Page 1

Description of Surfaces in Parallel Coordinates

by Linked Planar Regions

Chao-Kuei Hung1and Alfred Inselberg2

1CSIE Department, Shu-Te University, Kaoshiung, Taiwan

ckhung@mail.stu.edu.tw

2School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

aiisreal@math.tau.ac.il

Abstract. Anoverviewofthemethodologycoverstherepresentation(i.e.

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

presentedenablingefficientvisualdetectionof surfaceproperties.Thepar-

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.

1Introduction

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 [1] and the forthcoming textbook [2]).

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

2.1

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

(1)

R. Martin, M. Sabin, J. Winkler (Eds.): Mathematics of Surfaces 2007, LNCS 4647, pp. 177–208, 2007.

c ? Springer-Verlag Berlin Heidelberg 2007

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178C.-K. Hung and A. Inselberg

mm

x2

x1

?

¯?

x

y

¯ X1

¯X2

x2

¯ X1

¯X2

xx1

d

A1

A2

¯A2

¯A1

a1

a2

a1

ma2+ b

ma1+ b

a2

y

¯? = (

d

1−m,

b

1−m)

? : 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

¯? :

?

d

1 − m,

b

1 − m

?

,

(2)

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

linear transformation:

? : [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

blunders.

1Denoted for a planar point by a triple between ().

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Description of Surfaces in Parallel Coordinates by Linked Planar Regions 179

¯?i,k

¯?j,k

¯?i,j

¯Xi

¯Xj

¯Xk

(k − 1,pk,1)

(k − 1,pk,2)

(j − 1,pj,1)

(i − 1,pi,1)

x

(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

2.2Lines 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.

(4)

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 [3]). 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

¯?i,j:

j − i

1 − mi,j

+ (i − 1),

bj−i

1 − mi,j

?

(5)

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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

the axes.

points are visualized. The representation of a line in terms of N − 1 points still

holds when some xiare constant [4] and occurs in the hypercube’s representation

Fig. 5.

2.3 Planes, 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

12

23. To simplify, here ¯ y1

12= ¯ y1, ¯ y2

23= ¯ y2with the other two points on the

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Description of Surfaces in Parallel Coordinates by Linked Planar Regions181

x1

x3

x2

x

y

A

¯X1

¯X2

¯X3

y1

¯Y1

¯Y2

¯ y1

¯ y2

¯A

y2

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).

x1

¯X1

¯X1

x

x

¯X2

¯X2

¯X5

¯X3

¯X3

¯X4

(1,0,0)

(0,0,0,1,0)

0

¯X1

0

0

1

1

1

x2

¯X2

A = (0,0) B

¯A

¯C

x1

AB

x

¯D

¯B

x2

y

x3

y

C = (1,1)

CD

D

y

(a)

(b)

(c)

Fig.5. (a) Square (b) Cube in R3(c) Hypercube in R5– all edges have unit length

Page 6

182C.-K. Hung and A. Inselberg

having¯Y1.¯Y2as its ? coordinate system and¯A its origin. Clearly the same ar-

gument can be extended to any dimension concluding that a hyperplane in RN,

which is an N −1 dimensional flat, can be represented by a N −1 vertical lines

and a polygonal line representing one of its points. Conversely, a set of coplanar

points chosen on a grid, such as the one formed by lines parallel to a coordinate

system, is represented by polygonal lines whose pattern specifies two vertical

lines as shown in Fig. 4 (right).

The representation of a hypercube Fig. 5 is a preview of coming attractions. A

square (a) is displayed in Cartesian and ?-coords dually mapping a vertex A into

a line¯A and an edge AB into a point AB. A cube (b) in 3-D, positioned in the

first quadrant with a vertex at the origin and axes-parallel sides of unit length,

is represented by twice the square’s pattern the vertices’ coordinates being all

ordered triples of 0’s and 1’s. The edges are found by the intersection of the

polygonal lines representing the two vertices. For example, the two points at

(1,0), (2,0) represent the edge on the vertices (0,0,0), (1,0,0). The polygonal

lines corresponding to these vertices together with the¯ X2,¯ X3 axes represent

the faces (planes) at x = 0, x = 1. Here the two vertical lines representing

the planes coincide with the parallel axes. All the cube’s vertices, edges and

faces are ascertained (i.e. are “visible”) from the pattern where any interior

polygonal line (i.e. within the rectangle x ∈ [0,2], y ∈ [0,1]) represents one

of the cube’s interior points. The representation of a hypercube in R5, with one

vertex at the origin and axes-parallel edges, is 4 times the square’s pattern and all

vertices, edges and faces of all order can be determined from the picture and best

viewed interactively the same way as for the 3-D cube. The pattern’s repetition

pleasingly reveals the hypercube’s symmetries and relation to the square. and

the first an instance of a hypersurface’s representation by linked planar regions.

The ease of generalization from 3-D to higher dimensions is “built-in” in ?-coords

and also applies to the representation of polytopes [5] and much more complex

objects to be seen in the ensuing.

Recursion. While it is easy detecting coplanarity of points on a regular grid,

for randomly chosen coplanar points no pattern is apparent Fig. 6 (left). A new

approach is called for [6]. Rather than its points, an M dimensional hyperplane

is represented in terms of its M − 1 flats which are in turn represented by

their M − 2 flats and so on. Starting from its points, a recursive construction

algorithm builds the hyperplane’s flats increasing the dimension by one at each

step. Let’s see how it works for the points on the plane π in Fig. 6 (left). Lines

on π are constructed from pairs of polygonal lines providing the line¯L of the

3 point collinearity described above and shown in Fig. 2. The result seen on

Fig. 6 (right) is stunning. All the¯L lines intersect at a point which turns out to be

characteristic of coplanarity but not enough to specify the plane π. Translating

the first axis¯ X1to the position¯ X1?, one unit to the right of the¯ X3 axis and

repeating the construction, for the¯ X2,¯ X3,¯ X1? coordinate system, yields a second

point seen in Fig. 7 (left). These two points suffice to identify the plane π.

They are denoted by ¯ π123= ¯ π0? , ¯ π1?23= ¯ π1? having 3 indices, since the plane’s

equation involves 3 variables, and are distinguished from the points representing

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Description of Surfaces in Parallel Coordinates by Linked Planar Regions 183

a line which have 2 indices. Two more points are similarly obtained (translating

the¯ X2and then¯ X3axes 3 units to the right). From the 4 points, Fig. 7 (right),

the plane’s equation can be read from the picture! Just the two points are needed

to characterize the plane uncluttered by polygonal lines.

A few words are in order about the nice geometry behind this result. The

family of planes, called super-planes (abbr. sp):

πs: (x3− x2) + k(x2− x1) = 0 , k =d2− d3

d2− d1

.

(6)

are on the line u through the points (0,0,0) and (1,1,1). The ratio k is de-

termined by the axes spacing where the¯ Xiare placed at the distance difrom

the y-axis. In this axes system points of the associated sp, determined by k,

are represented by straight lines (try it). With the y and¯ X1 axes coincident,

d1= 0, d2= 1, d3= 2 specifying the sp (called the “first super-plane”) given

by πs

straight lines in the¯ X1,¯ X2,¯ X3coordinate system and all points so represented

belong to πs

by a single point; the intersection of two straight lines (representing points of

πs

line ? ⊂ π being on the 3 points¯?12,¯?13,¯?23. Since it is represented by a straight

rather than a polygonal line, L = π ∩ πs

a pair of polygonal lines determines another L?∈ pi ∩ πs

are on the line ?π = π ∩ πs

Fig. 6 (right).

The translation ¯ X1 →

πs

1?

1: x1− 2x2+ x3= 0 . As already mentioned, its points are represented by

1. Hence a line on πs

1even though it is 3-dimensional is represented

1). Further, the line¯L in the 3-pt-collinearity, Fig. 2, represents a point on the

1. Another line ??⊂ π constructed from

1. The points L,L?

1, seen in Fig. 8, represented by the single point in

¯ X1? , d3 = 0 → d3? = 3 specifies the second sp

x1+ x2− 2x3 = 0 , corresponding to a 120orotation of πs

:

1→ πs

1?

Fig.6. (left) The polygonal lines on the first 3 axes represent a randomly chosen set of

coplanar points. (right) Seeing coplanarity! Constructing lines on the plane, with the

3 point collinearity property, yields intersection at a point.

Page 8

184C.-K. Hung and A. Inselberg

y

x

¯X2

¯X3

¯X1?

¯X2?

¯X3?

¯ π1?2?3

¯ π231?

¯ π1?2?3?

3c1

3c2

3c3

¯X1

¯H

¯ π123

c0

c1+c2+c3

Fig.7. (Left)The two points where the lines intersect uniquely determine a plane π

in 3-D.(Right) From four points, constructed by consecutive axes translations, the

normalized (i.e. c1+c2+c3 = 1) coefficients of π : c1x1+c2x2+c3x3 = c0 can be read

from the picture!

u

x1

x2

x3

?π

??

π

π

πs

1

πs

1?

H

Fig.8. The intersections of a plane π with the two super-planes πs1 and πs

are the two lines ?π ,??

Fig. 7 (right), and provide its representation.

1?. These

π which specify the plane, represented by two points as in

about u. Repeating the construction for the¯ X2,¯ X3,¯ X1? axes system produces

the second point in Fig. 7 (left) and equivalently the second intersecting line

??

π. In retrospect for the constructions in Fig. 6 and Fig. 7, just two intersecting

π= π ∩ πs

1? seen in Fig. 8. The lines ?π,??

πintersect on u and specify the plane

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Description of Surfaces in Parallel Coordinates by Linked Planar Regions185

Fig.9. Randomly selected points on a hyperplane π5(a 5-flat) in R6. The ¯ π12 , ¯ π23

portions of the 1-flats ⊂ π5constructed (right) from the polygonal lines. No pattern is

evident. nor for the subsequent construction (not shown) of the points ¯ πkrepresenting

2 and 3-flats of π5.

lines are needed to determine the points. The x coordinates of the 3rd and 4th

points in Fig. 7 are obtained from x2? = 6−(x0? +x1?) , x3? = 3+x0? where xi?

denotes the x coordinate of ¯ πi?.

The gist is that while coplanar points can not be visually detected, a plane can

be recognized from its lines in R3and a hyperplane from its N −2-flats in RN.

In general, a hyperlane in N-dimensions is represented uniquely by N −1 points

each with N indices. The algorithm constructs these points recursively, raising

the dimensionality by one at each step, as is done here starting from points (0-

dimensional) constructing lines (1-dimensional) and so on. An example is shown

in Figs. 9 and 10 constructing the representing points for a hyperplane in R6.

All the nice higher dimensional projective dualities like point ↔ hyperplane,

rotation ↔ translation etc hold. Further, a multidimensional object, represented

in ?-coords, can still be recognized after it has been acted on by projective

transformations (i.e. translation, rotation, scaling and perspective). The recur-

sive construction and its properties are at the heart of the ?-coords methodology.

Page 10

186 C.-K. Hung and A. Inselberg

Fig.10. This is it! On the left are the ¯ π4

from the polygonal lines joining ¯ π3

that the original points in Fig. 9(left) are on a 5-flat in R6. The remaining points of

the representation are obtained in the same way and all 5 representing points of π5

plus the two additional ones as in Fig. 7 (right) are seen on the right. The coefficients

of π5’s equation are equal to 6 times the distance between sequentially indexed points.

12345 , ¯ π4

2345, ¯ π3

23456 of a 4-flat ⊂ π5constructed

3456 of two 3-flats in π5. This shows

1234, ¯ π3

Translation ↔ Rotation. The rotation ↔ translation correspondence be-

tween ? and Cartesian coordinates works for flats in RN. It is illustrated in

Fig. 11 for R3with the rotation of a plane π2about a line π1. The superscripts

indicate the flats’ dimensionality. When it is clear from the context, the super-

scripts are dropped for the dimensionality is apparent from the number of indices.

For any plane π the four points ¯ π123, ¯ π231? ;, ¯ π31?2? , ¯ π1?2?3? lie on a horizontal

line¯H. For the line π1, as previously mentioned, the 3 points ¯ π12, ¯ π13, ¯ π23are

on the same line¯L. It is useful to include the representation of the line π1in

terms of the other triples of axes. Namely, the points

¯ π1?2¯ π1?3¯ π2?3on line¯L?

The point ¯ π123is on the intersection of¯H with the line¯L and similarly

¯ π231? =¯H ∩¯L?, ¯ π31?2? =¯H ∩¯L??, ¯ π1?2?3? =¯H ∩¯L???.

A rotation of the plane π2about the line π1becomes a vertical translation of

the line¯H with ¯ π123translated along the line¯L; similarly the differently indexed

, ¯ π12? ¯ π13? ¯ π2?3? on line¯L??

, ¯ π12? ¯ π13? ¯ π2?3? on line¯L???.

(7)

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Description of Surfaces in Parallel Coordinates by Linked Planar Regions187

¯L?

¯L??

¯L

¯H

¯ π13

¯ π231?

¯ π12

¯ π1?2

¯ π1?2?

X

¯ π2?3?

¯ π31?2?

¯ π2?3

¯L???

¯ π1?3?

¯ π1?3

¯ π1?2?3?

¯ π23

¯ π123

Fig.11. Rotation of a 2-flat (plane) about a 1-flat(line) in R3corresponds to a trans-

lation of its representing points (with 3 indices) on the horizontal line¯ H along the

lines¯L ,¯L?,¯L??,¯L???joining the representing points (with 2 indices) representing the

2-flat.

points are translated along their corresponding lines¯L?,¯L??and¯L???as dictated

by eq. (7). At any instance, the (directed) distance between adjacent points is

equal to the coefficients of the plane’s equation as pointed out in Fig. 7 (right).

Where pairs of¯L-lines intersect one coefficient is zero and π2is perpendicular

to a principal 2-plane. Specifically, at

¯H ∩¯L ∩¯L?,

¯H ∩¯L?∩¯L??, c2= 0 and π2⊥ x1x3− plane ,

¯H ∩¯ L??∩¯L???, c3= 0 and π2⊥ x1x2− plane .

Consider π as an oriented plane with normal vector N, whose components

are the coefficients ci. As vertically translating¯H crosses a pairwise intersection

of¯L a ci changes sign. So after traversing all 3¯L intersections the oriented

plane completes a 1800rotation i.e. the normal N → −N. This is an informal

description of the “flipping lemma” revisited later in subsection 5 in connection

with the non-orientability of the M¨ obius strip.

c1= 0 and π2⊥ x2x3− plane ,

2.4 Curves

The image ¯ c of a point-curve

c : F(x1,x2) = 0 ?→ ¯ c : f(x,y) = 0(8)

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188C.-K. Hung and A. Inselberg

can be obtained as a line-curve the lines images of points of c. The better way

is to map the tangents of c directly into points obtaining a point-curve image ¯ c

whose point coordinates are given by:

⎧

⎪

⎪

⎩

⎨

x = d

∂F/∂x2

(∂F/∂x1+∂F/∂x2)

,

y =(x1∂F/∂x1+x2∂F/∂x2)

(∂F/∂x1+∂F/∂x2)

.

(9)

obtained directly from eq. (2) with m = −∂F/∂x1

The zeros of the denominator provide the ideal points, if any, of ¯ c where the

tangent has slope 1. For conics F is quadratic and the the partial derivatives in

eq. (9) are linear. So (x1,x2) ?→ (x,y) is a M¨ obius transformaton and so is its

inverse [7]. Therefore, the inverse relation of (x1,x2) in terms of (x,y) is also

quadratic showing that

∂F/∂x2with inter-axes distance d.

Conics ↔ Conics .

(10)

generalizing to higher-dimensional quadric hypersurfaces. Curves play a crucial

role in the ensuing being the boundaries of the linked planar regions representing

surfaces. For algebraic curves, as in Fig. 12, Pl¨ ucker’s formulae [8] are very

useful. A couple of examples, Figs. 12 and 13, are in order.The dualities cusp

↔ inflection-point, crossing-point ↔ bitangent hold and in fact carry over to

hypersurfaces.

−505

−5

−4

−3

−2

−1

0

1

2

3

4

5

X1

X2

−505

−5

−4

−3

−2

−1

0

1

2

3

4

5

X1’ X2’

X

Y

Fig.12. The image of the curve c : F(x1,x2) = x3

4 (determined from Pl¨ ucker’s formula). This is an example of the crossing-point ↔

bitangent duality.

1+ x2

1− 3x1x2 = 0 has degree

Page 13

Description of Surfaces in Parallel Coordinates by Linked Planar Regions189

−6−4−20246

−5

−4

−3

−2

−1

0

1

2

3

4

5

X1

X2

−505

−5

−4

−3

−2

−1

0

1

2

3

4

5

X1’ X2’

X

Y

Fig.13. The image of c : x2 = sin(x1) for x1 ∈ [−2π,2π]. A periodic curve has image

symmetric about the x-axes. This example illustrates the ip – inflection point ↔ cusp

duality. The ip at x1 = ±π are mapped into the two cusps, and the ip at the origin

having slope m = 1 is mapped to the ideal point along the x-axis.

π

¯ σ231?

¯ σ123

x

¯X1

¯X2

¯X3

y

x1

x3

x2

σ

P

Fig.14. A surface σ ∈ E is represented by two planar regions ¯ σ123 , ¯ σ231?. consists of

the pairs of points representing all its tangent planes.

3Surface Representation – Formulation

The representation is developed for the family E of smooth hypersurfaces

in RN

which are the envelopes of their tangent hyperplanes When the

Page 14

190 C.-K. Hung and A. Inselberg

higher-dimensional generalizations are clear the discussion is confined to R3. Fig.

14. At each point P ∈ σ, the tangent plane π is mapped to the two planar points

¯ π123 , ¯ π231?. Collecting the points by their indices for all the tangent planes

yields two planar regions ¯ σ123(x,y) , ¯ σ1?23(x,y) one each for the index triples.

The regions need to be linked via a matching algorithm which selects the pairs

of points (one from each region) representing valid tangent planes of σ. The two

linked regions form the representation ¯ σ of the surface. This is the extension of

the linking already due to the indexing of the representing points for lines and

hyperplanes. The intent is to reconstruct the surface from its representation. The

manner and extent to which this is possible, i.e. when the ?-coords mapping is

invertible, are the issues studied. Formally, for a surface σ ∈ E given by

σ : F(x) = 0 , x = (x1,x2,x3)(11)

its representation in ?-coords is

σ ?→ ¯ σ = (¯ σ123(x,y) , ¯ σ1?23(x,y)) ⊂ P2× P2

where use of the projective plane P2allows for the presence of ideal points. In the

notation the link between the two regions is indicated by placing them within

the ( , ) which are omitted when the discussion pertains to just the region(s).

The functions used are assumed to be continuous in all their variables together

with such of its derivatives as are involved in the discussion.

The gradient vector of F, ∇F = (∂F

to the surface σ at P and so the the tangent plane π of σ at the point P0(x0),

(x0) = (x0

(12)

∂x1,∂F

∂x2,∂F

∂x3)??

P, at the point P is normal

1,x0

2,x0

3), is given by

π : ∇F · (x − x0) =

3

?

i=1

(xi− x0

i)∂F

∂xi(x0

1,x0

2,x0

3) = 0 .

(13)

At times it is preferable to describe a surface in terms of 2 parameters as

σ : F(s,t) = F(x) = 0 ,

x = x(s,t) , s ∈ Is, t ∈ It,

(14)

where Is, Itare intervals of R with the representing points of π

¯ πi?(s,t) = (∇F · di, ∇F · (x0) , ∇F · u) ,

The equivalent description for hypersurfaces in RNrequires N − 1 parameters.

This parametric form is due to Gauss. The vector description is in many ways

simpler, clarifying the surface’s structure and higher-dimensional generalization,

is due to the mathematical physicist W.J. Gibbs who was an early advocate for

visualization in science2.

i = 0,1.

(15)

2In 1873 he wrote the two papers: “Graphical Methods in the Thermodynamics of Flu-

ids” and “A Method of Geometrical Representation of the Thermodynamics Prop-

erties of Substances by Means of Surfaces”.

Page 15

Description of Surfaces in Parallel Coordinates by Linked Planar Regions 191

Stated explicitly the point ?→ pair-of-points mapping,

x ∈ σ ?→ π ?→ (¯ π0?, ¯ π1?) = (¯ π123, ¯ π1?23) = ((x0?,y),(x1?,y)) ,

the x = x0? and x?= x1? are the x-coordinates of ¯ π1?23, ¯ π1?23respectively. The y

is the same for both points. In terms of the gradient’s components, Fi = ∂F/∂xi

(16)

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

⎨

x0? =

F2+2F3

F1+F2+F3,

y =x1F1+x2F2+x3F3

F1+F2+F3

,

x1? =3F1+F2+2F3

F1+F2+F3

.

(17)

These transformations are the direct extension of the 2-D point ↔ point curve

transformations eq. (9) A word of caution, when the inter-axes distance d ?= 1,

the right-hand-sides of x0? and x1? above need to be multiplied by d and 2d

respectively (see eq. (9)).

3.1Boundary Contours

The construction of the regions ¯ σ, and in particular their boundary ∂¯ σ, is

greatly facilitated by the next lemma. To guarantee that the intersections below

are non-empty the origin is, translated if necessary, to an interior point of the

surface σ.

Lemma 1. (Boundary of ¯ σ) For σ a surface in R3, ∂¯ σ123is the image of σ ∩ πs

That is ∂¯ σ123= σ ∩ πs

Proof. With reference to Fig. 15 each point ¯ ρ123∈ ¯ σ123is a representing point of

a plane ρ tangent to the surface σ at a point R. In fact, ¯ ρ123=¯?ρwhere ρ∩πs

?ρ. In addition, for a plane π tangent to σ at a point P where ¯ π123∈ ∂¯ σ123then

the corresponding ?π= π ∩ πs

at ¯ π123=¯?123⇒ P ∈ πs

due to the smoothness of ∂σ and, therefore, continuity of ∂¯ σ123, every tangent

to ∂¯ σ123represents a point of πs

∂¯ σ1?23= σ ∩ πs

The boundary ∂¯ σ is simply the image of two curves: the intersections of σ with

the first two superplanes πs

point P ∈ σ, ¯ π123∈ ∂¯ σ123and ¯ π1?23∈ ∂¯ σ1?23⇔ P ∈ σ ∩ πs

Lemma 2. (Boundary of ¯ σ in RN) For a σ ⊂ RN, ∂¯ σ is composed of

N −1 curves which are the images of the intersections of σ with the first N −1

superplanes.

0?.

0? and similarly ∂¯ σ1?23= σ ∩ πs

1?.

0? =

0?, P ∈ ?π. The image¯P is a line tangent to ∂¯ σ123

0?. In short, every point of ¯ σ123represents a line of πs

0?

0? and ∂σ. Hence ∂¯ σ123= σ ∩ πs

0? and similarly

1? . The smoothness of ∂σ ensures the continuity of ∂¯ σ123.

0?,πs

1?. Note further that for a plane π tangent at a

0? ∩ πs

1?.

An algebraic surface is one described by a polynomial equation providing an

important special case of Lemma 2.

Page 16

192C.-K. Hung and A. Inselberg

σ

tangent planes

σ

πs

0?

πs

0?

π

R

?π

P

?ρ

¯ π1?23

¯ σ1?23

¯P

¯ π123

¯ σ123

¯ ρ123

¯ ρ1?23

ρ

Fig.15. Formation of boundary contour

Corollary 1. (Boundary of an algebraic surface) The boundary ¯ σ of an alge-

braic surface σ of degree n is composed of N −1 algebraic curves each of degree

n(n − 1) or less.

The corresponding known result in Algebraic Geometry is that the dual of a non-

singular algebraic surface of degree n has degree n(n − 1)(N−1)[9], [10]. Here

the boundary representations can be found with the aid of Pl¨ ucker’s formulae.

From this lemma we can conclude that the for quadric surfaces (i.e. those whose

degree is 2) their boundary contours are conics.

Corollary 2. (Ideal points) An ideal point on the boundary ∂¯ σ is the image of

a tangent plane, at a point of σ, parallel to u.

Proof. It is an immediate consequence of eq. (15) that ideal points occur in the

image when the 3rd component is zero. That is the tangent plane’s normal is

normal to u or the tangent plane is parallel to u.

The ideal points map into asymptotes determining the branches of the boundary

curves. The boundary ∂¯ σ is the image of space curves contained in the super-

planes. A space curve can be represented as the image of its tangent lines. In

turn, the tangent t at point P ∈ c can be represented by N − 1 doubly indexed

points say¯ti−1,iso that ¯ c consists of N − 1 planar curves ¯ ci−1,iformed by the

¯ti−1,i; two such curves for R3. Each ¯ ci−1,i is the image of the projection of c

on the 2-plane xi−1,i The fun part is that, in our case, these two curves are

coincident since they are the images of a curve in a super-plane say c ⊂ πs

where each tangent t at a point P ∈ c is a line in πs

0?

1, and hence¯t12=¯t23.

Page 17

Description of Surfaces in Parallel Coordinates by Linked Planar Regions193

4Developable Surfaces

A smooth surface σ in R3generally has a 2-parameter family of tangent planes so

that its ?-coord representationconsists of two indexed planar regions ¯ σ123, ¯ σ1?23.

Of particular interest is the special case when σ has a 1-parameter family of tan-

gent planes. Then the ¯ σ123 , ¯ σ1?23 degenerate into two curves. Surfaces with

this property are precisely the class of developables, that is they can be un-

folded(developed) into a plane without streching or contracting. Examples are

cylinders, cones and their generalizations as well as more complex surfaces like

the helicoid in Fig. 20.

As we shall see the simplicity of developable surfaces results in many pleasant

properties of their ?-coord representations. A simple procedure exists to recon-

struct the surface given an indexed pair of monotonic pieces of its representation.

These pieces can then be strung together with some additional auxiliary pairs of

marks. The representation is indeed unique for a large subclass of developable

surfaces covering complex enough surfaces even if we restrict our attention to

cases where the number of marks is finite.

4.1 Reconstruction

The reconstruction procedure can be summarised in a few sentences. Pick a point

on one representing curve and find the other corresponding point (point at the

same height) on the other representing curve. Then two tangent lines at these

point determine a ruling of the surface enabling its reconstruction.

Consider a cone σ and two of its tangent planes π(t0) and π(t) as shown on

the left of Fig. 16 where the family of tangent planes of σ is parameterized by t.

On the right is its ?-coord representation, a pair of curves ¯ σ123and ¯ σ231? instead

of a pair of regions since σ is a developable surface. The secant line l(t,t0)

X

Y

Z

π(t)

π(t0)

X

Y

X1X2X3 X1’

o

¯ π123(t0)

o

¯ π123(t)

x

x

¯ σ123

¯ σ231?

η1(t,t0)

η2(t,t0)

η3(t,t0)

l(t,t0)

Fig.16. The intersection of the two tangent planes π(t) and π(t0) approaches the ruling

of tangency ρ(t0) at π(t0) as the tangent plane π(t) approaches π(t0). Accordingly, the

secant line l(t,t0) joining the representation of these two tangent planes approaches

the tangent line of ¯ σ123 at the point ¯ π(t0).

Page 18

194C.-K. Hung and A. Inselberg

X

Y

X1 X2X3 X1’

o

¯ π∗

o¯ π∗

231?

123

X

Y

X1X2 X3X1’

o

o

o

o

o

o

*

¯ ρ(t0)12

*

¯ ρ(t0)23

η∗

1

η∗

2

η∗

3

ζ∗

1?

ζ∗

2

ζ∗

3

Fig.17. The reconstruction procedure: Draw the tangent lines at the two corresponding

points representing a tangent plane π(t0) to σ (left), find the intercepts of each line

with the correct set of axes (either 123 or 231’), and finally join the corresponding pair

of intercepts to obtain the two end points of the polygonal line representing a ruling

ρ(t0). (right).

joining ¯ π123(t0) and ¯ π123(t) necessarily represents the intersection (a straight

line) of the two tangent planes π123(t0) and π123(t) because of duality. Recalling

the point representation of lines on the superplane πs

point in space η(t,t0) ≡ π(t) ∩ π(t0) ∩ πs

this secant line with the¯ X1,¯ X2, and¯ X3axes. As t approaches t0, the line of

intersection approaches ρ(t0) the line of tangency of π(t0) to σ and also a ruling

of σ. In the ?-coord plane this corresponds to ¯ π123(t) approaching ¯ π123(t0) and

the secant line l(t,t0) approaching the tangent line of ¯ σ123at ¯ π123(t0). Let’s write

η∗≡ limt→t0η(t,t0) = ρ(t0) ∩ πs

π(t0) with the special plane πs

them off the intercepts.

Up to now we have focused on only one of the two representing curves of

σ and obtained η∗. Repeating the same arguments on the other representing

curve ¯ σ231?, we similarly obtain another point ζ∗≡ limt→t0ζ(t,t0) = ρ(t0)∩πs

the intersection of the ruling of tangency with the other special plane. Fig. 17

shows the ?-coord representation of η∗as the three intercepts (η∗

that of ζ∗as the three intercepts (ζ∗

by joining η∗and ζ∗in space, which translates to the multi-dimensional line

construction in the ?-coord plane as illustrated in Fig. 3. A note of caution: the

ζ∗

to be translated to the¯ X1axis before joining it with η∗

We have just seen how to recover the developable surface from its ?-coord

representation. This procedure is clearly general for it does not depend on any

property of the cones. In fact we have simultaneously proved the uniqueness of

the ?-coord representation for a developable surface provided its representing

curves are monotonic with respect to the y axis.

1the coordinates of the

1are found by reading the intercepts of

1, the intersection of the ruling of tangency at

1, and write its coordinates (η∗

1,η∗

2,η∗

3) as we read

1?,

1,η∗

2,η∗

3), and

1,ζ∗

2,ζ∗

3). The ruling is now readily obtained

1intercept was obtained on the¯ X1? but in order to construct the ruling, it has

2.

Page 19

Description of Surfaces in Parallel Coordinates by Linked Planar Regions 195

4.2Resolution of Ambiguity

It turns out that with the help of a few additional marks in the picture, the

uniqueness result still holds valid given much weaker requirement on the rep-

resenting curves than the obvious monotonic constraint as mentioned in the

previous section.

For a cone, a cylinder, or any developable surface σ whose representation

curves ¯ σ123 and ¯ σ231? do not have any self-intersection points, the reconstruc-

tion procedure can be applied to a given pair of starting points ¯ π(t)123 ¯ π(t)231?

at t = t0 and repeated as this pair of points are traced smoothly along the

two curves. This means that, for example, when ¯ π(t)123 reaches an extremal

point and enters the other Y -monotonic branch on ¯ σ123, we choose the corre-

sponding ¯ π(t)231? that also goes into the other Y -monotonic branch on ¯ σ231?.

Thus there is no ambiguity in this case even if the representing curves are not

Y -monotonic. Even for developable surfaces each of whose ¯ σ123 and ¯ σ231? has

some self-intersection points, there is no ambiguity as to which branch to fol-

low at the intersections, as long as the branches don’t share the same tangent

line except in pairs, since we insist on smooth tracing. Therefore the ?-coord

representation of developable surfaces, barring those whose representations have

self-tangential points, is unique given just one pair of starting points in addition

to ¯ σ123and ¯ σ231?.

For a σ whose representation has self-tangential points, we break ¯ σ123 and

¯ σ231? at these self-tangential points into relatively simple pieces, each pair of

which can be uniquely traced to reconstruct the entire surface, provided the

pairing is known, say given by pairs of corresponding points each for one pair

of pieces. In a way we are patching the developable surfaces from its pieces by

such pairs of corresponding points (which of course cannot be self-tangential

points themselves). If the number of pieces is finite, say k, then we don’t even

need the pairing information to be given a priori. A O(k2) pairing algorithm

can be applied to find all possible surfaces as a result of different pairings arising

from ambiguity. As is easily verified, the worst case scenario is achievable but

is very unlikely since it implies that a non-negligible proportion of these pieces

share not only the same number of extremal points but also the same extremal

y-coordinates occurring in exactly the same sequence. As a side note we mention

that self-tangency includes the case where a curve has two or more pairs of

branches sharing the same asymptote as we have to take points at infinity into

account.

We can conclude that the ?-coord representation of developable surfaces is

unique and provides sufficient information for recovering the surfaces for almost

all practical purposes. The reconstruction is still possible even without a priori

pairing information, though at a quadratically higher cost, for all but the surfaces

whose representing curves have an infinite number of self-tangencies that break

the curves into similar pieces, an infinite number of which have the same number

of extremal points and the same extremal y-coordinates occurring in exactly the

same sequence. It also provides valuable pointers for treating more complex

surface classes.

Page 20

196C.-K. Hung and A. Inselberg

4.3 Higher Dimensions

A developable hypersurface σ in higher dimensions is the envelope of a one-

parameter family of hyperplanes. Each of these hyperplanes is one of its tangent

hyperplane. All the results in this section are generalizable to developable hy-

persurfaces in dimensions N > 3. These generalized results are stated without

proof below for readers who would like to turn these theories into practical

implementations.

Denote the N −1 representing curves as ¯ σגwhere ג ∈ I is one of the indexing

sets {1,2,...,N},{2,3,...,N,1?},...,{N −1,N,1?,2?,...,(N −2)?}. If the ¯ σג’s

are monotonic with respect to the y-axis, then σ can be reconstructed as follows:

1. Draw a horizontal line to intersect the ¯ σג’s at N − 1 points ¯ π(t0)ג which

together specify a tangent hyperplane π(t0).

2. Draw N − 1 tangent lines lגat each ¯ π(t0)גto each ¯ σג. Each of these lines

specifies the intersecting point of the “ruling” (an (N −2)-flat) of σ at π(t0)

with the superplane πs

3. Find the N intercepts of each of the N − 1 lגwith the corresponding axes

to obtain the coordinates of these points of intersection.

4. Draw horizontal lines in order to bring all numbers to the first set of axes

{¯ X1,...,¯ XN}.

5. Follow the recursive construction procedure in Sec. 2.3 to obtain the (N−2)-

dimensional ruling.

גof the corresponding index.

In general when the representing curves are not necessarily monotonic with

respect to the y-axis, we break the curves into k traceable pieces at self-tangential

Fig.18. A general cylinder illustrating the developable ↔ curve duality. Here the two

inflection points represent the ruling formed by cusps in R3. The equivalent hypersur-

face in RNis represented by N − 1 such curves all with inflection points at the same

value of y.

Page 21

Description of Surfaces in Parallel Coordinates by Linked Planar Regions 197

Fig.19. The two leaves of the surface in the previous figure are extended so that there

is a bitangent plane, tangent to two rulings. It is represented by two coinciding crossing

points for R3and N −1 for RN. This occurs together with the inflection points in each

of the representing curves.

Fig.20. Developable helicoid and its two representing curves. The two points on the

right (one on each curve) represent the tangent plane shown on the left determined by

the two line intersections with the first πs

such curves represent a developable helicoid in RN.

0? and second πs

1? super-planes. Again N − 1

points. The representation is unique if one (N − 1)-tuple of points of equal y-

value is given for matching one set of pieces, for a total of k such tuples. The

reconstruction algorithm still works most of the time by producing a maximum

Page 22

198C.-K. Hung and A. Inselberg

of O(kN−1) possible alternative hypersurface even if the matching tuples are

not given a priori. It fails only when the representing curves have an infinite

number of self-tangencies that break the curves into similar pieces, an infinite

number of which have the same number of extremal points and the same extremal

y-coordinates occurring in exactly the same sequence.

Curve dualities have analogues for developables in R3and in higher dimensions

as illustrated in Figs. 18, 19. A develobable helix and its representation are seen

in Fig. 20.

5 Ruled Surfaces

Developable surfaces are a subset of the more general class of ruled surfaces R

which are generated by a one parameter family of lines. A ruled surface can be

created by the motion of a line called ruling. Let C : y = y(s) be a curve and

g(s) a unit vector in the direction of a ruling passing through a point y(s) of C.

A point P : x(s,v) on the surface is then given by

x(s,v) = y(s) + vg(s)

G(y?,v,g?) = (y?+ vg?)×g ?= 0

∀ (s,v) ,

(18)

The curve C : y = y(s) is called the base curve or directix. The rulings are the

curves s = constant. Ruled surfaces used as architectural elements are not only

beautiful but also have great structural strength. There is a wonderful collection

of ruled many other kinds of surfaces in [11].

With G(y?,v,g?) = 0 eq. (18) describes developables which are cones for

y constant and cylinders for g(s) constant. Allowing G(y?,v,g?) ?= 0, permits

ruled surfaces to twist unlike developables. The tangent plane πP at a point

P on a ruled surface ρ contains the whole ruling rP on P. For another point

Q ∈ ρ the tangent plane πQ still contains rp. Whereas all points on a ruling

of a developable have the same tangent plane, moving along points on a ruling

r of a ruled surface causes the corresponding tangent planes to rotate about r

Fig.21. The saddle (left) is a doubly-ruled surface. One of the two regions representing

it on the left. Note the conic (parabolic) boundary.

Page 23

Description of Surfaces in Parallel Coordinates by Linked Planar Regions199

this being the essential difference and the basis for the ?-coords representation

of ruled surfaces.

A surface can be doubly-ruled in the sense that any one of its points is on two

lines completely contained in the surface (i.e. it can be generated by either one

of two sets of moving lines). An elegant result ([12] p. 42) is that in R3the only

doubly ruled surfaces are the quadrics: hyperboloid of one sheet and the saddle

(hyperbolic paraboloid).

The representing curve ¯ σ0? of a developable σ can be obtained as the envelope

of the family of lines¯R0? on the points ¯ r12, ¯ r13, ¯ r23for each ruling r of σ. Similarly

¯ σ1? is the envelope of the lines¯R1? on ¯ r1?2, ¯ r1?3, ¯ r23. A matched pair of points

¯ π0? ∈ ¯ σ0? , ¯ π1? ∈ ¯ σ1? represents the single plane containing r ⊂ σ and tangent to

everyone of its points. By contrast, a plane tangent to a ruled surface ρ at a point

contains a full ruling r but as the point of tangency is translated continuously

along r the tangent plane continuously rotates about r. In ?-coords the points

¯ π0?, ¯ π1continuously translate in tandem (with the same y coordinate) along the

corresponding lines¯R0?,¯R1? representing the rotating tangent planes along r –

see Fig. 11. An example is the saddle Fig. 21 (left) which, since it is a quadric,

is representing regions with conic boundaries. The first region and the lines¯R0?

tangent to the parabolic boundary (their envelope) is shown on the right.

Theorem 1. Representation of Ruled Surfaces – A. A ruled surface ρ is rep-

resented by the regions ¯ ρj? , j = 0,1 containing the families of lines Rj= {¯Rj}

whose envelopes are the boundaries ∂¯ ρj? specified by Lemma 1.

¯S?

¯ π0?

¯ π1?

(SA)1?

(SA)1?

¯R?

¯S

¯ r23

¯R

(SA)0?

Fig.22. The saddle SA is represented by the complements of the two shaded regions

((SA)0? and (SA)1?) having parabolic and hyperbolic boundaries respectively. The

points ¯ π0?, ¯ π1? representing a tangent plane and the ruling r ⊂ π are constructed with

the indicated matching algorithm. A different ruling on π can be constructed since SA

is doubly-ruled.

Page 24

200C.-K. Hung and A. Inselberg

To emphasize, a developable is represented by two curves which are the en-

velopes of the families of lines Rj? (formed from the representation of its rulings

as described above). A ruled surface ρ is represented by two regions ¯ ρj? ,j = 0,1

whose boundaries ∂¯ ρj? are also the envelopes of the lines Rj? (obtained in the

same way from the representation of its rulings), together with the lines Rj?.

Equivalently, it is helpful to consider ∂¯ ρj? as a line-curve and ¯ ρj? the region

covered by its tangents. The region’s structure enables the construction of the

matching algorithm for choosing pairs of points, representing the surface’s tan-

gent planes. By the way, the boundary curves deriving from the rulings of

a developable differ from those obtained from the rulings of a ruled surface

which do not necessarily have the same y range. These details are clarified with

Fig. 22 showing the saddle’s full representation.

A famous ruled surface is the M¨ obius strip described by:

x = y(θ) + vg(θ) , −1

y(θ) = (cosθ)ˆ e1+ (sinθ)ˆ e2,

g(θ) = (sin1

2< v <1

2, 0 ≤ θ ≤ 2π ,

(19)

2θcosθ)ˆ e1+ (sin1

2θsinθ)ˆ e2+ (cos1

2θ)ˆ e3,

It is non-orientable in the sense that tracing the normal vector at a moving

point along a loop, specified by its directrix the circle y(θ) = cosθˆ e1+ sinθˆ e2,

the normal flips by 1800from its original direction when the traversal is com-

pleted. The strip has only one side, its structure is elucidated in Fig. 24 (right)

showing a ruling moving along the directrix and twisting at an angle θ/2 from

the vertical, where θ is the angle swept along the directrix, intersecting the

central axis in 3 positions inverting its orientation by 1800by the time it com-

pletes a circuit along the twisted wonder we see in Fig. 24 (left). How does this

Fig.23. Representations of various ruled surfaces. The straight lines reveal that the

surfaces are ruled and partially outline the regions’ boundaries. A tangent plane is

represented by the two linked points.

Page 25

Description of Surfaces in Parallel Coordinates by Linked Planar Regions 201

θ

g(θ)

θ

2

y(θ)

x3

x1

x2

Fig.24. M¨ obius strip surface and its structure(left) A ruling traversing the circular

directrix intersects the central axis three times in one complete circuit

surface appear in ?-coords? The straight lines, characteristic of ruled surfaces,

show up producing dazzling patterns for various orientations Fig. 25. Of par-

ticular interest is the search for a pattern characterizing non-orientability. In

Fig. 26 (right) is the image of a closed circuit on the strip; a pair of curves with

Fig.25. Representation of a Mobius strip. For two orientations the straight lines (left

and center) show that it is a ruled surface. On the right the, cusp (being the dual of

inflection point) of the “bird-like” pattern may represent the non-orientable twisting

resembling a “3-D inflection point”. Two linked points represent a tangent plane.

Page 26

202 C.-K. Hung and A. Inselberg

Fig.26. Visualizing non-orientability. (left) M¨ obius strip (thanks and acknowledgment

to [11]) placed on one of the principal planes say x2x3. Note the 3 folds (which do need

to be symmetrical) resulting in 3 tangents planes perpendicular to the x2x3 plane.

(right) Traversal of a closed circuit on the M¨ obius strip represented in ?-coords by

two curves having exactly 3 crossing points each corresponding to a tangent plane

perpendicular to the x2x3 plane.

3 intersections. Each intersection point represents a tangent plane π for which

¯ π0? and ¯ π1? coincide, that is π is perpendicular to the x2x3 principal plane. A

nice way to understand the twisting is suggested in Fig. 26 (left) where the

strip is placed on a plane, in this case the x2x3, creating three “folds” which

do not have be symmetrically positioned as the ones shown. At each of the

folds the tangent plane is perpendicular and flips with respect to the x2x3plane

– review again the flipping associated with the traversal of a crossing point

Fig. 11 where one coefficient changes sign. After traversing all 3 crossing points

the tangent plane’s orientation differs by 1800(all three coefficients changed sign)

from that at the start of the circuit. Returning to the representing curves, there

are exactly three intersection points corresponding to tangent planes normal to

the x2x3plane and these must represent the tangent planes at each of the three

folds. Crossing the three intersections there is the ensuing flip associated with

the strip’s non-orientability. For a closed circuit on a doubly-twisted M¨ obius

strip, the representing curves the ¯ π0? and ¯ π1? coincide at six intersection points

corresponding to six folds and so on. It appears then that a pattern characteris-

tic of non-orientability has been found whose verification relies on interactively

tracing a closed circuit on the representing curves. The picture, of course, is a

not proof but may provide insight leading to a proof. If true the generalization

to RNis as before with N −1 such curves with N crossing points corresponding

to the N coefficients of a tangent hyperplane and pN crossing points for an N

dimensional M¨ obius strip with p twists.

Page 27

Description of Surfaces in Parallel Coordinates by Linked Planar Regions203

Fig.27. Representation of a sphere centered at the origin (left) and after a translation

along the x1 axis (right) causing the two hyperbolas to rotate in opposite directions.

As in previous figures the two linked points represent a tangent plane.

Here is another thought: twisting a curve about its tangent at a point creates

an inflection point. So are the cusps seen in Fig. 25 (right), which are the dual

of inflection points, together with the surrounding pattern in the second curve

an inkling of an non-orientable twist; like a “3-D inflection point” produced by

twisting and then joining a curved strip? Though imprecise this depiction, in the

spirit of visualization, is intuitive stimulating ideas. Does the bird-like pattern

represent the M¨ obius strip’s non-orientability? We sum up the representation

results thus far.

Theorem 2. (Representation of developable and ruled surfaces in RN)

A surface σ ⊂ RNand represented by N − 1 regions ¯ σ ⊂ P2is

1. ruled but not developable ⇔ all points of ¯ σ are exterior with respect to

the boundary of its N − 1 regions,

2. developable ⇔ ¯ σ consists of N − 1 curves, the boundaries of ¯ σ, all having

the same y-range,

6More General Surfaces

A sphere in R3is represented by two hyperbolas as seen in Fig. 27 consisting of

two hyperbolas. Due to their symmetry the hypercube, as in Fig. 5 for R5and

a hypersphere in RNare represented by N − 1 identical patterns. Unlike ruled

surfaces, here the representing points are interior to the hyperbola; this is an

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204C.-K. Hung and A. Inselberg

easy consequence of convexity. Ellipsoids and paraboloids of one sheet are also

represented by hyperbolas. A bounded convex surface (abbr. by bcs) surface in

R3has a pair of tangent (or supporting planes) planes in any direction includ-

ing the direction of u; the line through (0,0,0) and (1,1,1) and according to

corollary 2 the boundaries of each of its representing regions has two asymptotes

resembling hyperbolas as in Fig. 28 (left). For easy reference such curves are

called generalized hyperbolas and abbreviated by gh. Similarly, a bounded con-

vex hypersurface in RNis represented by N − 1 gh regions each with a pair of

asymptotes. Conversely, consider a bounded hypersurface whose representation

at a specific orientation consists only of gh regions. By the the translation ↔ ro-

tation duality, evident in Fig. 27, the regions remain gh for all orientations. The

representation of an unbounded convex hypersurface (abbr. ucs) also consists of

gh regions unless it has some (say m) ideal points in the direction of u in which

case m of its representing regions are parabola-like (i.e. have one ideal point –

called generalized-parabola and abbr. gp) and N − m − 1 are gh. Without enter-

ing into details we conjecture that the gp regions have vertical ideal points. This

yields a powerful result, important in many applications, enabling the detection

and viewing of convexity in RN.

Theorem 3. (Convexity) A hypersurface σ ⊂ RNis bounded and convex ⇔

for an orientation it is represented by N−1 gh regions. If a representation has at

least one gp region and the rest are gh regions then σ is unbounded and convex.

Whereas the interior points of the hyperbolic regions properly matched represent

the sphere’s tangent planes there is another interesting matching which gives a

feel about the sphere’s interior. Consider the representation of concentric spheres

(centered at the origin) with radii r ranging between a minimum rm and rM

Fig.28. Representation of a non-quadric convex surface by gh regions (left) and a

non-convex one with a bump (right)

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Description of Surfaces in Parallel Coordinates by Linked Planar Regions205

Fig.29. A non-convex surface with 3 dimples (depressions) corresponding to the swirls

in the representation (right)

and maximum values. As r increases the hyperbolas’ vertices and the rest of the

curve rise linearly with r enlarging the region in between the upper and lower

branches. Let a pair of points at y = yabe given on the hyperbolic boundaries

representing a tangent plane πaof one of the spheres. Its radius a is found from

the vertices of the hyperbolas, the points are on, which are at y = ±?2/3a.

Draw the vertical lines on the pair of given points. Any other pair of points at

Fig.30. Astroidal surface and representations for two orientations. The 3 crossing-

points pattern persists as the orientation is changed.

Page 30

206C.-K. Hung and A. Inselberg

y = yc represents a tangent plane πc parallel to πa on the sphere with radius

c = yc/yaa. A similar analysis for any bounded convex hypersurface reveals that

as it expands the distance between the branches of the gh representing regions

increases.

It is instructive to examine the representation of surfaces having small devi-

ations – “perturbations” – from convexity. In Fig. 28 on the right is a surface

σbwith a small bump. Its orientation was interactively varied in every imagin-

able way and significantly its representations like the one shown were invariably

markedly different than a gh. Even for the orientations where the intersections

of σbwith the super-planes, see Corollary 2, are bounded and convex the rep-

resenting regions are not gh. No less significant is that the representing regions

are not chaotic having very interesting shapes continuously varying with chang-

ing orientation. Proceeding, a surface σdwith three small dimples – depression

being considered here as a perturbation of convexity “opposite” to that in σb

– shown in Fig. 29 is explored next. Interesting representation patterns with a

“swirl” corresponding to each dimple are seen with varying orientations. Again

it is rather remarkable that for a multitude of orientations none of the repre-

sentations consists of just two hyperbolas as may naively be expected.when the

intersection with the super-planes is a bounded convex set. There seem to be

“invariants” like the “swirls” ↔ “dimples” worth further investigation.

The representation of the asteroidal surface [11] in Fig. 30 has a pattern with

3 crossing points which persists in several orientations. Upon careful scrutiny

only at the mid-point do curves from the two regions intersect. Could this point

represent a tangent plane approaching one of the surface’s vertices and becoming

a bitangent plane in the limit? Another association that comes to mind is of an

exterior supporting plane touching the surface at exactly three vertices. The

significant finding from these examples is that, the ?-coords representation even

of complex non-convex surfaces in various orientations is not chaotic but consists

of discernible patterns corresponding to their properties which may also lead to

new surface classifications. So based on this discussion is what happened to the

claim the representing boundaries are the images of the surface’s intersection

with the super-planes? This question is connected with some deeper aspects in

the theory of envelopes [13]. Briefly, the envelope of a family of planar curves is

a “reasonable” curve as long as the curves are all on the same side of envelope.

These correspond to the boundary curves prescribed by Lemma 2.

Families of Proximate Planes and Proximate Developables. A central

problem in many applications e.g. Statistics(Regression), Geometric Modeling

(Tolerancing), Computer Vision (Picture Reconstruction) etc. is concerned with

“proximity” of planes. Let a a family of proximate planes be generated by

Π = Π : {c1x1+ c2x2+ c3x3= c0,

ci∈ [c−

(20)

i,c+

i],i = 0,1,2,3}.

In Fig. 31 (left) we see the two point clusters generated by randomly choosing

planes in Π and plotting their two point representation. Closeness is apparent

Page 31

Description of Surfaces in Parallel Coordinates by Linked Planar Regions207

00.5

1

2

1.5

3.02.5

0

0.2

0.4

0.6

0.8

1

(+,−,−,−)

(+,+,−,−)

(−,+,−,−)

(+,−,−,−)

(−,+,−,−)

(+,+,−,−)

(−,+,+,+)(−,+,+,+)

(−,+,−,+)

(−,+,+,−)

(−,−,+,+)

(+,−,+,−)

(−,−,+,−)

(+,−,−,+)

(−,−,+,+)

(−,−,−,+)

Fig.31. (Left) Pair of point clusters representing close planes.(Right) The hexagonal

regions (interior) are the regions containing the points ¯ π123 (left) and ¯ π1?23 for the

family of planes with c0 = 1. For variable c0 the regions (exterior) are octagonal with

two vertical edges.

and the distribution of the points is not chaotic. The precise polygonal patterns

are shown on the right. Not only it is possible to see near coplanarity but also

see, estimate and compare errors. Families of proximate hyperplanes in RNare

represented by N − 1 convex polygons having 2(N + 1) edges constructed by a

nice algorithm with O(N) computational complexity [14]. The family Π forms a

non-convex difficult to visualize surface in R3yet its image in ?-coords consists of

two convex octagons which are easy to work with. By dividing the polygons into

triangles a point in one polygon can be matched to another in the second which

represent a valid plane in Π. This generalizes completely to RNand prompts

another idea.

In our surroundings there are lots of objects that are “nearly” developable

and many parts are manufactured by deforming flat sheets. This and the simple

?-coordinate representation motivate the study of “approximate developables”

as for the proximate planes above, by introducing perturbations in the devel-

opables’ equation allowing |(y?+vg?)×g| = ? ?= 0 and there may be other useful

definitions. The conjecture is that families of “nearly developables” are repre-

sented in ?-coords by curved “strips” with properties “close” to those discovered

by the duality (of developables) with space curves and amenable to an intuitive

treatment. New classes of surfaces (patches), with their N-D generalizations, easy

to visualize and work with interactively may emerge suitable for applications like

Geometric Modeling.

Engineering drawings depict a 3-D object in terms of 2 principal projections

which are pointwise linked i.e. 2 linked regions. In ?-coords an object is not

represented by its points but by its tangent planes. Unlike a point, the tangent

plane at the point also provides local directional information like the normal

direction. The ?-coords representation depicts the object in terms of 2 linked

regions formed by its tangent planes containing much more information than the

point projections. Cavities, like the dimples in Fig. 29 are hidden in a projection

but can not hide in the ?-coords representation. This and other properties like

Page 32

208C.-K. Hung and A. Inselberg

non-orientability and proximity are revealed leading to their higher dimensional

generalizations. We rest our case for considering, working, studying and enjoying

this surface representation.

Acknowledgements

We are grateful to David Adjiashvili who wrote the software constructing the

representation of many surfaces presented here and to the referee for valuable

suggestions.

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