12

Notation and List of Symbols

Alphabetic

ℵ0,ℵ1,ℵ2, . . . cardinality of infinite sets

, r, s, . . . lines are usually denoted by lowercase italics Latin letters

A,B,C, . . . points are usually denoted by uppercase Latin letters

π, ρ, σ, . . . planes and surfaces are usually denoted by lowercase Greek letters

N positive integers; natural numbers

R real numbers

C complex numbers

5 real part of a complex number

6 imaginary part of a complex number

R2 Euclidean 2-dimensional plane (or space)

P2 projective 2-dimensional plane (or space)

RN Euclidean N-dimensional space

PN projective N-dimensional space

Symbols

General

∈ belongs to

� such that

∃ there exists

∀ for all

⊂ subset of

∪ set union

∩ set intersection

A. Inselberg, Parallel Coordinates, DOI 10.1007/978-0-387-68628-8, © Springer Science+Business Media, LLC 2009

532 Notation and List of Symbols

∅ empty st (null set)

2A power set of A, the set of all subsets of A

ipipip inflection point of a curve

n principal normal vector to a space curve

n unit normal vector to a surface

t vector tangent to a space curve

b binormal vector

τ torsion (measures “non-planarity” of a space curve)

× cross product

→ mapping symbol

A→ B correspondence (also used for duality)

⇒ implies

⇔ if and only if

<,> less than, greater than

|a| absolute value of a

|b| length of vector b

‖ ‖ norm, length

L1 L1 norm

L2 L2 norm, Euclidean distance∑

summation

Jp set of integers modulo p with the operations + and ×≈ approximately equal

�= not equal

‖ parallel

⊥ perpendicular, orthogonal

( , , ) homogeneous coordinates — ordered triple representing apoint in P

2; point coordinates

[ , , ] homogeneous coordinates — ordered triple representing aline P

2; line coordinates

p(A,B,C,D, . . . )P� perspectivity between the lines p and p′ with respect to a

p′(A′, B ′, C′,D′, . . . ) point P

Notation and List of Symbols 533

p � p′ projectivity between the lines p and p′dfdx

derivative of f with respect to x, also written as f ′(x)∂� boundary of a region �∂F∂xi

partial derivative of F with respect to xi , also denoted Fxiand when context is clear, Fi

∇F gradient of F

Specific to Parallel Coordinates (In the order occuring in the text)

‖-coords abbreviation for parallel coordinate system, parallelcoordinates

A image of A in ‖-coords, representation of A

X1, . . . , XN axes for parallel coordinate system forN -dimensional space (eitherEuclidean or Projective)

i,j point representing the linear relation between xi and xj ; (N − 1)points with two indices appear in the representation of a line in R

N

(or PN )

πp p-flat in RN , a plane of dimension 1 ≤ p ≤ (N − 1) in R

N

π1 a one-flat is a line, also denoted by

spspsp abbreviation for superplane, a 2-flat whose points are representedby straight lines in ‖-coords

πs denotes a superplane

πNs denotes a superplane in RN

E family of superplanes, named after J. Eickemeyer, who firstdiscovered them

E also denotes the class of surfaces who are the envelope of theirtangent (hyper) planes

EG class of surfaces contained in E whose representing contours canbe obtained by Lemma 9.3.1

di directed distance of Xi from the y-axis

d0N (see next), standard axis spacing, when the X1 is coincident with

the y-axis and inter-axis distance is one unit for all axes

diN axes spacing where the X1 axis is placed one unit after the XN and

all axes up to and including Xi are placed successively one unitapart and the rest of axes remain in the standard axes spacing.

534 Notation and List of Symbols

πs1 first superplane, with the standard axes spacing

πs1′ second superplane, the X1 placed one unit to the right of XNthe remaining axes are in standard axes spacing.

πsi′ is the ith superplane with axes spacing di

Nπ123, π1′123 two triple indexed points representing the plane (2-flat) π ⊂ R

3,also denoted by π123 = π0′, π1′23 = π1′ or 123, 1′23 when π isclear from the context; π1′23 is based on the axes spacing d1

3π123,... ,N = π0′ point with (N +1) indices representing a hyperplane (of dimen-

sion N − 1)), and similarly πi′ for the representing points basedon axes spacing di

Ngconicsgconicsgconics generalized conics, sections of a double cone in R

3 whose 2bases are the same bounded convex set in R

2

bc, ubc, ghbc, ubc, ghbc, ubc, gh like conics there are 3 kinds of gconics: bounded convex set (bc),unbounded convex set (ubc) and generalized hyperbola (gh)

) outer union

* inner intersection

12

Bibliography

[1] A.E. Abbot. A Romance of Many Dimensions. Dover, New York (originallypublished in 1884 as ‘Flatland by A Square’, Seeley & Co., London), 1992.

[2] N. Adrienko and G. Adrienko. Constructing Parallel Coordinates Plots forProblem Solving. in Proc. 1st Inter. Symp. on Smart Graphics, A. Butz,A. Krueger, P. Oliver and M. Zhou (eds.), ACM Press 9–14, New York,2001.

[3] C. Ahlberg and B. Shneiderman. Visual Information Seeking: Tight Cou-pling of Dynamic Query Filters with Starfield Displays. Proc. CH94 Conf.,313–317, ACM Press, New York, 1994.

[4] D. Asimov. The grand tour: A tool for viewing multidimensional data.SIAM J. Sci. & Stat. Comp., 6:128–143, 1985.

[5] T. Avidan and S. Avidan. Parallax – a data mining tool based on parallelcoordinates. Comput. Statit., 13–1:65, 1998.

[6] S. Axelson. Understanding Intrusion Detection Through Visualization.Ph.D. Thesis, CS Dept. Chalmers Univ., Gotesborg, Sweden, 2005.

[7] F. Ayres Jr. Projective Geometry. Shaum’s Outlines on Math., McGraw-Hill, New York, 1967.

[8] T.F. Banchoff. Beyond the Third Dimension. Scientific American Library,New York, 1990.

[9] E.W. Bassett. Ibm’s ibm fix. Industrial Computing, 14(41):23–25, 1995.

[10] A.F. Beardon. The Geometry of Discrete Groups. Springer-Verlag,New York, 1983.

[11] B. Bederson and B. Shneiderman. (eds.) The Craft of Information Visu-alization: Readings and Reflections. Morgan-Kaufman, San Francisco,2003.

[12] J. Bergen, J. Borwein and P. Borwein. Pi: A Source Book. Springer-Verlag,New York, 1997.

[13] M. Berger. Geometry I & II. Corrected second printing, Springer-Verlag,New York, 1994.

[14] J. Bertin. Semiologie Graphique. First pub. 1967, Ecole des Hautes Etudesen Science, Paris, 1999.

536 Bibliography

[15] D.M. Bloom. Linear Algebra and Geometry. Cambridge University Press,Cambridge, 1979.

[16] B. Bollobas. Graph Theory. Springer-Verlag, New York, 1979.[17] V.G. Boltyanskii. Envelopes, R.B. Brown, translator (original in Russian).

Pergamon Press, New York, 1964.[18] M.F. Brannan, D.A. Esplen and J.J. Gray. Geometry. Cambridge University

Press, New York, 1999.[19] O.S. Brodetsky. A First Course in Nomography (first published in 1920).

G. Bell and Sons, London, 1949.[20] S. Bulthoff and H.H. Edelman. Psychophysical support for a two-

dimensional view interpolation theory of object recognition. Proc. Nat.Acad. Sci., 89-Psychology: 60–4, 1992.

[21] J. Canny and J. Reif. New Lower Bound Techniques for Robot MotionPlanning Problems, in Proc. of 28th Symp. on Found. of Comp. Sci. IEEEComp. Soc., Washington, D.C., 1987.

[22] S. Card, J. MacKinlay and B. Shneidermanm. (eds.) Readings inInformation Visualization. Morgan Kaufman, San Francisco, 1999.

[23] A. Chatterjee. Visualizing Multidimensional Polytopes and Topologies forTolerances. Ph.D. thesis, Dept. Comp. Sci., Univ. of S. Calif., 1995.

[24] A. Chatterjee, P.P. Das and S. Bhattacharya. Visualization in linear pro-gramming using parallel coordinates. Pattern Recognition, 26–11:1725–36,1993.

[25] C. Chen. Information Visualization: Beyond the Horizon, 2nd ed. Springer,New York, 2004.

[26] C.H. Chen. Generalized association plots for information visualization: Theapplication of the convergence of iteratively formed correlation matrices.Statistica Sica, 12:1–23, 2002.

[27] C.H. Chen and J.A. Chen. Interactive diagnosis plots for multidimensionalscaling with applications in psychosis disorder data analysis. StatisticaSica, 10:665–691, 2000.

[28] E.H. Chi. A Framework for Visualizing Information. Kluwer, Amsterdam,2002.

[29] H. Choi and H. Lee. PCAV: Internet Attack Visualization in ParallelCoordinates, LNCS 3783, 454–466. Springer-Verlag, New York, 2005.

[30] T. Chomut. Exploratory Data Analysis in Parallel Coordinates. M.Sc.thesis, Dept. Comp. Sci., UCLA, 1987.

[31] S. M. Cohan and D.C.H. Yang. Mobility analysis in parallel coordinates.J. Mech. & Mach., 21:63–71, 1986.

Bibliography 537

[32] G. Conti. Security Data Visualization. No Starch Press, San Francisco,2007.

[33] D. Conversano, D. Di Benedetto and R. Siciliano. The Clockworktree:Visual Multivariating Splitting. Cladag 2003, Book of Short Papers 113–7,Clueb Publisher, Bologna, 2003.

[34] D. Cook and A. Buja. Manual controls for high-dimensional dataprojections. J. of Comp. & Graph. Stat., 6–4, 464–480, 1997.

[35] R. Courant. Differential and Integral Calculus vol. 2. Blackie & Son,London 171–183, 1946.

[36] R. Courant and H.E. Robins. What is Mathematics? Oxford UniversityPress, New York, 1953.

[37] D. Cox, J. Little and D. O’Shea. Ideals, Varieties, and Algorithms. 2ndedition, Springer-Verlag, New York, 1997.

[38] H.S.M. Coxeter. Introduction to Geometry. John Wiley, New York, 1969.

[39] H.S.M. Coxeter. Projective Geometry. Second Ed., John Wiley, New York,1974.

[40] D.H. Deterding. Speaker Normalization for Automatic Speech Recognition.Ph.D. thesis, Cambridge University, 1989.

[41] B. Dimsdale. Conic Transformations – IBM LASC Tech. Rep. G320-2713.IBM LA Scientific Center, 1981.

[42] B. Dimsdale. Conic Transformations and Projectivities – IBM LASC Tech.Rep. G320-2753. IBM LA Scientific Center, 1984.

[43] J.A. Dykes, A.M. MacEachren and M.J. Kraak. (eds). Exploring Geo-visualization. Elsevier, Amsterdam, 2005.

[44] J.A. Dykes and D.M. Mountain. Seeking structure in records of spatio-temporal behaviour: visualization issues, efforts and applications. Comput.Statit. & Data Anal., 43 (Data Visualization II special ed.):581–603, 2003.

[45] R. Edsall. The parallel coordinate plot in action: Design and use forgeographic visualization. Comput. Statit. & Data Anal., 43–4:605–619,2003.

[46] P. Ehrenfest. In What Way Does it Become Manifest in the FundamentalLaws of Physics that Space has Three Dimensions? – Appeared in 1917 –in Collected Scientific Papers of P. Ehrenfest, Klein, M.J. ed., 400–409.North-Holland Publishing Company, Interscience Publishers, New York,1959.

[47] J. Eickemeyer. Visualizing p-flats in N-space using Parallel Coordinates.Ph.D. thesis, Dept. Comp. Sci., UCLA, 1992.

538 Bibliography

[48] P.L. Eisenhart. A Treatise on the Differential Geometry of Curves andSurfaces. Dover Publications, New York, 2004.

[49] S. El Mejdani, R. Egli and F. Dubeau. Old & new straight-line detectors,descriptions & comparisons. Pattern Recognition, 41:1845–66, 2008.

[50] A. Ellis and G. Dix. Enabling automatic clutter reduction in parallelcoordinates. Trans. on Vis. & Comp. Grap., 12–5:717–724, 2006.

[51] N. Elmquist, J. Stasko and P. Tsingas. Datameadow: A visual canvas foranalysis of large-scale multivariate data. Proc. IEEE Vast Conf., to appear,2007.

[52] L. Ernstrom. A plucker formula for the singular projective varieties.Commut. Algebra., 5–9:2897–2901, 1997.

[53] K. Esbensen, et al. Multivariate Analysis, 1994, Camo AS, Trondheim,Norway.

[54] S. Even. Graph Algorithms. Computer Science Press, Rockville MD, 1979.[55] M.G. Ewing. On the definition of inflection point. Amer. Math. Monthly, 45:

681–683, 1938.[56] F. Fallside and A.J. Robinson. A Dynamic Connectionist Model of Phoneme

Recognition. Proc. of 1st European Neural Network Conf. (nEURO), 1988.[57] U.M. Fayad, G. Piatesky-Shapiro, P. Smyth and R. Uthurusamy. Advances

in Knowledge Discovery and Data Mining. AAAI/MIT Press, CambridgeMass., 1996.

[58] U.M. Fayad, G. Grinstein and A. Wierse. Information Visualization in DataMining and Knowledge Discovery. Morgan Kaufmann, San Francisco,2001.

[59] R. Finsterwalder. A Parallel Coordinate Editor as a Visual Decision Aidin Multi-Objective Concurrent Control Engineering Environment 119–122.IFAC CAD Contr. Sys., Swansea, UK, 1991.

[60] P. Fiorini and A. Inselberg. Configuration Space Representation in ParallelCoordinates. IEEE Conf. Rob. & Aut., 1215–1220, 1989.

[61] G. Fischer. Plane Algebraic Curves. Amer. Math. Soc. Press, Providence,2001.

[62] M. Friendly. Extending mosaic displays: Marginal, conditional, and partialviews of categorical data. J. of Comp. & Graph. Stat., 8–3:373–95, 1999.

[63] M. Friendly and al. Milestones in Thematic Cartography.www.math.yorku.ca/scs/SCS/Gallery/milestones/, 2005.

[64] LORAL FSD. Test Report for the Quick-Look Flight Demonstration of theIntraFormation Positioning System, Document Nos. 93-A37-002, Contractf33615-90-C-3609. LORAL Federal Systems Co, Oswego, 1994.

Bibliography 539

[65] M. O. Fua, Y.H. Ward and E.A. Rundensteiner. Hierarchical ParallelCoordinates for Exploration of Large Datasets – Proc.of Conf. on Vis.,43–50. IEEE Comp. Soc. Press, Los Alamitos, CA, 1999.

[66] G.W. Furnas and A. Buja. Prosection views: Dimensional inference throughsections and projections. J. Comp. & Graph. Stat., 3:323–85, 1994.

[67] C. Gennings, K.S. Dawson, W.H. Carter and R.H. Myers. Interpretingplots of a multidimensional dose-response surface in parallel coordinates.Biometrics, 46:719–35, 1990.

[68] W.J. Gibbs. Graphical methods in the thermodynamics of fluids.Conn. Acad., II-1:309–342, 1873.

[69] W.J. Gibbs. A method of geometrical representation of the thermodynamicproperties of substances by means of surfaces. Conn. Acad., II-2:382–404,1873.

[70] L. Girardin and D. Brodbeck. Visual Approach for Monitoring Logs. 12thSystems Adm. Conf., Boston, 1998.

[71] A. Goel. Vizcraft: A Multidimensional Visualization Tool for AircraftConfiguration Design. IEEE Vis Conf., 425–8, 1999.

[72] H. Goldstein. Classical Mechanics, 2nd ed. Addition-Wesley, ReadingMass., 1980.

[73] E. Goursat. A Course in Mathematical Analysis vol. 1. Ginn & Co,New York, 19xx.

[74] M.J. Greenberg. Euclidean and Non-Euclidean Geometries, Developmentand History. W.H. Freeman and Co., 1974.

[75] W.H. Guggenheimer. Differential Geometry. McGraw-Hill, New York,1963.

[76] H.S. Guyford and J.J. Haggerty. Flight – Life Science Library. Time Inc.,New York, 1965.

[77] F. Harary. Graph Theory. Addison-Wesley, Reading, Mass., 1969.

[78] J. Harris. Algebraic Geometry. Springer-Verlag, New York, 1992.

[79] H. Hauser. Parallel Sets: Visual Analysis of Categorical Data. Proc. IEEEInfovis, 2005.

[80] M.A. Hearst. Clustering versus faceted categories for information explo-ration. Comm. ACM, 49–4:59–61, 2006.

[81] N. Helfman. Navigating in Visual Models of Decision Support Systems.M.Sc. thesis, Dept. Comp. Sci., Bar Ilan University, Israel, 2000.

[82] J. Helly. Applications of Parallel Coordinates to Complex System Designand Operation. Proc. Nat. Comp. Grap. Assoc. vol. III, 541–546, 1987.

540 Bibliography

[83] P. Hertzog. Visualizations to Improve Reactivity towards Security Inci-dents inside Corporate Networks, SIGSAC Conf. Proc., 95–102. ACM,New York, 2006.

[84] B. Hetzler, P. Whitney, L. Martucci and J. Thomas. Multi-faceted InsightThrough Interoperable Visual Information Analysis Paradigms in Proc.IEEE on Info Vis pp. 137–144. IEEE Comp. Soc., Los Alamitos, CA,1998.

[85] D. Hilbert and S. Cohn-Vossen. Geometry and the Imagination. Chelsea,New York, 1983.

[86] W. Hodge and D. Pedoe. Methods of Algebraic Geometry, Vol. II.Cambridge Univ. Press, Cambridge, 1952.

[87] P. Hoffman and G. Grinstein. Multidimensional Information Vis. for DataMining with Applications for Machine Learning Classifiers. In Info. Vis.for KDD, Morgan-Kaufman, 2000.

[88] P. Hoffman, G. Grinstein, K. Marx and I. Grosse. DNA Visual and AnalyticData Mining. Proc. IEEE Infovis, pp. 437–441, 1997.

[89] Q. Huanin, C. Wing-Yi, X. Anbang, C. Kai-Lun, L. Kai-Hon and G. Ping.Visual analysis of the air pollution problem in hong kong. Trans. on Vis. &Comp. Grap., 13–6:1408–15, 2007.

[90] H.P. Hudson. Ruler & Compass in ‘Squaring the Circle’ by Hopson et al.Chelsea, New York, 1953.

[91] M.Y. Huh and K.Y. Song. DAVIS: A java based data visualization system.Comp. Stat., 17–3:411–423, 2002.

[92] C.K. Hung and D. Ierardi. Constructing Convex Hulls of Piecewise SmoothObjects. Univ. of S. Calif. CS Dept. Tech R. 94–583, 1994.

[93] C.K. Hung and A. Inselberg. Parallel Coordinate Representation of SmoothHypersurfaces. USC Tech. Report #CS-92-531, Los Angeles, 1992.

[94] C.K. Hung and A. Inselberg. Description of Surfaces in Parallel Coordi-nates by Linked Planar Regions, Mathematics of Surfaces XII, 177–208,LNCS 4647. Springer-Verlag, New York, 2007.

[95] C.B. Hurley and R.W. Olford. Pairwise Display of High DimensionalInformatin via Eulerial Tours and Hamiltonian Decompositions. to appear,2008.

[96] IEEE. The faa’s advanced automation program — special issue. IEEEComputer J., 20–2, 1987.

[97] A. Inselberg. N-Dimensional Graphics, LASC Tech. Rep. G320-2711,140 pages. IBM, 1981.

Bibliography 541

[98] A. Inselberg. Intelligent Instrumentation and Process Control, Proc. 2ndIEEE Conf. on AI Appl., 302–307. IEEE Comp. Soc., Los Alamitos, CA,1985.

[99] A. Inselberg. The plane with parallel coordinates. Visual Computer,1:69–97, 1985.

[100] A. Inselberg. Parallel Coordinates: A Guide for the Perplexed, in HotTopics Proc. of IEEE Conf. on Visualization, 35–38. IEEE Comp. Soc.,Los Alamitos, CA, 1996.

[101] A. Inselberg. Multidimensional Detective, in Proc. of IEEE InformationVisualization ’97, 100–107. IEEE Comp. Soc., Los Alamitos, CA, 1997.

[102] A. Inselberg. Visual data mining with parallel coordinates. Comput. Statit.,13–1:47–64, 1998.

[103] A. Inselberg and T. Avidan. The Automated Multidimensional Detective, inProc. of IEEE Information Visualization ’99, 112–119. IEEE Comp. Soc.,Los Alamitos, CA, 1999.

[104] A. Inselberg, M. Boz and B. Dimsdale. Planar Conflict Resolution Algo-rithm for Air-Traffic Control and the One-Shot Problem, in IBM PASC Tech.Rep. G320-3559. IBM Palo Alto Scientific Center, 1991.

[105] A. Inselberg and B. Dimsdale. Parallel Coordinates: A Tool For VisualizingMultidimensional Geometry, in Proc. of IEEE Conf. on Vis. ’90, 361–378.IEEE Comp. Soc., Los Alamitos, CA, 1990.

[106] A. Inselberg and B. Dimsdale. Multidimensional lines i: Representation.SIAM J. of Applied Math., 54–2:559–577, 1994.

[107] A. Inselberg and B. Dimsdale. Multidimensional lines ii: Proximity andapplications. SIAM J. of Applied Math., 54–2:578–596, 1994.

[108] A. Inselberg and C.K. Hung. A New Representation of Hypersurfaces Basedon Their Tangent Hyperplanes. In Course Notes for Tutorial #16 on Multi-dimensional Visualization 105–112, SIGGRAPH94, ACM Publications,1994.

[109] A. Inselberg and P.L. Lai. Visualizing Families of Close Planes, 66. Proc.5th Asian Conf. on Stat., Hong Kong, 2005.

[110] A. Inselberg, M. Reif and T. Chomut. Convexity algorithms in parallelcoordinates. J. ACM, 34:765–801, 1987.

[111] T. Itoh, et al. Hierarchical visualization of network intrusion detection data.IEEE Comp. Grap. & Appl., 26–2:40–47, 2006.

[112] M.Y. Ivory and M.A. Hearst. Statistical profiles of highly-rated web sites.IEEE Inter. Comp., 6–2:55–63, 2002.

542 Bibliography

[113] J. Johansson, P. Ljung, M. Jern and M. Cooper. Revealing Structure withinClustered Parallel Coordinates Displays, Proc. IEEE Infovis, 125–132.IEEE Comp. Soc., Los Alamitos, CA, 2005.

[114] J. Johansson, P. Ljung, M. Jern and M. Cooper. Revealing structure invisualizations of dense 2d and 3d parallel coordinates. Inf. Vis. Pal-Grave-Mcmillan Journ., 5:125–136, 2006.

[115] C. Jones. Visualization and Optimization. Kluwer Academic Publishers,Boston, 1996.

[116] M.W. Kantor, A.R. Liebler, E.S. Payne and E.E. Shult. (eds.) Finite Geome-tries, Buildings, and Related Topics. Clarendon Press Oxford, Oxford,1990.

[117] F. Karteszi. Introduction to Finite Geometries. North-Holland, Amsterdam-Oxford, 1976.

[118] D.A. Keim. Designing pixel-oriented visualization techniques: Theory andapplications. Comp. Graphics & Applic., 6–1:59–78, 2000.

[119] D.A. Keim and H.P. Kriegel. Visualization techniques for mining largedatabases: A comparison. Trans. Knowl. and Data Engr., 8–6:923–938,1996.

[120] M. Kipouros, T. Mleczko and M. Savill. Use of Parallel Coordinatesfor Post-Analyses of Multiobjective Aerodynamic Design Optimization inTurbomachinery (to appear). Proc. 4th AIAA Meet., 2008.

[121] M. Kline. Mathematical Thought from Ancient to Modern Times. OxfordUniv. Press, New York, 1972.

[122] R. Kriz. Visual Thinking by Scientists.www.sv.vt.edu/classes/ESM4714/Gen-Prin/vizthink.html, 2007.

[123] N. Kumasaka and R. Shibata. High Dimensional Data Visualization theTextile Plot. Proc. Compstat 2006 Psysica-Verlag, 581–9, 2006.

[124] P.L. Lai and C. Fyfe. A family of canonical correlation analysis networks.Neural Proc. Let., 14–2:93–105, 2001.

[125] F. Lauro, N.C. Palumbo and A. Iodice D’Enza. New Graphical SymbolicObjects Representation in Parallel Coordinates, in Between Data Scienceand Applications (M. Schader et al. eds). Springer, Heidelberg, 2003.

[126] C.L. Lay. Linear Algebra and its Applications. Updated Edition, Addison-Wesley, New York, 2000.

[127] R.O. Lejeune. Government Provided Complex Scenarios for the AdvancedAutomated System Design Competition Phase, MTR-85W240. MITRE Co,McLean, Virginia, 1985.

Bibliography 543

[128] M.M. Lipschutz. Differential Geometry. Shaum’s Outlines on Math.,McGraw-Hill, New York, 1969.

[129] J. Martin and G. Kennedy. Using Curves to Enhance Parallel Coordi-nates Visualization. Proc. IEEE Infovis, London, IEEE Comp. Soc., LosAlamitos, CA, 10–16, 2003.

[130] Mathcurve.com. Great Website for Surfaces.www.mathcurve.com/surfaces/surfaces.shtml, 2006.

[131] T. Matskewich, A. Inselberg and M. Bercovier. Approximated Planes inParallel Coordinates. Proc. of Geom. Model. Conf., St. Malo, VanderbiltUniv. Press, 257–266, 2000.

[132] B.H. McCormick, T.A. Defanti and M.D. Brown. Visualization in ScientificComputing. Computer Graphics 21–6, ACM SIGGRAPH, New York, 1987.

[133] N. Megido. On the Complexity of some Geometric Problems in UnboundedDimension, in IBM Res. Rep. RJ5744(58033). IBM Research, 1987.

[134] D. Michie, D.J. Spiegelhalter and C.C. Taylor. Machine Learning, Neuraland Statistical Classification. Ellis Horwood series in AI, 1994.

[135] T. Mihalisin, J. Timlin and J. Schwegler. Visualizing multivariate functions,data and distributions. IEEE Comp. Graph. and Appl., 1991.

[136] T.M. Mitchell. Machine Learning. McGraw-Hill, 1997.

[137] J.W. Moon. Various Proofs of Cayley’s Formula for Counting Trees,Seminar in Graph Theory, F. Harary (ed.). Holt, Rinehart & Winston,1967.

[138] J.R. Munkries. Topology, 2nd. ed. Prentice-Hall, New Jersey, 2000.

[139] J. Nakano and K. Honda. Three-dimensional parallel coordinates plotused for variable selection. Proc. COMPSTAT 2006, (eds. A. Rizzi andM. Vichi), Springer, 187–195, 2006.

[140] J.R. Newman. World of Mathematics. Simon and Schuster, New York,1956.

[141] M. d’Ocagne. Coordonnees parallels et axiale. Gautier-Villars, Paris,1885.

[142] M. d’Ocagne. Traite de Nomographie. Gautier-Villars, Paris, 1899.

[143] W.F. Osgood. Advanced Calculus. Macmillan, New York, 186–194, 1925.

[144] E. Otto. Nomography. Translated from 1956 Polish ed. by J. Smolska.Macmillan, New York, 1963.

[145] D. Pedoe. Circles. Pergamon, New York, 1957.

[146] P.A. Penna and R.R. Patterson. Projective Geometry and its Application toComputer Graphics. Prentice-Hall, New Jersey, 1986.

544 Bibliography

[147] Y. Pham, B. Cai and R. Brown. Visualization Techniques for TongueAnalysis in Traditional Chinese Medicine. Proc. Med. Imag. SPIE 14–19,2004.

[148] F.P. Preparata and M.I. Shamos. Computational Geometry. Springer-Verlag, New York, 1985.

[149] J.R. Quinlan. C4.5: Programs for Machine Learning. Morgan Kaufman,1993.

[150] J. Riordan. An Introduction to Combinatorial Analysis. John Wiley,New York, 1958.

[151] H. Rosenbaum and R. Schumann. Chances and Limits of Progression inVisualization, in SimVis. Proc. Sim. & Vis., Magdeburg, Germany, 2007.

[152] B.A. Rosenfeld. A History of Non-Euclidean Geometry. Springer-Verlag,New York, 1988.

[153] F. Rossi. Visual Data Mining and Machine Learning, in 14th Europ. ESANNProc., 251–264. ESANN, Bruges Belgium, 2006.

[154] P. Samuel. Projective Geometry. Springer-Verlag, New York, 1988.

[155] M. Schall. Diamond and ice: Visual exploratory data analysis tools.Perspective, J. of OAC at UCLA, 18(2):15–24, 1994.

[156] C. Schmid and H. Hinterberger. Comparative Multivariate Vis. AcrossConceptually Different Graphic Displays, in Proc. of 7th SSDBM. IEEEComp. Soc., Los Alamitos, CA, 1994.

[157] O. Schreier and Sperner. Projective Geometry of n Dimensions. ChelseaPublishing Company, New York, 1985.

[158] N. Shang and L. Breiman. Distribution based trees are more accurate.Proc. of 1st Inter. Conf. on Neural Info. Proc. (ICONIP96), 133, 1996.

[159] B. Shneiderman. Leonardo’s Laptop. MIT Press, Boston, 2002.

[160] B. Shneiderman and C. Plaisant. Designing the User Interface (4th Edition).Addison Wesley, Reading MA, 2005.

[161] C.H. Sisam and W.F. Atchison. Analytic Geometry, third ed. Holt, Rinehartand Winston, New York, 1955.

[162] D.J. Sobiski. Collision Detection and Collision Avoidance, Document Nos.196A798. LORAL Federal Systems Co, Oswego, NY, 1995.

[163] D.M.Y. Sommerville. An Introduction to the Geometry of N Dimensions.(First publ. 1929), Dover Publications, New York, 1958.

[164] R. Spence. Information Visualization: Design for Interaction, 2nd ed.Prentice Hall, 2007.

[165] I. Stewart. Flatterland. Perseus, New York, 2001.

Bibliography 545

[166] D.J. Struik. Lectures on Classical Differential Geometry 2nd. Ed. Dover(Addison-Wesley 1961), New York, 1988.

[167] D.F. Swayne, D. Cook and A. Buja. XGobi: Interactive Dynamic Graphicsin the X Window System. JCGS, 7–1, 113–130, 1998.

[168] D.F. Swayne, (T. Lang) Duncan, A. Buja and D. Cook. Ggobi: evolvingfrom Xgobi into an extensible framework for interactive data visualization.J. of Comp. & Graph. Stat., 43:423–444, 2003.

[169] J.L. Synge and A. Schild. Tensor Calculus, 2nd ed. University of TorontoPress, Toronto, 1956.

[170] V. Tontodonato. Symbolic Objects Representation, Ph.D. thesis. Dept. ofMath and Stat., Univ. Federico II, Napoli, 2002.

[171] M. Tory, S. Potts and T. Moller. A parallel coordinates style interface forexploratory volume visualization. Trans. on Vis. & Comp. Grap., 11–1:71–80, 2005.

[172] E.R. Tufte. The Visual Display of Quantitative Information. Graphic Press,Connecticut, 1983.

[173] E.R. Tufte. Envisioning Information. Graphic Press, Connecticut, 1990.

[174] E.R. Tufte. Visual Explanation. Graphic Press, Connecticut, 1996.

[175] L.A. Tweedie, R. Spence, H. Dawkes and Su H. Externalizing AbstractMathematical Models, Proc. CHI, Vancouver, Can., 406–412. ACM Press,1996.

[176] A.R. Unwin. Requirements for interactive graphics software for exploratorydata analysis. Comp. Stat., 14:7–22, 1999.

[177] A.R. Unwin, M. Theus and H. Hofmann. (Eds.) Graphics of Large Datasets.Springer, New York, 2006.

[178] F.B. Viegas, M. Wattenberg and D. Kushal. Studying Cooperation and Con-flict between Authors with history flow Visualizations in Proc. of SIGHCI onHuman Factors in Computing Systems, 575–582. ACM Press, New York,2004.

[179] R.J. Walker. Algebraic Curves. Springer, New York, 1978.

[180] W.A. Walker. What is a point of inflection? Amer. Math. Monthly, 63:182–183, 1956.

[181] M.O. Ward. XmdvTool: Integrating multiple methods for visualizing multi-variate data, Proc. IEEE Conf. on Vis., CA, 326–333. IEEE Comp. Soc.,Los Alamitos, CA, 1994.

[182] C. Ware. Information Visualization: Perception and Design, 2nd ed.Morgan Kaufman, San Francisco, 2004.

546 Bibliography

[183] C.E. Weatherburn. Differential Geometry of Three Dimensions. CambridgeUniv. Press, Cambridge, 1955.

[184] E. Wegman. Hyperdimensional data analysis using parallel coordinates.J. Amer. Stat. Assoc., 85:664–675, 1990.

[185] P.C. Wong and R.D. Bergeron. 30 Years of Multidimensional Multivari-ate Visualization in Scientific Visualization: Overviews, Methodologies &Techniques, G.M. Nelson, H. Mueller and H. Hagen (eds.), 3–33. IEEEComp. Soc., Los Alamitos, CA, 1997.

[186] V.O. Ya. Some integral calculus based on euler characteristic, lect. notes.Amer. Math. Monthly, 1346:127–138, 1988.

[187] L. Yang. Pruning & visualizing generalized association rules in parallelcoordinates. IEEE Know. & Data Engr., 15(1):60–70, 2005.

[188] H. Ye and Z. Lin. Speed-up simulated annealing by parallel coordinates.Euro J. of OR, 173:59–71, 2006.

[189] X. Yin, et al. VisFlowConnect. Comp. Sec., ACM Workshop on Visua-lization, Washington, D.C. 26–34, 2004.

12

Index

Abbot A.E., 2Adjiashvili D., xxiii, 376air traffic control

angle between superplanes, 133collision avoidance, 103–106conflict intervals, 106conflict resolution, 108–113information display, 101particle fields, 103–113

angle between two planes, 133approximate quadric hypersurfaces, 359–

365Archimedes, xvastroidal surface, 370Atariah, D., 322automatic classification, 406Avidan S. and T., xxiiiaxes

axis translation → rotation, 133permutations, 122spacing, 125

Banchoff T.F., 2Bertin J., xxvBinniaminy M., 322Boz M., xxiiibump, 370

Chatterjee A., xxiii, 250, 326Chen C., xxvChen C.C., xxvCohen L., xxiii, 376Cohen S., 429Cohen-Ganor S., 85

displaying several lines, 429–442collinearity construction algorithm

example in R6, 173

collinearity construction algorithm RN ,

170collinearity property p-flats in R

N , 169

computer graphicstransformations and matrices, 31–35

conoids, 352parabolic conoid, 352Plucker’s conoid, 352Whitney’s umbrella, 352

construction algorithmsin R

3, 136–159half spaces, 136intersecting planes, 138line on a plane, 136parallel plane on point, 151plane line intersection, 145rotation ↔ translation, 151special planes, 141the four indexed points, 138

in R4, 174–179

intersecting hyperplanes, 175the five indexed points, 174

convex surfaces, 367course

prerequisites, xxiiprojects, xixsyllabus, xviii–xxii

curves, 195–241algebraic curves, 208

Plucker class formula, 230Plucker curves, 229–230examples, 208–211

conicsclassification of transforms, 220–

226ideal points, 226–227the Indicator, 226transforms via Mobius transforma-

tion, 217conics ↔ conics, 216–229convex sets and their relatives, 231–

241convex upward and downward, 197curve plotting, 216cusps and inflection points, 199–204

548 Index

dualitybitangent ↔ double−point , 230

envelope, 43exponential, 212gconics – generalized conics, 231–241

convex hull with curved edges, 235convex-hull construction, 235gconics ↔ gconics, 232–241inner intersection, 238operational dualities, 238outer union, 238outer union ↔ inner intersection,

240Inselberg transformation, 206line, 43oscillatory, 212plotting, 207point, 43point-curves ↔ point-curves, 204–

241point-curves and line-curves, 195–

196, 204space curves, 216trigonometric, 212

d’Ocagne, xvidata mining, xvi, xvii, xxi, 378–427

classification, xvii, xxigrand vision, 5guidelines and strategies, 4introduction, 379–382Piatesky-Shapiro G., xxvZaavi J., xxv

decision support, xxi, 359, 416–418Desargue

theorem, 10developable surfaces

characteristic point, 312duality with space curves, 312general cone, 341helicoid, 342reconstruction, 319reconstruction from its rulings, 321representation, 314ruling, 312rulings of cylinders, 326start points, 319

symmetry and coincindence, 333tangent planes, 311

dimensionality, 2dimple, 370Dimsdale B., xxiii, 361, 375

conics, 217hyperplanes, 165

duality, 10R

2 – slope and x-coordinate relation,51

bitangent ↔ double point , 230cusp ↔ inflection point, 199–204gconics – generalized conics

operational dualities, 238outer union ↔ inner intersection,

240–241in 2-D point ↔ line, 15in 3-D point ↔ plane, 15point ↔ line in P

2, 49–62point↔ line

horizontal position of , 51line → point linear transformation,

53point → line linear transformation,

54point-curves and line-curves, 55projective plane, 49recognizing orthogonality, 58rotation ↔ translations, 57transformations, 56–60

Ehrenfest P., 2Eickemeyer

representation of p-flats, 167representation of hyperplanes, 166superplanes – E, 125

Eickemeyer J., xxiii, 125ellipsoid, 367envelopes, 55, 118, 185–194

formulation and computation, 186necessary and sufficient conditions,

188of families of curves, 190–193one-parameter family of curves, 185singural points, 189

Index 549

Euclidelements, 8parallels axiom, 15

exploratory data analysis – EDA, xvii, xxi,382–383

data display, 380hundreds of variables, 400interactivity, 380pattern recognition, 379simple case study, 383–390software, 4software requirements, 4testing assumptions, 401

exploratory data analysis – EDA, xxiiiexterior point, 349

financial dataset, 392, 401Fiorini P., xxiii

geometry, 7–48affine, 48constructions

compass alone, 8first algorithms, 8Poncelet and Steiner conditions, 8

Egyptians, 7Euclid

elements, 8Greeks, 7origins, 7parallelism, 2projective, 8–45transformations

invariants, 45–48geovisualization, xxivGibbs W.J., 333Giles N., 236GIS, xxivgold, high price

correlation with currencies, 397grand tour

Asimov D., xxivGreenshpan O., 460–475Grinstein G., xxvgroup, 46

half planeoriented, 197

half spaces, 136Hearst M., xxvHelfman N.

decision support, xxiiihelicoid (developable), 342Hinterberger H.

comparative visualization, xxivdata density analysis, xxiv

homogeneous coordinates, 22–43analytical proofs, 39–43for finite geometries, 24on a line, 25

Hung C.K., xxiii, 314, 326, 335, 336, 375convex-hull with curved edges, 235surface representation decomposition,

451–460Hurwitz A., xxiiihypercube, 297hyperellipse, 362hyperellipsoids, 360

ILMduality in P

2

point ↔ line, 51, 56, 58geometry

Desargues’s theorem, 42Pappus’s theorem, 42point ↔ line duality, 19

multidimensional linesgeneral case, 83indexed points – adjacent and base

variable, 67three-point collinearity, 71

planesrotation ↔ translation for indexed

points, 122, 153incidence matrix, 300intelligent process control, 363–365

domain, 363instrument panel, 364learning, 365rule, 363

interaxis distancesrecursive notation, 299

Itai T., 322

550 Index

Keim D., xxivKosloff Z., 322Kriz R., 2

Lifshits Y., 373

Mobius strip, 354Makarov I., 373Matskewich T., xxiii, 250, 273, 279

proximate hyperplanes construction,279

proximate hyperplanes representation,273

multidimensional lines3-point collinearity, 127air traffic control, 100–113construction algorithms, 79–82

example, 80displaying several lines, 429–442distance and proximity, 88–100L2 minimum distance – monotonic-

ity with dimension, 97–100L2 versus L1 minimum distance, 99constrained L1 distance, 92–96intersecting lines, 88minimum distance between lines,

91–100nonintersection, 89–100

indexed points3-point collinearity, 70–72arbitrary parametrization, 75dummy index, 77identification, 82

representation, 63–88adjacent variables, 63–65base variable, 65convenient display of a line with

positive slopes, 83convenient display of several lines,

85general case, 75–78line on a plane, 119transforming positive to negative

slopes, 83–85two-point form, 68–70two-point form continued, 78

rotations ↔ translations duality, 87separation in the xy plane

above and below relations, 149

nearly developable surfaces, 343network

visualization and analysis, 460nomography, xvinonlinear models, 416–418nonorientability

in R2, 149

projective plane P2, 13

visualization, 358nonorientable surface, 354

paraboloid, 367parallel coordinates

P2, 49–62bitangent ↔ double-point duality,

230air traffic control, 100–113algebraic curves, 229–230applications, xxiv, 5automatic classification, 406automatic classifiers, 4bare essentials, 427complex-valued functions, 476–488conics, 216–229curves, 55, 195–241

curve plotting, 207–216cusp ↔ inflection-point duality, 199–

204data mining, 378–427decision support, 416–418definition, 3enhancing with curves, xxivexploratory data analysis – EDA, 382–

383atomic queries, 385guidelines, 383–390multidimensional detective, 382–

383requirements, 381

first proposed, 2gconics – generalized conics

operational dualities, 238

Index 551

hypercube, 61in 3-D, xxivInselberg transformation, 206multidimensional lines, 63–113

representation, 70network visualization and analysis,

460–475notation in R

2

bar, point, line, 55orthogonality, 58parallax, 4parallelism, 51permutations for adjacencies, 388planes, p-flats, and hyperplanes, 115–

183point ↔ line duality

queries, 387properties, 3proximate planes and hyperplanes,

243–295recursive construction algorithm, 4representation mapping I, 73–75separation in R

2 and R3, 149

separation in the xy plane, 196software, 4starplots, xxivsurfaces in R

N , 297–377visual data mining – EDA, 4

Pascal, 44pattern recognition, 1pi (π ), 7planes, p-flats, and hyperplanes

approximate planes and flats, 179axis spacing, 125axis translation → rotation

rotation angle in R3, 133

coefficients and distance betweenadjacent points, 134

collinearity construction algorithmexample in R

6, 173collinearity construction algorithm

RN , 170

coplanarityin industrial data, 117indexed points, 125near coplanarity, 179vertical lines pattern, 116

determining multiple planes, 180

line on a plane, 119planes in R

3, 115planes on a line, 122point on a plane, 118recursive construction, 164recursive construction algorithm R

N ,170

representationp-flat in R

N , 167augmented, 132by two indexed points in R

3, 128collinearity property forp-flats, 169hyperplane in R

N , 166hyperplanes and p-flats in R

4, R5,

168indexed points, 124indexed points in R

N , 164planar coordinates, 118recursive construction, 164recursive construction algorithm,

127unification for p-flats 0 ≤ p < N ,

168vertical lines, 115

rotation ↔ translation indexed points,151

rotation ↔ translation vertical lines,120

separating points clusters on differentplanes, 443–451

separation in R3 – points and planes,

149superplanes sp, 125

in R4, 159

uper-planes spin R

3, 126point ↔ line duality in P

2, 49–62projective geometry, 9–45

3-point collinearity, 24analytic, 22–43

proofs, 39–43augmented lines, planes, space, 12axioms, 10, 13–16conics, 43cross-ratio, 27

invariant under perspectivity, 27Desargue’s theorem, 12duality, 9, 16–19

552 Index

finite geometries, 19–21fundamental transformations, 26–29homogeneous coordinates, 9, 22–43

analytical proofs, 39–43for finite geometries, 24on a line, 25

ideal elements, 11incidence relation, 10line coordinates, 23linearizability of projective transfor-

mations, 29–31Pappus’s theorem and its dual, 18Pappus’s theorem generalizations, 44

Brianchon’s theorem dual of Pas-cal’s theorem, 45Pascal theorem for conics, 44

perpectivity and projectivity, 28perspectivity, 26primitives, 10projections and sections, 10projective plane model, 10–13

Hilbert, 12projective plane nonorientability, 13

projective transformationsin P

2, 32in P

3, 33linearized, 31–35perspective, 32, 34rotation, 32, 33scaling, 32translation, 32, 33

proximate planes and hyperplanes, 243–295

Chatterjee, A., 250Matskewich, T., 250neighborhood, 244outline of development, 253proximity of hyperplanes, 251–295

adding the c0, 280–285components and structure of �,

257–263conditions for� being bc, uc, or gh,

267–274costruction of �, 263–275formulation of the problem, 251–

254full image of NH, 275–279

matching points and navigation,287–295properties of fN , 254–257region �, 257–275region crossing x axis, 287some examples, 283–295

proximity of lines and line neighbor-hoods, 244–251

topology for proximity, 243–245

quasars data set, 390–392queries

atomic, 385compound, 401zebra, 397

recursive construction algorithm, 127general in R

N , 164Reif M., xxiiirepresentation

astroidal representation, 370bump, 370convex surfaces, 369dimple, 370Mobius strip, 358sphere, 367torus, 370

representation mappingversion II, 180

Rivero J., xxiiirotation, 158rotation ↔ translation points, 151–158rotation ↔ translation vertical lines, 120rotations

coordinate system in R3, 36

in R2, 32

in R3, 33

ruled surfacesconoids, 352Mobius strip, 354representation, 349saddle, 347

saddle, 347Satellite data set, 383–390separation in R

2 – points and lines, 149

Index 553

separation in R3 – points and planes (new

approach), 146separation in R

3 – points and planes (oldapproach), 149

separation in the xy planeabove and below relations, 196convex upward or downward, 197oriented half plane, 196

Shahaf N., 443–451Shneiderman B., xxvSinger Y., 460–475Snow J., xvSpence R., xxvsphere, 367Spivak M., 376surfaces

representation decomposition and devel-opable quadrics, 451–460

surfaces in RN , 297–377

approximate quadrics, 359–365decision support, 359hyperellipsoids, 360intelligent process control, 362–365

boundary contours, 304–310developable surfaces, 310–346

ambiguities and uniqueness, 316–319characteristic point, 312duality with space curves, 312general, 342–346general cone, 341helicoid, 342Hung C.K., 311reconstruction, 319–325, 341reconstruction from its rulings, 321representation, 314ruling, 312rulings of cylinders, 326specific developables, 325–341specific developables, cones, 336–

341specific developables, cylinders,

326–336symmetry and coincidence, 333tangent planes, 311

formulation, 301–304general

matching algorithm, 372–377

hypercube, 297matching algorithm, 301, 303, 311,

348, 351, 352, 372more general surfaces, 365–377nearly developable surfaces, 343normal vector, 302preliminaries, 297–300representation, 301

algebraic surface, 306astroidal surface, 370boundaries, 305boundaries - example, 307boundaries - the class EG, 305boundaries of quadrics, 306boundaries of ruled surfaces, 348,

351boundaries, ideal points, 305convex surface boundaries, 366convex surfaces, 367, 369coordinates of representing points,

303Mobius strip, 358ruled surfaces, 347saddle, 347sphere, 367torus, 370

representation of developable andruled, 358

representing points relations, 300ruled surfaces, 346–359

conoids, 352Mobius strip, 354representation, 349saddle, 347

to see C2, 476–488

torus, 370transformations

invariants, 45linear, 46linear fractional, 30linearizability of projective transfor-

mations, 31Mobius, 30

homomorphic to group of nonsingu-lar matrices, 217

554 Index

visualizationArchimedes, xvearly success of, xvemergence of, xvinformation, 1multidimensional, xvi, 1science, 2, 333scientific, 1

VLSI data set, 401–406

Ward M., xxvWare C., xxv

Yaari Y., 476–488

zero defects, 401–405