点在多边形内算法的实现
点在多边形内算法的实现
cheungmine
2007-9-22
本文是采用射线法判断点是否在多边形内的C语言程序。多年前，我自己实现了这样一个算法。但是随着时间的推移，我决定重写这个代码。参考周培德的《计算几何》一书，结合我的实践和经验，我相信，在这个算法的实现上，这是你迄今为止遇到的最优的代码。
这是个C语言的小算法的实现程序，本来不想放到这里。可是，当我自己要实现这样一个算法的时候，想在网上找个现成的，考察下来竟然一个符合需要的也没有。我对自己大学读书时写的代码没有信心，所以，决定重新写一个，并把它放到这里，以飨读者。也增加一下BLOG的点击量。
首先定义点结构如下：
/* Vertex structure */
typedef
struct
{
double x, y;
} vertex_t;
本算法里所指的多边形，是指由一系列点序列组成的封闭简单多边形。它的首尾点可以是或不是同一个点（不强制要求首尾点是同一个点）。这样的多边形可以是任意形状的，包括多条边在一条绝对直线上。因此，定义多边形结构如下：
/* Vertex list structure – polygon */
typedef
struct
{
int num_vertices; /* Number of vertices in list */
vertex_t *vertex; /* Vertex array pointer */
} vertexlist_t;
为加快判别速度，首先计算多边形的外包矩形（
rect_t），判断点是否落在外包矩形内，只有满足落在外包矩形内的条件的点，才进入下一步的计算。为此，引入外包矩形结构rect_t和求点集合的外包矩形内的方法vertices_get_extent，代码如下：
/* bounding rectangle type */
typedef
struct
{
double min_x, min_y, max_x, max_y;
} rect_t;
/* gets extent of vertices */
void
vertices_get_extent (const vertex_t* vl, int np, /* in vertices */
rect_t* rc /* out extent*/ )
{
int i;
if (np > 0){
rc->min_x = rc->max_x = vl[0].x; rc->min_y = rc->max_y = vl[0].y;
}else{
rc->min_x = rc->min_y = rc->max_x = rc->max_y = 0; /* =0 ? no vertices at all */
}
for(i=1; i<np; i++)
{
if(vl[i].x < rc->min_x) rc->min_x = vl[i].x;
if(vl[i].y < rc->min_y) rc->min_y = vl[i].y;
if(vl[i].x > rc->max_x) rc->max_x = vl[i].x;
if(vl[i].y > rc->max_y) rc->max_y = vl[i].y;
}
}
当点满足落在多边形外包矩形内的条件，要进一步判断点（
v）是否在多边形（vl：np）内。本程序采用射线法，由待测试点（v）水平引出一条射线B（v，w），计算B与vl边线的交点数目，记为c，根据奇内偶外原则（c为奇数说明v在vl内，否则v不在vl内）判断点是否在多边形内。
具体原理就不多说。为计算线段间是否存在交点，引入下面的函数：
（
1）is_same判断2（p、q）个点是（1）否（0）在直线l（l_start，l_end）的同侧；
（
2）is_intersect用来判断2条线段（不是直线）s1、s2是（1）否（0）相交；
/* p, q is on the same of line l */
static
int is_same(const vertex_t* l_start, const vertex_t* l_end, /* line l */
const
vertex_t* p,
const
vertex_t* q)
{
double dx = l_end->x - l_start->x;
double dy = l_end->y - l_start->y;
double dx1= p->x - l_start->x;
double dy1= p->y - l_start->y;
double dx2= q->x - l_end->x;
double dy2= q->y - l_end->y;
return ((dx*dy1-dy*dx1)*(dx*dy2-dy*dx2) > 0? 1 : 0);
}
/* 2 line segments (s1, s2) are intersect? */
static
int is_intersect(const vertex_t* s1_start, const vertex_t* s1_end,
const
vertex_t* s2_start, const vertex_t* s2_end)
{
return (is_same(s1_start, s1_end, s2_start, s2_end)==0 &&
is_same(s2_start, s2_end, s1_start, s1_end)==0)? 1: 0;
}
下面的函数
pt_in_poly就是判断点（v）是（1）否（0）在多边形（vl：np）内的程序：
int
pt_in_poly ( const vertex_t* vl, int np, /* polygon vl with np vertices */
const
vertex_t* v)
{
int i, j, k1, k2, c;
rect_t rc;
vertex_t w;
if (np < 3)
return 0;
vertices_get_extent(vl, np, &rc);
if (v->x < rc.min_x || v->x > rc.max_x || v->y < rc.min_y || v->y > rc.max_y)
return 0;
/* Set a horizontal beam l(*v, w) from v to the ultra right */
w.x = rc.max_x + DBL_EPSILON;
w.y = v->y;
c = 0; /* Intersection points counter */
for(i=0; i<np; i++)
{
j = (i+1) % np;
if(is_intersect(vl+i, vl+j, v, &w))
{
c++;
}
else if(vl[i].y==w.y)
{
k1 = (np+i-1)%np;
while(k1!=i && vl[k1].y==w.y)
k1 = (np+k1-1)%np;
k2 = (i+1)%np;
while(k2!=i && vl[k2].y==w.y)
k2 = (k2+1)%np;
if(k1 != k2 && is_same(v, &w, vl+k1, vl+k2)==0)
c++;
if(k2 <= i)
break;
i = k2;
}
}
return c%2;
}
本想配些插图说明问题，但是，
CSDN的文章里放图片我还没用过。以后再试吧！实践证明，本程序算法的适应性极强。但是，对于点正好落在多边形边上的极端情形，有可能得出2种不同的结果。
下面是python的版本：
下面是python的版本：
#!/usr/bin/python
#-*- coding: UTF-8 -*-
#
# pip.py
# point in polygon
#
# 2016-01-07
#
# point(pt) is inside polygon(poly)
########################################################################
# gets extent of vertices vl
#
def extent_vertices(vl):
min_x = 0.0
min_y = 0.0
max_x = 0.0
max_y = 0.0
i = 0
for (x, y) in vl:
if not i:
(min_x, min_y) = (x, y)
(max_x, max_y) = (x, y)
i = 1
else:
if x < min_x:
min_x = x
if y < min_y:
min_y = y
if x > max_x:
max_x = x
if y > max_y:
max_y = y
return (min_x, min_y, max_x, max_y)
# points p, q are on the same of line l
#
def same_side(l_start, l_end, p, q):
dx, dy = float(l_end[0] - l_start[0]), float(l_end[1] - l_start[1])
dx1, dy1 = float(p[0] - l_start[0]), float(p[1] - l_start[1])
dx2, dy2 = float(q[0] - l_end[0]), float(q[1] - l_end[1])
d = (dx * dy1 - dy * dx1) * (dx * dy2 - dy * dx2)
if d >= 0:
return 1
else:
return 0
# are line segments s1, s2 intersect ?
#
def intersect_segs(s1_start, s1_end, s2_start, s2_end):
if not same_side(s1_start, s1_end, s2_start, s2_end) and not same_side(s2_start, s2_end, s1_start, s1_end):
return 1
else:
return 0
# is point pt(x, y) in polygon poly
# returns:
# 0 : not in
# 1 : in
def pt_in_poly(pt, poly):
np = len(poly)
if np < 3:
return 0
(x, y) = pt
(min_x, min_y, max_x, max_y) = extent_vertices(poly)
if x < min_x or x > max_x or y < min_y or y > max_y:
return 0
# set a horizontal beam line (pt, w) from pt to the ultra right
w = (max_x + max_x*0.1 + 1, y)
c = 0
for i in range(0, np):
j = (i+1) % np
if pt == poly[i]:
return 1
elif intersect_segs(poly[i], poly[j], pt, w):
c = c + 1
elif poly[i][1] == w[1]:
k1 = (np + i - 1) % np
while (k1 != i and poly[k1][1] == w[1]):
k1 = (np + k1 - 1) % np
k2 = (i+1) % np
while (k2 != i and poly[k2][1] == w[1]):
k2 = (k2 + 1) % np
if k1 != k2 and not same_side(pt, w, poly[k1], poly[k2]):
c = c + 1
if k2 <= i:
break
i = k2
return c % 2
epsilon = 0.0000001
assert pt_in_poly((0, 0), [(-1, -1), (-1, 1), (1, 1), (1, -1)]) == 1, "(1) appliaction error"
assert pt_in_poly((-1, -1), [(-1, -1), (-1, 1), (1, 1), (1, -1)]) == 1, "(2) appliaction error"
assert pt_in_poly((-1, 1), [(-1, -1), (-1, 1), (1, 1), (1, -1)]) == 1, "(3) appliaction error"
assert pt_in_poly((-1 - epsilon, -1), [(-1, -1), (-1, 1), (1, 1), (1, -1)]) == 0, "(4) appliaction error"
assert pt_in_poly((-1 + epsilon, -1 + epsilon), [(-1, -1), (-1, 1), (1, 1), (1, -1)]) == 1, "(5) appliaction error"
assert pt_in_poly((-1 + epsilon, -1 + epsilon), [(-1, -1), (-1, 1), (1, 1), (1, -1)]) == 1, "(6) appliaction error"
assert pt_in_poly((0, 0), [(-1, -1), (-1, 1), (1, 1), (1, 0), (2, -1)]) == 1, "(7) appliaction error"
assert pt_in_poly((0, 0), [(-1, -1), (-1, 1), (1, 1), (1, 0), (2, 1), (2, -1)]) == 1, "(8) appliaction error"
assert pt_in_poly((0, 0), [(-1, -1), (-1, 1), (1, 1), (1, 0), (2, 1), (2, 0), (2, -1)]) == 1, "(9) appliaction error"
assert pt_in_poly((0, 0), [(-1, -1), (-1, 1), (1, 1), (1, 0), (2, 0), (3, 0), (2, -1)]) == 1, "(10) appliaction error"
assert pt_in_poly((-2, 0), [(-1, -1), (-1, 1), (1, 1), (1, 0), (2, 0), (3, 0), (2, -1)]) == 0, "(11) appliaction error"
assert pt_in_poly((3, 0), [(-1, -1), (-1, 1), (1, 1), (1, 0), (2, 0), (3, 0), (2, -1)]) == 1, "(12) appliaction error"