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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