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12 Notation and List of Symbols 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 ip inflection point of a curve n principal normal vector to a space curve nˆ 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 L L norm, Euclidean distance 2 2 summation Jp set of integers modulo p with the operations + and × ≈ approximately equal = not equal parallel ⊥ perpendicular, orthogonal (,,) homogeneous coordinates — ordered triple representing a point in P2; point coordinates [,,] homogeneous coordinates — ordered triple representing a line 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 system, parallel coordinates A¯ image of A in -coords, representation of A ¯ ¯ X1,... ,XN axes for parallel coordinate system 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 dimension 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 envelope 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 hyperplane (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 cone in R3 whose 2 bases are the same bounded convex set in R2 bc, 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. 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