2D Polygons( Poygon） CGAL 4.13 -User Manual

1 Introduction

A polygon is a closed chain of edges. Several algorithms are available for polygons. For some of those algorithms, it is necessary that the polygon is simple. A polygon is simple if edges don't intersect, except consecutive edges, which intersect in their common vertex.

The following algorithms are available:

• find the leftmost, rightmost, topmost and bottommost vertex.
• compute the (signed) area.
• check if a polygon is simple.
• check if a polygon is convex.
• find the orientation (clockwise or counterclockwise)
• check if a point lies inside a polygon.

All those operations take two forward iterators as parameters in order to describe the polygon. These parameters have a point type as value type.

The type Polygon_2 can be used to represent polygons. Polygons are dynamic. Vertices can be modified, inserted and erased. They provide the algorithms described above as member functions. Moreover, they provide ways of iterating over the vertices and edges.

The Polygon_2 class is a wrapper around a container of points, but little more. Especially, computed values are not cached. That is, when the Polygon_2::is_simple() member function is called twice or more, the result is computed each time anew.

 多边形是一个闭合的边的链。多边形有多个算法。对于 其中的一些算法，要求多边形是简单的。多边形是简单的，如果其所有边除相邻边的共同顶点处外都不相交。

下列的算法可用：

　　（1）查找最左侧、最右侧、最上方、最下方顶点

　　（2）计算（）面积

　　（3）检查多边形是不是简单的

　　（4）检查多边形是不是凸的

　　（5）求其方向（顺时针或逆时针）

　　（6）检查一个点是否在多边形中

2 Examples

2.1 The Polygon Class

The following example creates a polygon and illustrates the usage of some member functions.

创建一个多边形并使用一些成员函数
File Polygon/Polygon.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>

#include <CGAL/Polygon_2.h>

#include <iostream>

 

typedef CGAL::Exact_predicates_inexact_constructions_kernel K;

typedef K::Point_2 Point;

typedef CGAL::Polygon_2<K> Polygon_2;

using std::cout; using std::endl;

 

 

int main()

{

Point points[] = { Point(0,0), Point(5.1,0), Point(1,1), Point(0.5,6)};

Polygon_2 pgn(points, points+4);

 

// check if the polygon is simple.

cout << "The polygon is " <<

(pgn.is_simple() ? "" : "not ") << "simple." << endl;

 

// check if the polygon is convex

cout << "The polygon is " <<

(pgn.is_convex() ? "" : "not ") << "convex." << endl;

 

return 0;

}

Figure 15.1 A polygon and some points

2.2 Algorithms Operating on Sequences of Points

The following example creates a polygon and illustrates the usage of some global functions that operate on sequences of points.

创建一个多边形并使用全局函数来操纵其点的序列
File Polygon/polygon_algorithms.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>

#include <CGAL/Polygon_2_algorithms.h>

#include <iostream>

 

typedef CGAL::Exact_predicates_inexact_constructions_kernel K;

typedef K::Point_2 Point;

using std::cout; using std::endl;

 

void check_inside(Point pt, Point *pgn_begin, Point *pgn_end, K traits)

{

cout << "The point " << pt;

switch(CGAL::bounded_side_2(pgn_begin, pgn_end,pt, traits)) {

case CGAL::ON_BOUNDED_SIDE :

cout << " is inside the polygon.\n";

break;

case CGAL::ON_BOUNDARY:

cout << " is on the polygon boundary.\n";

break;

case CGAL::ON_UNBOUNDED_SIDE:

cout << " is outside the polygon.\n";

break;

}

}

 

int main()

{

Point points[] = { Point(0,0), Point(5.1,0), Point(1,1), Point(0.5,6)};

 

// check if the polygon is simple.

cout << "The polygon is "

<< ( CGAL::is_simple_2(points, points+4, K()) ? "" : "not ")

<< "simple." << endl;

 

check_inside(Point(0.5, 0.5), points, points+4, K());

check_inside(Point(1.5, 2.5), points, points+4, K());

check_inside(Point(2.5, 0), points, points+4, K());

 

return 0;

}

2.3 Polygons in 3D Space

Sometimes it is useful to run a 2D algorithm on 3D data. Polygons may be contours of a 3D object, where the contours are organized in parallel slices, generated by segmentation of image data from a scanner.

In order to avoid an explixit projection on the xy plane, one can use the traits class Projection_traits_xy_3 which is part of the 2D and 3D Linear Geometric Kernel.

有时在3D数据中运行2D算法也是有用的。多边形可能是3D对象的轮廓，它由一个扫描仪通过对图像数据进行分段生成，轮廓以平行的片段组织。
File Polygon/projected_polygon.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>

#include <CGAL/Projection_traits_yz_3.h>

#include <CGAL/Polygon_2_algorithms.h>

#include <iostream>

 

typedef CGAL::Exact_predicates_inexact_constructions_kernel K;

typedef K::Point_3 Point_3;

 

int main()

{

Point_3 points[4] = { Point_3(0,1,1), Point_3(0,2,1), Point_3(0,2,2), Point_3(0,1,2) };

bool b = CGAL::is_simple_2(points,

points+4,

CGAL::Projection_traits_yz_3<K>());

if (!b){

std::cerr << "Error polygon is not simple" << std::endl;

return 1;

}

return 0;

}