Version 5.0 The LEDA User Manual

Without this assistance, reals automatically compute a weaker bound of |Ẽ| ≥ 2−56k−161
for Ẽ ≠ 0 by [15].
We turn to the line segment intersection problem next. Assume that all endpoints have
k–bit integer homogeneous coordinates. This implies that the intersection points have
homogeneous coordinates (X, Y, W ) where X, Y and W are (4 k + 3) - bit integers. The
Bentley–Ottmann plane sweep algorithm for segment intersection [65] needs to sort points
by their x–coordinates, i.e., to compare fractions X 1 /W 1 and X 2 /W 2 where X 1 , X 2 , W 1 , W 2
are as above. This boils down to determining the sign of the 8k + 7 bit integer X 1 ∗ W 2 −
X 2 ∗W 1 . If all variables X i , W i are declared real then their sign test will be performed quite
efficiently. First, an interval approximation is computed and then, if necessary, bigfloat
approximations of increasing precision. In many cases, the interval approximation already
determines the sign. In this way, the user of the data type real gets nearly the efficiency
of a hand-coded floating point filter [35, 66] without any work on his side. This is in
marked contrast to [35, 66] and will be incorporated into [65].