It can be done by comparing only relative extents and signs ofĪppropriate coefficients, as shown in Figure 3 (see Part I for detailed procedures The determination of the types (and corresponding quotient sequences) sufficesįor that purpose. The values of the quotients, as shown in Figures 3b and 3c, are not really necessaryįor the essentially qualitative kind of analysis we are most interested in here. Note that the exact diagrammatic constructions for The same applies to the cut by the Ox2 axis See Part I) is the same as the arrangement of various types of solutions of theĮxpression (2.1) along that cut, and the quotients of the a1 ⋅ x1 ♦ b relation indicateīorders of these solutions along the cut. Sets on the axis (as defined by the quotient sequence of the a1 ⋅ x1 ♦ b relation, Representing the solution sets of (2.1), the arrangement of one-dimensional solution I.e., a one-dimensional expression, whose quotients are equal to the intersections Special case of a boundary line going through the origin is not included here.Īs mentioned already, that case belongs to the separate analysis of intermediateĪs was indicated above, setting x2 = 0 in the expression (2.1), we obtain a1 ⋅ x1 ♦ b, Parallel lines, because (α1 α2 −) (α1 α2 +), see Figure 2c again. There are eight boundary lines, in four pairs of Namely, the lines (α1 − β ) and (α1 + β ) pass through Q1βα1, while the lines (−α2 β )Īnd (+α2 β ) pass through Q2βα2. (− + −) = L1 S2, see the tableĪt least two boundary lines pass through every quotient point on the axes. Will be also denoted by quotient pairs Q1βα1 Q2βα2, e.g. Obtained by setting x2 = 0 or x1 = 0 in the original expression (2.1). In Part I for definitions) of one-dimensional relations a1 ⋅ x1 ♦ b and a2 ⋅ x2 ♦ b, 1–20) of the paper, an analysis of the two-dimensional relational expressionĪ1 ⋅ x1 + a2 ⋅ x2 ♦ b, where ♦ ∈, is called aīoundary line for the expression (2.1), and denoted as (α1 α2 β ).Īs it is obvious, boundary lines intersect the Ox1 and Ox2 axes at the pointsī β / a1α1 = Q1βα1 and b β / a2α2 = Q2βα2, respectively, i.e., at the quotients (see Lemma 4.1
Using the results obtained for the one-dimensional case in Part I (Reliable Computingĩ (1) (2003), pp. Świȩtokrzyska 21, 00–049 Warsaw, Poland, e-mail: 8 September 2001 accepted: 11 June 2002)Ībstract. Institute of Fundamental Technological Research, Polish Academy of Sciences, Part II: The Two-Dimensional Case and Generalization to