2D Snap Rounding

**Figure: **An arrangement of segments before (a) and after (b)
SR (hot pixels are shaded)

In a snap-rounded arrangement, the distance between a vertex and a non-incident edge can be extremely small compared with the width of a pixel in the grid used for rounding. ISR is a modification of SR which makes a vertex and a non-incident edge well separated (the distance between each is at least half-the-width-of-a-pixel). However, the guaranteed quality of the approximation in ISR degrades. Figure 19.3 depicts the results of SR and ISR on the same input. Conceptually, the ISR procedure is equivalent to repeated application of SR, namely we apply SR to the original set of segments, then we use the output of SR as input to another round of SR and so on until all the vertices are well separated from non-incident edges. Algorithmically we operate differently, as this repeated application of SR would have resulted in an efficient overall process. The algorithmic details are given in [HP02].

Our package supports both schemes, implementing the algorithm described in [HP02]. Although the paper only describes an algorithm for ISR, it is easy to derive an algorithm for SR, by performing only the first rounding level for each segment.

The input to the program is a set $$*S* of $$*n* segments,
$$*S={s _{1},...,s_{n}}* and the output is a set $$

There are three template parameters: *Traits* is the underlying geometry,
i.e., the number type used and the coordinate representation.
*InputIterator* is the type of the iterators that point to the first
and after-the-last elements of the input. Finally, *OutputContainer* is the
type of the output container.

Since the algorithm requires kernel functionalities such as the rounding to the
center of a pixel, a special traits class must be provided. The precise
description of the requirements is given by the concept
*SnapRoundingTraits_2*. The class *Snap_rounding_traits_2* is a model of
this concept.

**Figure: **An arrangement of segments before (a), after SR (b)
and ISR (c) (hot pixels are shaded).

The following example generates an ISR representation of an arrangement of four line segments. In particular it produces a list of points that are the vertices of the resulting polylines in a plane tiled with one-unit square pixels.

#include <CGAL/basic.h> #include <CGAL/Cartesian.h> #include <CGAL/Quotient.h> #include <CGAL/MP_Float.h> #include <CGAL/Snap_rounding_traits_2.h> #include <CGAL/Snap_rounding_2.h> typedef CGAL::Quotient<CGAL::MP_Float> Number_type; typedef CGAL::Cartesian<Number_type> Kernel; typedef CGAL::Snap_rounding_traits_2<Kernel> Traits; typedef Kernel::Segment_2 Segment_2; typedef Kernel::Point_2 Point_2; typedef std::list<Segment_2> Segment_list_2; typedef std::list<Point_2> Polyline_2; typedef std::list<Polyline_2> Polyline_list_2; int main() { Segment_list_2 seg_list; Polyline_list_2 output_list; seg_list.push_back(Segment_2(Point_2(0, 0), Point_2(10, 10))); seg_list.push_back(Segment_2(Point_2(0, 10), Point_2(10, 0))); seg_list.push_back(Segment_2(Point_2(3, 0), Point_2(3, 10))); seg_list.push_back(Segment_2(Point_2(7, 0), Point_2(7, 10))); // Generate an iterated snap-rounding representation, where the centers of // the hot pixels bear their original coordinates, using 5 kd trees: CGAL::snap_rounding_2<Traits,Segment_list_2::const_iterator,Polyline_list_2> (seg_list.begin(), seg_list.end(), output_list, 1.0); int counter = 0; Polyline_list_2::const_iterator iter1; for (iter1 = output_list.begin(); iter1 != output_list.end(); ++iter1) { std::cout << "Polyline number " << ++counter << ":\n"; Polyline_2::const_iterator iter2; for (iter2 = iter1->begin(); iter2 != iter1->end(); ++iter2) std::cout << " (" << iter2->x() << ":" << iter2->y() << ")\n"; } return(0); }

This program generates four polylines, one for each input segment. The exact output follows:

Polyline number 1: (0/4:0/4) (12/4:12/4) (20/4:20/4) (28/4:28/4) (40/4:40/4) Polyline number 2: (0/4:40/4) (12/4:28/4) (20/4:20/4) (28/4:12/4) (40/4:0/4) Polyline number 3: (12/4:0/4) (12/4:12/4) (12/4:28/4) (12/4:40/4) Polyline number 4: (28/4:0/4) (28/4:12/4) (28/4:28/4) (28/4:40/4)

The package is supplied with a graphical demo program that opens a window,
allows the user to edit segments dynamically, applies a selected
snap-rounding procedures, and displays the result onto the same window
(see *<CGAL_ROOT>/demo/Snap_rounding_2/demo.C*).

Next: Reference Manual

CGAL Open Source Project.
Release 3.2.1.
13 July 2006.