Version 5.0 The LEDA User Manual

14.30 Generalized polygons with circular edges (
r circle gen polygon )
1. Definition
The data type r circle polygon is not closed under boolean operations, e.g., the set difference
of a polygon P and a polygon Q nested in P is a region that contains a “hole”.
Therefore we provide a generalization called r circle gen polygon which is closed under
(regularized) boolean operations (see below).
A formal definition follows: An instance P of the data type r circle gen polygon is a
regular polygonal region in the plane. A regular region is an open set that is equal to the
interior of its closure. A region is polygonal if its boundary consists of a finite number of
r circle segments.
The boundary of an r circle gen polygon consists of zero or more weakly simple closed
polygonal chains. Each such chain is represented by an object of type r circle ploygon.
There are two regions whose boundary is empty, namely the empty region and the full
region. The full region encompasses the entire plane. We call a region trivial if its
boundary is empty. The boundary cycles P 1 , P 2 , . . . , P k of an r circle gen polygon are
ordered such that no P i is nested in a P j with i < j.
#include < LEDA/geo/r circle gen polygon.h >
2. Types
r circle gen polygon :: KIND { EMPTY, FULL, NON TRIVIAL }
describes the kind of the polygon: the empty set, the full plane or
a non-trivial polygon.
r circle gen polygon ::CHECK TYPE { NO CHECK, SIMPLE, WEAKLY SIMPLE, NOT
WEAKLY SIMPLE }
used to specify which checks should be applied and also describes
the outcome of a simplicity check.
r circle gen polygon :: RESPECT TYPE { DISREGARD ORIENTATION, RESPECT
ORIENTATION }
used in contructors to specify whether to force a positive orientation
for the constructed object (DISREGARD ORIENTATION ) or to
keep the orientation of the input (RESPECT ORIENTATION ).
3. Creation
r circle gen polygon P ;
creates an empty polygon P .