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Home , Flat (geometry), Line coordinates, List of symbols, Notation, Parallel (operator), Plane (geometry)

12

Notation and List of

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 are usually denoted by lowercase Greek letters N positive integers; natural numbers R real numbers C complex numbers 5 real part of a 6 imaginary part of a complex number R2 Euclidean 2-dimensional (or ) 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 ∪ union ∩ set

A. Inselberg, Coordinates, DOI 10.1007/978-0-387-68628-8, © Springer Science+Business Media, LLC 2009 532 and

∅ empty st (null set) 2A power set of A, the set of all subsets of A ip inflection of a n principal vector to a space curve nˆ unit normal vector to a t vector to a space curve b binormal vector τ torsion (measures “non-planarity” of a space curve) × product → mapping A → B correspondence (also used for ) ⇒ implies ⇔ if and only if <, > less than, greater than |a| of a |b| of vector b  , length L1 L1 norm L L norm, Euclidean 2 2 summation Jp set of integers modulo p with the operations + and × ≈ approximately equal = not equal  parallel ⊥ , orthogonal (,,) — ordered triple representing a point in P2; point coordinates [,,] homogeneous coordinates — ordered triple representing a P2; line coordinates P p(A,B,C,D,...) 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 df dx derivative of f with respect to x, also written as f (x) ∂ boundary of a region ∂F partial derivative of F with respect to x , also denoted F ∂xi i xi and 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 , parallel coordinates A¯ image of A in -coords, representation of A ¯ ¯ X1,... ,XN axes for parallel for N-dimensional space (either Euclidean 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 RN (or PN ) πp p-flat in RN , a plane of 1 ≤ p ≤ (N − 1) in RN π1 a one-flat is a line, also denoted by sp abbreviation for superplane, a 2-flat whose points are represented by straight lines in -coords πs denotes a superplane πNs denotes a superplane in RN E family of superplanes, named after J. Eickemeyer, who first discovered them E also denotes the class of surfaces who are the of their tangent (hyper) planes EG class of surfaces contained in E whose representing contours can be obtained by Lemma 9.3.1 ¯ di directed distance of Xi from the y-axis 0 ¯ dN (see next), standard axis spacing, when the X1 is coincident with the y-axis and inter-axis distance is one unit for all axes i ¯ ¯ d axes spacing where the X1 axis is placed one unit after the XN and N ¯ all axes up to and including Xi are placed successively one unit apart and the rest of axes remain in the standard axes spacing. 534 Notation and List of Symbols

s π1 first superplane, with the standard axes spacing s ¯ ¯ π1 second superplane, the X1 placed one unit to the right of XN the remaining axes are in standard axes spacing. s i πi is the ith superplane with axes spacing dN 3 π¯123, π¯1 123 two triple indexed points representing the plane (2-flat) π ⊂ R ,

also denoted by π¯123 =¯π0 , π¯1 23 = π1 or 123, 1 23 when π is ¯ 1 clear from the context; π1 23 is based on the axes spacing d3 π¯123,... ,N =¯π0 point with (N + 1) indices representing a (of dimen- sion N − 1)), and similarly π¯i for the representing points based i on axes spacing dN gconics generalized conics, sections of a double in R3 whose 2 bases are the same bounded in R2 bc, ubc, gh like conics there are 3 kinds of gconics: bounded convex set (bc), unbounded convex set (ubc) and generalized (gh) ) outer union * inner intersection 12

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Index

Abbot A.E., 2 computer graphics Adjiashvili D., xxiii, 376 transformations and matrices, 31Ð35 air traffic control conoids, 352 between superplanes, 133 parabolic conoid, 352 collision avoidance, 103Ð106 Plucker’s¬ conoid, 352 conflict intervals, 106 Whitney’s umbrella, 352 conflict resolution, 108Ð113 construction algorithms information display, 101 in R3, 136Ð159 particle fields, 103Ð113 half , 136 angle between two planes, 133 intersecting planes, 138 approximate hypersurfaces, 359Ð line on a plane, 136 365 parallel plane on point, 151 , xv plane line intersection, 145 astroidal surface, 370 ↔ translation, 151 Atariah, D., 322 special planes, 141 automatic classification, 406 the four indexed points, 138 Avidan S. and T., xxiii in R4, 174Ð179 axes intersecting hyperplanes, 175 axis translation → rotation, 133 the five indexed points, 174 permutations, 122 convex surfaces, 367 spacing, 125 course prerequisites, xxii projects, xix Banchoff T.F., 2 syllabus, xviiiÐxxii Bertin J., xxv curves, 195Ð241 Binniaminy M., 322 algebraic curves, 208 Boz M., xxiii Plucker¬ class formula, 230 bump, 370 Plucker¬ curves, 229Ð230 examples, 208Ð211 conics Chatterjee A., xxiii, 250, 326 classification of transforms, 220Ð Chen C., xxv 226 Chen C.C., xxv ideal points, 226Ð227 Cohen L., xxiii, 376 the Indicator, 226 Cohen S., 429 transforms via Mobius¬ transforma- Cohen-Ganor S., 85 tion, 217 displaying several lines, 429Ð442 conics ↔ conics, 216Ð229 construction algorithm convex sets and their relatives, 231Ð example in R6, 173 241 collinearity construction algorithm RN , convex upward and downward, 197 170 curve plotting, 216 collinearity property p-flats in RN , 169 cusps and inflection points, 199Ð204 548 Index

duality and coincindence, 333 bitangent ↔ double−point, 230 tangent planes, 311 envelope, 43 dimensionality, 2 exponential, 212 dimple, 370 gconics Ð generalized conics, 231Ð241 Dimsdale B., xxiii, 361, 375 convex hull with curved edges, 235 conics, 217 convex-hull construction, 235 hyperplanes, 165 gconics ↔ gconics, 232Ð241 duality, 10 inner intersection, 238 R2 Ð slope and x-coordinate relation, operational dualities, 238 51 outer union, 238 bitangent ↔ double point, 230 ↔ outer union inner intersection, ↔ inflection point, 199Ð204 240 gconics Ð generalized conics Inselberg transformation, 206 operational dualities, 238 line, 43 outer union ↔ inner intersection, oscillatory, 212 240Ð241 plotting, 207 in 2-D point ↔ line, 15 point, 43 in 3-D point ↔ plane, 15 point-curves ↔ point-curves, 204Ð point ↔ line in P2, 49Ð62 241 point↔ line point-curves and line-curves, 195Ð horizontal of ¯,51 196, 204 line → point linear transformation, space curves, 216 53 trigonometric, 212 point → line linear transformation, 54 d’Ocagne, xvi point-curves and line-curves, 55 data mining, xvi, xvii, xxi, 378Ð427 , 49 classification, xvii, xxi recognizing , 58 ↔ grand vision, 5 rotation translations, 57 guidelines and strategies, 4 transformations, 56Ð60 introduction, 379Ð382 Piatesky-Shapiro G., xxv Zaavi J., xxv decision support, xxi, 359, 416Ð418 Ehrenfest P., 2 Desargue Eickemeyer , 10 representation of p-flats, 167 developable surfaces representation of hyperplanes, 166 characteristic point, 312 superplanes Ð E, 125 duality with space curves, 312 Eickemeyer J., xxiii, 125 general cone, 341 ellipsoid, 367 , 342 envelopes, 55, 118, 185Ð194 reconstruction, 319 formulation and computation, 186 reconstruction from its rulings, 321 necessary and sufficient conditions, representation, 314 188 ruling, 312 of families of curves, 190Ð193 rulings of , 326 one-parameter family of curves, 185 start points, 319 singural points, 189 Index 549

Euclid half plane elements, 8 oriented, 197 parallels , 15 half spaces, 136 exploratory data analysis Ð EDA, xvii, xxi, Hearst M., xxv 382Ð383 Helfman N. data display, 380 decision support, xxiii hundreds of variables, 400 helicoid (developable), 342 interactivity, 380 Hinterberger H. pattern recognition, 379 comparative visualization, xxiv simple case study, 383Ð390 data density analysis, xxiv software, 4 homogeneous coordinates, 22Ð43 software requirements, 4 analytical proofs, 39Ð43 testing assumptions, 401 for finite geometries, 24 exploratory data analysis Ð EDA, xxiii on a line, 25 exterior point, 349 Hung C.K., xxiii, 314, 326, 335, 336, 375 convex-hull with curved edges, 235 surface representation decomposition, financial dataset, 392, 401 451Ð460 Fiorini P., xxiii Hurwitz A., xxiii , 297 hyperellipse, 362 hyperellipsoids, 360 geometry, 7Ð48 affine, 48 constructions ILM compass alone, 8 duality in P2 first algorithms, 8 point ↔ line, 51, 56, 58 Poncelet and Steiner conditions, 8 geometry Egyptians, 7 Desargues’s theorem, 42 Pappus’s theorem, 42 elements, 8 point ↔ line duality, 19 Greeks, 7 multidimensional lines origins, 7 general case, 83 parallelism, 2 indexed points Ð adjacent and base projective, 8Ð45 variable, 67 transformations three-point collinearity, 71 invariants, 45Ð48 planes geovisualization, xxiv rotation ↔ translation for indexed Gibbs W.J., 333 points, 122, 153 Giles N., 236 , 300 GIS, xxiv intelligent process control, 363Ð365 gold, high price domain, 363 correlation with currencies, 397 instrument panel, 364 grand tour learning, 365 Asimov D., xxiv rule, 363 Greenshpan O., 460Ð475 interaxis Grinstein G., xxv recursive notation, 299 group, 46 Itai T., 322 550 Index

Keim D., xxiv ↔ translations duality, 87 Kosloff Z., 322 separation in the xy plane Kriz R., 2 above and below relations, 149

Lifshits Y., 373 nearly developable surfaces, 343 network visualization and analysis, 460 Mobius¬ strip, 354 nomography, xvi Makarov I., 373 nonlinear models, 416Ð418 Matskewich T., xxiii, 250, 273, 279 nonorientability proximate hyperplanes construction, in R2, 149 279 projective plane P2,13 proximate hyperplanes representation, visualization, 358 273 nonorientable surface, 354 multidimensional lines 3-point collinearity, 127 air traffic control, 100Ð113 paraboloid, 367 construction algorithms, 79Ð82 parallel coordinates example, 80 P2, 49Ð62 displaying several lines, 429Ð442 bitangent ↔ double-point duality, distance and proximity, 88Ð100 230 L2 minimum distance Ð monotonic- air traffic control, 100Ð113 ity with dimension, 97Ð100 algebraic curves, 229Ð230 L2 versus L1 minimum distance, 99 applications, xxiv, 5 constrained L1 distance, 92Ð96 automatic classification, 406 intersecting lines, 88 automatic classifiers, 4 minimum distance between lines, bare essentials, 427 91Ð100 complex-valued functions, 476Ð488 nonintersection, 89Ð100 conics, 216Ð229 indexed points curves, 55, 195Ð241 3-point collinearity, 70Ð72 curve plotting, 207Ð216 arbitrary parametrization, 75 cusp ↔ inflection-point duality, 199Ð dummy index, 77 204 identification, 82 data mining, 378Ð427 representation, 63Ð88 decision support, 416Ð418 adjacent variables, 63Ð65 definition, 3 base variable, 65 enhancing with curves, xxiv convenient display of a line with exploratory data analysis Ð EDA, 382Ð positive slopes, 83 383 convenient display of several lines, atomic queries, 385 85 guidelines, 383Ð390 general case, 75Ð78 multidimensional detective, 382Ð line on a plane, 119 383 transforming positive to negative requirements, 381 slopes, 83Ð85 first proposed, 2 two-point form, 68Ð70 gconics Ð generalized conics two-point form continued, 78 operational dualities, 238 Index 551

hypercube, 61 line on a plane, 119 in 3-D, xxiv planes in R3, 115 Inselberg transformation, 206 planes on a line, 122 multidimensional lines, 63Ð113 point on a plane, 118 representation, 70 recursive construction, 164 network visualization and analysis, recursive construction algorithm RN , 460Ð475 170 notation in R2 representation bar, point, line, 55 p-flat in RN , 167 orthogonality, 58 augmented, 132 parallax, 4 by two indexed points in R3, 128 parallelism, 51 collinearity property for p-flats, 169 permutations for adjacencies, 388 hyperplane in RN , 166 planes, p-flats, and hyperplanes, 115Ð hyperplanes and p-flats in R4, R5, 183 168 point ↔ line duality indexed points, 124 queries, 387 indexed points in RN , 164 properties, 3 planar coordinates, 118 proximate planes and hyperplanes, recursive construction, 164 243Ð295 recursive construction algorithm, recursive construction algorithm, 4 127 representation mapping I, 73Ð75 unification for p-flats 0 ≤ p

finite geometries, 19Ð21 matching points and navigation, fundamental transformations, 26Ð29 287Ð295 homogeneous coordinates, 9, 22Ð43 properties of fN , 254Ð257 analytical proofs, 39Ð43 region , 257Ð275 for finite geometries, 24 region crossing x axis, 287 on a line, 25 some examples, 283Ð295 ideal elements, 11 proximity of lines and line neighbor- incidence relation, 10 hoods, 244Ð251 line coordinates, 23 topology for proximity, 243Ð245 linearizability of projective transfor- mations, 29Ð31 quasars data set, 390Ð392 Pappus’s theorem and its dual, 18 queries Pappus’s theorem generalizations, 44 atomic, 385 Brianchon’s theorem dual of Pas- compound, 401 cal’s theorem, 45 zebra, 397 Pascal theorem for conics, 44 perpectivity and projectivity, 28 perspectivity, 26 recursive construction algorithm, 127 primitives, 10 general in RN , 164 projections and sections, 10 Reif M., xxiii projective plane model, 10Ð13 representation Hilbert, 12 astroidal representation, 370 projective plane nonorientability, 13 bump, 370 projective transformations convex surfaces, 369 in P2,32 dimple, 370 in P3,33 Mobius¬ strip, 358 linearized, 31Ð35 , 367 perspective, 32, 34 , 370 rotation, 32, 33 representation mapping scaling, 32 version II, 180 translation, 32, 33 Rivero J., xxiii proximate planes and hyperplanes, 243Ð rotation, 158 ↔ 295 rotation translation points, 151Ð158 rotation ↔ translation vertical lines, 120 Chatterjee, A., 250 rotations Matskewich, T., 250 coordinate system in R3,36 neighborhood, 244 in R2,32 outline of development, 253 in R3,33 proximity of hyperplanes, 251Ð295 ruled surfaces adding the c , 280Ð285 0 conoids, 352 components and structure of , Mobius¬ strip, 354 257Ð263 representation, 349 conditions for being bc, uc,orgh, saddle, 347 267Ð274 costruction of , 263Ð275 formulation of the problem, 251Ð saddle, 347 254 Satellite data set, 383Ð390 full image of NH, 275Ð279 separation in R2 Ð points and lines, 149 Index 553 separation in R3 Ð points and planes (new hypercube, 297 approach), 146 matching algorithm, 301, 303, 311, separation in R3 Ð points and planes (old 348, 351, 352, 372 approach), 149 more general surfaces, 365Ð377 separation in the xy plane nearly developable surfaces, 343 above and below relations, 196 normal vector, 302 convex upward or downward, 197 preliminaries, 297Ð300 oriented half plane, 196 representation, 301 Shahaf N., 443Ð451 algebraic surface, 306 Shneiderman B., xxv astroidal surface, 370 Singer Y., 460Ð475 boundaries, 305 Snow J., xv boundaries - example, 307 Spence R., xxv boundaries - the class EG, 305 sphere, 367 boundaries of , 306 Spivak M., 376 boundaries of ruled surfaces, 348, surfaces 351 representation decomposition and devel- boundaries, ideal points, 305 opable quadrics, 451Ð460 convex surface boundaries, 366 RN surfaces in , 297Ð377 convex surfaces, 367, 369 approximate quadrics, 359Ð365 coordinates of representing points, decision support, 359 303 hyperellipsoids, 360 Mobius¬ strip, 358 intelligent process control, 362Ð365 ruled surfaces, 347 boundary contours, 304Ð310 saddle, 347 developable surfaces, 310Ð346 sphere, 367 ambiguities and uniqueness, 316Ð torus, 370 319 representation of developable and characteristic point, 312 ruled, 358 duality with space curves, 312 representing points relations, 300 general, 342Ð346 ruled surfaces, 346Ð359 general cone, 341 conoids, 352 helicoid, 342 Mobius¬ strip, 354 Hung C.K., 311 representation, 349 reconstruction, 319Ð325, 341 saddle, 347 reconstruction from its rulings, 321 representation, 314 ruling, 312 rulings of cylinders, 326 to see C2, 476Ð488 specific developables, 325Ð341 torus, 370 specific developables, , 336Ð transformations 341 invariants, 45 specific developables, cylinders, linear, 46 326Ð336 linear fractional, 30 symmetry and coincidence, 333 linearizability of projective transfor- tangent planes, 311 mations, 31 formulation, 301Ð304 Mobius,¬ 30 general homomorphic to group of nonsingu- matching algorithm, 372Ð377 lar matrices, 217 554 Index

visualization Ward M., xxv Archimedes, xv Ware C., xxv early success of, xv emergence of, xv information, 1 Yaari Y., 476Ð488 multidimensional, xvi, 1 science, 2, 333 scientific, 1 VLSI data set, 401Ð406 zero defects, 401Ð405