Advanced Review

Natural Edward J. Wegman∗ and Yasmin H. Said

The natural homogeneous is the analog of the Cartesian coordinate system for projective . Roughly speaking a projective geometry adds an axiom that parallel lines meet at a point at infinity. This removes the impediment to -point duality that is found in traditional Euclidean geometry. The natural homogeneous coordinate system is surprisingly useful in a number of applications including computer graphics and statistical data visualization. In this article, we describe the axioms of projective geometry, introduce the formalism of natural homogeneous coordinates, and illustrate their use with four applications.  2010 John Wiley & Sons, Inc. WIREs Comp Stat 2010 2 678–685 DOI: 10.1002/wics.122

Keywords: projective geometry; crosscap; perspective; parallel coordinates; Lorentz equations

PROJECTIVE GEOMETRY Visualizing the is itself an intriguing exercise. One can imagine an ordinary atural homogeneous coordinates for projective Euclidean plane augmented by a set of points at geometry are the analog of Cartesian coordinates N infinity. Two parallel lines in Euclidean space would for ordinary Euclidean geometry. In two-dimensional meet at a point often called in elementary projective Euclidean geometry, we know that two points will geometry an ideal point. One could imagine that a always determine a line, but the dual statement, two pair of parallel lines would have an ideal point at each lines always determine a point, is not true in general end, i.e. one at −∞ and another at +∞. However, because parallel lines in two dimensions do not meet. there is only one ideal point for a set of parallel lines, The axiomatic framework for projective geometry not one at each end. The significance of axiom 6, adds the axiom that two lines in two dimensions will which may seem strange at first, can be thought of always meet at a point. Thus, in two-dimensional as follows. A can be thought of as projective geometry, two points determine a line an ordinary Euclidean line together with that extra and two lines determine a point. This introduces ideal point at ±∞. Of course, there is a one-to- a complete duality between lines and points. A one correspondence with the Euclidean line and the statement true about points and lines will also be real numbers. The ideal point is the left-over point true if the words points and lines are interchanged.1,2 that is not in one-to-one correspondence. Of course, The axioms for projective geometry are as there is also the set of ideal points, which is the follows: ideal line. There is an ideal point on the ideal line corresponding to every slope between −∞ and +∞, 1. There exists at least one line. hence for every point on the real line. There is one 2. On each line there are at least three points. extra point on the ideal line corresponding to an 3. Not all points lie on the same line. infinite slope. Thus axiom 6 holds whether the line is an ordinary Euclidean line plus an ideal point or the 4. Two distinct points lie on one and only one line. line is the ideal line. 5. Two distinct lines meet at one and only one Because of duality these axioms can be rewritten point. with the words, point and line interchanged. 6. There is a one-to-one correspondence between the real numbers and all but one point on a 1a. There exists at least one point. line. 2a. Through each point there are at least three lines. ∗Correspondence to: [email protected] Center for Computational Data Sciences, George Mason University, 3a. Not all lines pass through the same point. Fairfax, VA, USA 4a. Two distinct lines meet at one and only one DOI: 10.1002/wics.122 point (cf. 5 above).

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5a. Two distinct points lie on one and only one Line parallel to plane line (cf. 4 above). Ideal point 6a. There is a one-to-one correspondence between the real numbers and all but one line through a point.

(0,0) MODELS FOR A PROJECTIVE PLANE Projective plane An alternate way of imagining the projective plane is to visualize a hemisphere with its South Pole sitting FIGURE 1| Representation of the projective plane by a hemisphere at the origin of a Euclidean plane. Any point on the which can be deformed into a crosscap. ordinary Euclidean plane can be represented on the hemisphere by connecting that point to the center of the hemisphere with a straight line. The point where the line meets the hemisphere is the mapping of the point in the Euclidean plane into the hemisphere. Lines from the center of the hemisphere through the equator represent ideal points because they are parallel to the Euclidean plane. As described in Figure 1, antipodal points on the equator of the hemisphere represent the same ideal point and, hence, these points are identified in a topological sense. One can imagine deforming the equator in such a way that antipodal points are joined. See Figure 2 to illustrate this deformation partially completed. In Figure 3, we represent the completion of this FIGURE 2| Partially deformed hemisphere so that antipodal points deformation. This structure is called a crosscap and along the equator are approaching each other. represents a topological model of the projective plane. Figure 4 is a color and shaded rendering of a crosscap. Solving simultaneously, we have (C − C)z = 0. Since we know C − C = 0, ⇒ z = 0. Therefore, NATURAL HOMOGENEOUS (x, y, 0) represents an ideal point. COORDINATES The two-dimensional natural homogeneous coordinate system will be a triple (x, y, z). If z = 0, In ordinary Euclidean space, we have the Cartesian then we have an ideal point. Notice that if the origin, Coordinate system. We wish to develop an analog given by (0, 0) in ordinary Cartesian coordinates, is to Cartesian Coordinates for the projective plane. joined to a point (x, y), again in Cartesian coordinates, Consider the following equations: they determine a line and the ideal point (x, y,0)ison that line. The line has slope y/x. Hence, (x, y,0)isthe + + = Ax By C 0, ideal point corresponding to all lines with slope y/x.  = Ax + By + C = 0. For ordinary points, we want z 1sothat the ordinary equation Ax + By + C = 0 holds. Thus These are the ordinary linear equations for two the Cartesian point (x, y) is represented in natural homogeneous coordinates as (x, y, 1). However, if straight lines which are parallel. If we try to solve + + = + + = these equations simultaneously we obtain no solution. Ax By C 0, then also Apx Bpy Cp 0so However in the projective plane we know that the that (px, py, p) is also a valid representation of solution is the ideal point. We rewrite these equations (x, y). We may always rescale so that if we have = x y (x, y, z), z 0, then this is equivalent to z , z ,1 or as x y the Cartesian point z , z . Although this multiple representation for Cartesian points at first appears to Ax + By + Cz = 0, be a handicap, it is indeed a useful representation as  Ax + By + C z = 0. we shall see in our later examples.

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This definition can be extended in the obvious way for higher dimensional projective planes. An equation in xi, u1x1 + u2x2 + u3x3 = 0, is the equation of a line in the projective plane. Notice that the triple, (x1, x2, x3), represents a point. However, the values in the triple, [u1, u2, u3], are the coefficients of the line and hence represent the line. The natural homogeneous coordinates mirror the projective duality between points and lines. Notice that if u1 = u2 = 0, then the equation is satisfied by ideal points. In other words, u3x3 = 0 is the equation of the ideal line, u3 = 0. Definition: The set of all real number triples [u1, u2, u3], ui not all 0, are the natural homogeneous line coordinates in the . FIGURE 3| The completely deformed hemisphere with antipodal An equation in ui, u1x1 + u2x2 + u3x3 = 0, is the points identified. In this rendition, 2D view of a 3D structure, the equation of a point in the real projective plane. surfaces penetrate each other. However, embedded in a higher dimensional space these surfaces do not intersect. APPLICATIONS OF NATURAL HOMOGENEOUS COORDINATES Computer Graphics Application I—Representing Translations in Matrix Form In computer graphics applications,3 we are interested in representing translations, rotations, and scalings. Consider the situation with translations. If p = (x, y)T is a point in Cartesian coordinates and it is translated by an amount tx in the x direction and an amount ty  T in the y direction, then p = T(p) = (x + tx, y + ty) . T(p) is of course the sum of two vectors, but this oper- ation is not directly translatable into matrix notation. For rotations, consider a point p = (x, y)T which is given in polar coordinates by (r, γ )T. Thus x = r cos(γ )andy = r sin(γ ). If the point is rotated through an angle θ,thenx = r cos(γ + θ)and y = r sin(γ + θ); see Figure 5. But FIGURE 4| Crosscap rendered as a color shaded figure. (Reprinted with permission from Professor Paul Bourke, University of Western cos(γ + θ) = cos(γ )cos(θ) − sin(γ )sin(θ), Australia. http://local.wasp.uwa.edu.au/∼pbourke/geometry/). sin(γ + θ) = sin(γ )cos(θ) + cos(γ )sin(θ).

Definition: The set of all triples (x1, x2, x3)with Multiply by r so that not all xi zero are the natural homogenous coordinates + = − in the real projective plane where (x1, x2, x3)arethe r cos(γ θ) r cos(γ )cos(θ) r sin(γ )sin(θ), coordinates of or

 i. an ordinary point (x, y), if x = x1/x3 and x = x cos(θ) − y sin(θ). y = x2/x3, x3 = 0, Similarly, y = x sin(θ) + y cos(θ). This can be ii. the ideal point with slope m if x = 0and 3 writteninmatrixformas m = x2/x1, x1 = 0, x cos(θ) −sin(θ) x iii. the ideal point with infinite slope (vertical line) = . y sin(θ)cos(θ) y if x1 = x3 = 0.

680  2010 John Wiley & Sons, Inc. Volume 2, November/December 2010 WIREs Computational Statistics Natural homogeneous coordinates

Mirror about the x axis (x′, y′)       x 100 x y = 0 −10 y . (x, y) 1 001 1

Mirror about the y axis       x −100 x g y =  010 y . θ 1 001 1

These matrices are square and conformable. | FIGURE 5 Rotations in polar coordinates. Hence the operations may be concatenated. The nat- ural homogeneous coordinate representation forms a   basic matrix tool for elementary geometric transfor- For scaling, consider x = sxx and y = syy. These rescale in both the x and y directions. Scaling mations in computer graphics. For more details see 7 can easily be put into matrix form by Wegman and Carr.  x sx 0 x Computer Graphics Application  = . y 0 sy y II—Perspective One of the founding ideas of projective geometry The question is then how to put translation into arises from the artist’s notion of perspective. This, matrix form. We consider a natural homogeneous of course, is a crucial idea in computer graphics as coordinates formulation. well as the world of art.6 In the following discussion, we will use ei, i = x, y, z to represent the unit vectors Translation along the x, y,andz axes respectively. Consider a Consider the matrix right-handed coordinate system with the computer   screen located coincident with the ex − ey plane. Let T 10tx an object be located at point p = (x, y, z,1) while the   01ty . eye is located at the center of perspectivity along the T 001 ez axis at point (0, 0, −d,1) . Here we are writing all points in a three-dimensional natural homogeneous Then the natural homogeneous coordinate coordinate system. From the diagram in Figure 6, T representation  of the Cartesian point (x, y) can be ey x written as y so that

1 p (x, y, z, 1)        x 10tx x       p′ = (x′, y′, 0, 1) y = 01ty y . 1 001 1

(0, 0, z, 1) Center Rotation ez −       (0, 0, d, 1) x cos(θ) −sin(θ)0 x (x′, 0, 0, 1) y = sin(θ)cos(θ)0 y . 1 0011 (x, 0, z, 1)

Scaling        ex x sx 00 x       y = 0 sy 0 y . FIGURE 6| Perspective representation using natural homogeneous 1 001 1 coordinates.

Volume 2, November/December 2010  2010 John Wiley & Sons, Inc. 681 Advanced Review wires.wiley.com/compstats by using similar triangles, we obtain the equation Because projective transformations are repre-  d = x d+z x . Hence the ex screen coordinate for the sented by nonsingular matrices, mathematical struc- projection of p onto the screen is x = (d · x)/(d + z). tures in Cartesian space are transformed into math- Similarly the ey screen coordinate for the projection ematical structures in parallel coordinate space. This of p onto the screen is given by y = (d · y)/(d + z). implies geometric structures such as ellipses in Carte- Notice this gives a singularity if z =−d. We can avoid sian space map into identifiable structures in parallel this by writing in natural homogeneous coordinates. coordinate space as well. However, in principle, there Of course at the screen, z = 0. Thus we have is no limit to the number of parallel axes we can draw         in parallel coordinate space. Hence, it is feasible to x d · x d 000 x represent much higher dimensional structures in par-          y   d · y  0 d 00 y allel coordinate space. As just suggested, the parallel p =  = = . z   0  0000 z coordinate representation enjoys some elegant duality 1 z + d 001d 1 properties with the usual Cartesian orthogonal coordi- nate representation. Consider a line L in the Cartesian L = + This matrix is singular as would be expected coordinate plane given by : y mx b and con- + T because there is no way of mapping the two-dimen- sider two points lying on that line, say (a, ma b) + T sional projection back into full three-dimensional and (c, mc b) . space. This is the perspective transformation of p For simplicity of computation we consider the  − − into p . Notice that the point (d · x, d · y,0,z + d)T ex ey Cartesian axes mapped into the ex ey parallel is equivalent to the natural homogeneous coordinate axes. We superimpose a Cartesian coordinate axes · · d x d y T et − eu on the ex − ey parallel axes so that the y parallel point ( z+d , z+d ,0,1) or the ordinary Cartesian point · · axis has the equation u = 1 while the x parallel axis ( d x , d y ,0)T. If indeed z =−d, then the transforma- z+d z+d and t axis share the equation u = 0 (Figure 7). The · · T tion yields (d x, d y,0,0) which is an ideal point + T − d·y point (a, ma b) in the ex ey Cartesian system = = / . T T in the z 0 plane with slope d·x y x This corre- maps into the line joining (a,0) to (ma + b,1) in sponds to a situation where the point p is in the plane T the et − eu coordinate axes. Similarly, (c, mc + b) of the viewer’s eye. maps into the line joining (c,0)T to (mc + b,1)T.It − Also note that if z < d, then the point p is a straightforward computation to show that these is behind the viewpoint and z + d < 0. Thus when −  two lines intersect at a point in the et eu plane given computing the position of p in ordinary Cartesian by L∗:(b(1 − m)−1,(1− m)−1)T. d·x d·y coordinates, both z+d and z+d are negative. Hence images behind the viewer are inverted and mirrored.

y

Parallel Coordinates (c, mc+b) The parallel coordinate representation is a method for visualizing data in high dimensional spaces.6 The basic idea is that because we run out of orthogonal axes after three, we give up orthogonality and replace the (a, ma+b) orthogonal axes with parallel axes. The parallel axes x may be drawn either all north–south or all east–west. ac We adopt the latter convention. T Consider a two-dimensional point (x1, x2) u which determines a point in a Cartesian coordinate system. In a parallel axis system that same point is represented by drawing a straight line between (ma+b,1) (mc+b,1) y the value x1 on the x1-axis and x2 on the x2-axis. This suggests that projective geometry comes into play because points in the Cartesian plane map into lines in the parallel coordinate plane. If we x, t consider both planes to be projective planes, then the (a, 0) (c, 0) transformation from Cartesian coordinate projective FIGURE 7| Mapping Cartesian points into lines in parallel plane into the parallel coordinate projective plane coordinate space and Cartesian lines into points in parallel coordinate becomes a projective transformation. space.

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Notice that this point in the parallel coordinate perspective. The latter yields the appropriate ideal plot depends only on m and b the parameters point when m = 1. A straightforward computation of the original line in the Cartesian plot. Thus shows that t = Ax with L∗ is the dual of L and we have the interesting   duality result that points in Cartesian coordinates 001 =  −  map into lines in parallel coordinates while lines A 0 10. − − in Cartesian coordinates map into points in parallel 1 10 coordinates. − T = − T For 0 < (1 − m)−1 < 1, m is negative and the In other words, (b,1,1 m) A[m, 1, b] . intersection occurs between the parallel coordinate Thus the transformation from lines in orthogonal axes. For m =−1, the intersection is exactly midway. coordinates to points in parallel coordinates is a A ready statistical interpretation can be given. For particularly simple projective transformation with the rather nice computational property of having only highly negatively correlated pairs, the dual line adds and subtracts. segments in parallel coordinates will tend to cross near Similarly, a point (x , x ,1)T expressedinnat- a single point between the two parallel coordinate 1 2 ural homogeneous coordinates maps into the line axes. The scale of one of the variables may be represented by [1, x − x , −x ]T in natural homo- transformed in such a way that the intersection occurs 1 2 1 geneous coordinates. Another straightforward com- midway between the two parallel coordinate axes putation shows that the linear transformation given in which case the slope of the linear relationship is by t = Bx or [1, x − x , −x ]T = B(x , x ,1)T where negative one. 1 2 1 1 2 − −1 − −1   In the case that (1 m) < 0or(1 m) > 1, 001 m is positive and the intersection occurs external to B =  1 −10 the region between the two parallel axes. In the special −100 case m = 1, this formulation breaks down. However, + T it is clear that the point pairs are (a, a b) and describes the projective transformation of points in + T (c, c b) . The dual lines to these points are the Cartesian coordinates to lines in parallel coordinates. −1 lines in parallel coordinate space with slope b and An interesting connection between A and B is that −1 −1 intercepts −ab and −cb respectively. Thus the BAT = ABT = I where I is the 3 × 3 identity matrix. duals of these lines in parallel coordinate space are Thus AT = B−1 and BT = A−1. −1 parallel lines with slope b . We thus append the Because these are nonsingular linear transfor- ideal points to the parallel coordinate plane to obtain mations, hence projective transformations, it follows a projective plane. These parallel lines intersect at the from the elementary theory of projective geometry that − ideal point in direction b 1. In the statistical setting, conics are mapped into conics. This is straightforward we have the following interpretation. For highly to see because an elementary quadratic form in the positively correlated data, we will tend to have lines original space, say xT Cx = 0 represents the general not intersecting between the parallel coordinate axes. conic. Clearly then because t = Bx, B nonsingular, By suitable linear rescaling of one of the variables, the we have x = B−1t,sothattT(B−1)TC(B−1t) = 0isthe lines may be made approximately parallel in direction quadratic form in the image space. An instructive com- − with slope b 1. putation involves computing the image of an ellipse In this case, the slope of the linear relationship ax2 + by2 − cz2 = 0witha, b, c > 0. The image in the between the rescaled variables is one. Of course, parallel coordinate space is c(t + u)2 − bu2 = av2,a nonlinear relationships will not respond to simple general hyperbolic form. linear rescaling. However, by suitable nonlinear It should be noted that the solution to this transformations, it should be possible to transform equation is not a locus of points, but the natural to linearity. homogeneous coordinates of a locus of lines, a line Recall now that the line L: y = mx + b is conic. The of this line conic is a point conic. ∗ − − mapped into the point L :(b(1 − m) 1,(1 − m) 1,1) In the case of this computation, the point conic in the in parallel coordinates. In natural homogeneous original Cartesian coordinate plane is an ellipse, the coordinates, L is represented by the triple [m, −1, b]T image in the parallel coordinate plane is as we have ∗ − and the point L by the triple (b(1 − m) 1, just seen a line hyperbola with a point hyperbola as (1 − m)−1,1)T or equivalently by (b,1,1− m)T. envelope. Notice the ability of natural homogeneous coordinates We mentioned the duality between points and to eliminate problems with singularities as we lines and conics and conics. It is worthwhile to point saw earlier in the computer graphics example on out two other nice dualities. Rotations in Cartesian

Volume 2, November/December 2010  2010 John Wiley & Sons, Inc. 683 Advanced Review wires.wiley.com/compstats coordinates become translations in parallel coordi- Multiplying out and collecting coefficients of nates and vice versa. Perhaps more interesting from a common variables, statistical point of view is that points of inflection in 2 − 2 2 2 + 2 + 2 − 2 + 2 Cartesian space become cusps in parallel coordinate (a11 c a21)x y z 2(a11v c a21a22) space and vice versa. Thus the relatively hard-to- = 2 2 − 2 2 2 detect inflection point property of a function becomes xt (c a22 v a11)t . the notably more easy to detect cusp in the paral- lel coordinate representation. Inselberg4 derives these But we also have duality relationships in detail. x2 + y2 + z2 = c2t2.

Special Relativity Equating coefficients, we obtain three equations One of the pesky problems with special relativity is in three unknowns the annoying situation of traveling at light speed.5 This leads to a singularity in the Lorentz equations (a2 − c2a2 ) = 1, which leads most people except science fiction writers 11 21 − 2 + 2 = to the conclusion that it is impossible to travel at light 2(a11v c a21a22) 0, speed. As we have seen before, natural homogeneous (c2a2 − v2a2 ) = c2. coordinates offer a way of getting around such nasty 22 11 singularities. This would correspond to the space in These may be solved simultaneously to obtain which we are living as being described by projective geometry rather than Euclidean geometry, which is the 1 1 premise of special relativity. But locally the projective a11 = = , 2 plane is indistinguishable from a Euclidean plane. 1 − v w c2 First let us consider a two-dimensional special − v − 2 relativity with one spatial dimension and one temporal c2 v/c a21 = = , 2 dimension. Assume there are two inertial platforms, 1 − v w one at rest and one moving at a constant speed v. c2 We will prime the moving inertial frame. That is 1 1 a22 = = . the primed letters are the coordinates in the moving 2 1 − v w inertial frame. We assume as follows: c2

 x = a11x + a12t, These solutions are easy to verify. Thus, the  two-dimensional Lorentz equations become t = a21x + a22t,  y = y,  x − vt x = ,  2 z = z. 1 − v c2  Our assumption about the relative motion of y = y,  =  the two inertial frames is that x 0 if and only z = z, if x = vt. In the above two-dimensional Lorentz t − xv equations we have 0 = a11vt + a12t so that 0 =  c2  t = . (a v + a )t ⇒ a =−a v ⇒ x = a (x − vt). The 2 11 12 12 11 11 1 − v premise of special relativity is that light moves at the c2 same speed in all frames of reference. Thus a pulse → of light in either frame will spread out spherically As mentioned before there is a problem as v c. according to Consider for the moment only the two-dimensional case and let us use natural homogeneous coordinates. 2 2 2 2 2 + + = − xv x y z c t , − t Then (x, t,1)=  x vt , c2 ,1 . Equivalently 2 2 2 2 2 2 2 x + y + z = c t . − v 1− v 1 2 2 c c   in natural homogeneous coordinates (x , t ,1)= (x − In the second equation, we may substitute in the 2 vt, t − xv , 1 − v ). At light speed, v = c, we obtain appropriate transformations so that c2 c2 an ideal point, (x − ct, t − x/c, 0). Supposing v > c, 2 − 2 + 2 + 2 = 2 + 2 v2 v2 a11(x vt) y z c (a12x a22t) . then > 1 ⇒ 1 − is an imaginary number. Thus c2 c2

684  2010 John Wiley & Sons, Inc. Volume 2, November/December 2010 WIREs Computational Statistics Natural homogeneous coordinates

Imaginary hyperspace. The model for a two-dimensional space is just a sphere with antipodal points on the equator identified, essentially two crosscaps stuck together. Two crosscaps joined are topologically equivalent to Hyperspace a Klein Bottle. For the full Lorentz equations the bottom plane (real space) would be a four-dimensional real pro- jective plane and likewise the top plane (hyperspace) would be a four-dimensional real projective plane Real parallel to the bottom plane. There would be a four- t dimensional hypersphere between the two. This gives an interesting twist to the notion of parallel universes.

Regular space x FIGURE 8| A model for two-dimensional projective plane for CONCLUSION special relativity using natural homogeneous coordinates. The use of analytic projective geometry and natural we could imagine two parallel projective planes, and homogeneous coordinates has a number of interesting a sphere between. The bottom plane would be a real implications for computer graphics, data visualization projective plane, with the South pole of the sphere rest- and other scientific applications. The ability to elimi- ing on (0, 0, 1) and the top plane would have its origin nate divide-by-zero singularities providing an essential resting on the North pole of the sphere (Figure 8). continuity on either side of the ‘singularity’ together The equator of the sphere would correspond to ideal with an ability to formulate computations in a matrix points, i.e. when v = c. Antipodal points on the sphere setting proves the natural homogeneous coordinate are identified. The entire upper half of the sphere is representation to be quite a valuable technique. In the imaginary plane, what science fiction writers call addition, by understanding the geometric framework hyperspace, while the bottom half is real space.Inthis in a somewhat different sense allows for gathering model there is no singularity between real space and potential insight.

ACKNOWLEDGEMENTS Figure 4 was provided by Professor Paul Bourke of the University of Western Australia. His website, http://local.wasp.uwa.edu.au/∼pbourke/geometry/, contains beautifully rendered images corresponding to many geometric concepts.

REFERENCES 1. Coxeter HSM. Projective Geometry. 2nd ed. (Soft 5. Robinson EA. Einstein’s Relativity in Metaphor and Cover). New York: Springer-Verlag; 2003. Mathematics. Upper Saddle River, NJ: Prentice Hall, 2. Fishback WT. Projective and Euclidean Geometry. New Inc.; 1990. York: John Wiley and Sons; 1962. 6. Wegman EJ. Hyperdimensional data analysis using par- 3. Foley JD, van Dam A, Feiner SK, Hughes J. Computer allel coordinates. Journal of the American Statistical Graphics: Principles and Practice. 2nd ed. Reading, MA: Association 1990, 85:664–675. Addison Wesley Publishing Co.; 1995. 7. Wegman EJ, Carr, DB. Statistical graphics and visual- 4. Inselberg A. Parallel Coordinates: Visual Multidimen- ization. In: Rao, CR, ed. Handbook of Statistics 9: sional Geometry and Its Applications. New York: Computational Statistics. Amsterdam: North Holland; Springer; 2009. 1993, 857–958

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