Version 5.0 The LEDA User Manual

14.24 Geometry Algorithms ( geo alg )
All functions listed in this section work for geometric objects based on both floating-point
and exact (rational) arithmetic. In particular, point can be replace by rat point, segment
by rat segment, and circle by rat circle.
The floating point versions are faster but unreliable. They may produce incorrect results,
abort, or run forever. Only the rational versions will produce correct results for all inputs.
The include-file for the rational version is rat geo alg.h, the include-file for the floating
point version is float geo alg.h, and geo alg.h includes both versions. Including both
versions increases compile time. An alternative name for geo alg.h is plane alg.h.
• Convex Hulls
list CONVEX HULL(const list& L)
CONVEX HULL takes as argument a list of points and returns the polygon
representing the convex hull of L. The cyclic order of the vertices
in the result list corresponds to counter-clockwise order of the vertices
on the hull. The algorithm calls our current favorite of the algorithms
below.
polygon CONVEX HULL POLY(const list& L)
as above, but returns the convex hull of L as a polygon.
list UPPER CONVEX HULL(const list& L)
returns the upper convex hull of L.
list LOWER CONVEX HULL(const list& L)
returns the lower convex hull of L.
list CONVEX HULL S(const list& L)
as above, but the algorithm is based on the sweep paradigm. Running
time is O(n log n) in the worst and in the best case.
list CONVEX HULL IC(const list& L)
as above, but the algorithm is based on incremental construction. The
running time is O(n 2 ) worst case and is O(n log n) expected case. The
expectation is computed as the average over all permutations of L. The
running time is linear in the best case.
list CONVEX HULL RIC(const list& L)
as above. The algorithm permutes L randomly and then calls the preceding
function.