Version 5.0 The LEDA User Manual

integer floor(const real& x)
integer ceil(const real& x)
returns the largest integer smaller than or equal to x.
returns the smallest integer greater than or equal to x.
rational small rational between(const real& x, const real& y)
returns a rational number between x and y with the smallest
available denominator. Note that the denominator does not
need to be strictly minimal over all possible rationals.
rational small rational near(const real& x, double eps)
5. Implementation
returns small rational between(x − eps, x + eps).
A real is represented by the expression which defines it and an interval inclusion I that
contains the exact value of the real. The arithmetic operators +, −, ∗, d √ take constant
time. When the sign of a real number needs to be determined, the data type first computes
a number q, if not already given as an argument to sign, such that |x| ≤ 2 −q implies x = 0.
The bound q is computed as described in [79]. Using bigfloat arithmetic, the data type
then computes an interval I of maximal length 2 −q that contains x. If I contains zero,
then x itself is equal to zero. Otherwise, the sign of any point in I is returned as the sign
of x.
Two shortcuts are used to speed up the computation of the sign. Firstly, if the initial
interval approximation already suffices to determine the sign, then no bigfloat approximation
is computed at all. Secondly, the bigfloat approximation is first computed only with
small precision. The precision is then roughly doubled until either the sign can be decided
(i.e., if the current approximation interval does not contain zero) or the full precision 2 −q
is reached. This procedure makes the sign computation of a real number x adaptive in
the sense that the running time of the sign computation depends on the complexity of x.
6. Example
We give two typical examples for the use of the data type real that arise in Computational
geometry. We admit that a certain knowledge about Computational geometry is required
for their full understanding. The examples deal with the Voronoi diagram of line segments
and the intersection of line segments, respectively.
The following incircle test is used in the computation of Voronoi diagrams of line segments
[17, 14]. For i, 1 ≤ i ≤ 3, let l i : a i x + b i y + c i = 0 be a line in two–dimensional space
and let p = (0, 0) be the origin. In general, there are two circles passing through p and
touching l 1 and l 2 . The centers of these circles have homogeneuos coordinates (x v , y v , z v ),
where
√
x v = a 1 c 2 + a 2 c 1 ± sign(s) 2c 1 c 2 ( √ N + D)