points¯?i,i+1, i =
1,...,8 which provide the complete description of ?. The point¯?12 being to the right
of¯ X2-axis ⇒ slope of ?12 is ∈ [0,1], ?67 has slope 1 since the corresponding lines are
parallel. The remaining ?i,i+1 have negative slopes with their¯?i,i+1 being in between
points are visualized. The representation of a line in terms of N − 1 points still
holds when some xiare constant  and occurs in the hypercube’s representation
2.3Planes, Hyperplanes and Recursion
Vertical Line Representation. While a line can be determined from its pro-
jections, even in 3-D it is not possible to identify a (full) plane from two arbi-
trary projections. By contrast, coplanarity is nicely characterized with ?-coords.
A plane π shown in Fig. 4(left), intersects the x1x2and x2x3planes at the lines
y1and y2respectively with A = y1∩ y2. Let Yibe the family of lines parallel
to yi, i = 1,2. Each yibeing a line in R3is represented by two points the ¯ yi
and ¯ yi
x-axis not shown. A non-orthogonal coordinate system on π is formed, using
the yias axes, so that a point P is determined as the intersection of two lines
parallel to y1and to y2respectively. As pointed out in subsection 2.1, each of
the families of lines Yiis represented by a vertical line¯Yicontaining the point
¯ yi. Actually¯Yi represent the projections of the parallel lines on the x1x2 and
x2x3planes respectively. By choosing a point A ∈ π a distinct plane is specified
23. To simplify, here ¯ y1
12= ¯ y1, ¯ y2
23= ¯ y2with the other two points on the
Description of Surfaces in Parallel Coordinates by Linked Planar Regions181
Fig.4. (left) A plane π in R3represented by two vertical lines and a polygonal line. A
set of coplanar grid points – note the two vertical lines pattern (right).
A = (0,0) B
C = (1,1)
Fig.5. (a) Square (b) Cube in R3(c) Hypercube in R5– all edges have unit length