ArrangementBasicTraits_2

Definition

A model of this concept must define nested Point_2 and X_monotone_curve_2 types, which represent planar points and continuous $$x-monotone curves (a vertical segment is also considered to be weakly $$x-monotone), respectively. The $$x-monotone curves are assumed to be pairwise disjoint in their interiors, so they do not intersect except at their endpoints.

The $$x-monotone curves may be unbounded, namely they may have unbounded ends that lie at infinity, or bounded, in which case their have finite endpoints are representable as Point_2 objects. An $$x-monotone curve may also have one unbounded end and one finite endpoint (e.g. a ray). If unbounded curves are supported, the requirements from the traits class are extended a bit, as described below. In particular, the traits class needs to support comparisons at infinity, where we interpret comparisons at infinity as follows: Suppose we wish to compare the $$y-position of the hyperbolas $$y₁ = (1)/(x) and $$y₂ = (2)/(x) at $$x = ; than since there exists $$x₀ such that for each finite $$x > x₀ we have $$y₂(x) > y₁(x) (in our case we can take $$x₀ = 0), we say that $$y₂ is above $$y₁ at infinity. Similarly, when comparing the $$x-positions of the vertical line $$x = 0 and the $$xy-minimal end of parabola $$y = (1)/(x) for $$x > 0 (which has a vertical asymptote at $$x = 0), we define that the line lies to the left of the hyperbola. Namely, comparing curves at their unbounded ends should return the comparison result EQUAL only if the curves overlap.

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