Find the Border UVALive - 3218(PSGL)

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Find the Border

 UVALive - 3218 

PSGL

技术分享
  1 #include <bits/stdc++.h>
  2 using namespace std;
  3 #define pb push_back
  4 const double eps = 1e-8;
  5 const int inf = 0x3f3f3f3f;
  6 
  7 int dcmp(double x) {
  8     if(fabs(x) < eps) return 0;
  9     return x < 0 ? -1 : 1;
 10 }
 11 
 12 struct Point {
 13     double x,y;
 14     Point (double x=0, double y=0): x(x), y(y) {}
 15 };
 16 typedef Point Vector;
 17 
 18 Vector operator + (Vector a, Vector  b) {
 19     return Vector(a.x+b.x, a.y+b.y);
 20 }
 21 Vector operator - (Point a, Point b) {
 22     return Vector(a.x-b.x, a.y-b.y);
 23 }
 24 Vector operator * (Vector a, double p) {
 25     return Vector(a.x*p, a.y*p);
 26 }
 27 Vector operator / (Vector a, double p) {
 28     return Vector(a.x/p, a.y/p);
 29 }
 30 // 理论上这个“小于”运算符是错的,因为可能有三个点a, b, c, a和b很接近(即a<b好b<a都不成立),b和c很接近,但a和c不接近
 31 // 所以使用这种“小于”运算符的前提是能排除上述情况
 32 bool operator < (Point a, Point b) {
 33     return dcmp(a.x-b.x)<0 || (dcmp(a.x-b.x)==0&&dcmp(a.y-b.y)<0);
 34 }
 35 bool operator == (Point a, Point b) {
 36     return dcmp(a.x-b.x)==0 && dcmp(a.y-b.y)==0;
 37 }
 38 double Dot(Vector a, Vector b) {
 39     return a.x*b.x + a.y*b.y;
 40 }
 41 double Cross(Vector a, Vector b) {
 42     return a.x*b.y - a.y*b.x;
 43 }
 44 double Length(Vector a) {
 45     return sqrt(Dot(a,a));
 46 }
 47 
 48 typedef vector<Point> Poly;
 49 
 50 bool SegProIn(Point a1, Point b1, Point a2, Point b2) {
 51     double c1 = Cross(b1-a1, b2-a1), c2 = Cross(b1-a1,a2-a1);
 52     double c3 = Cross(b2-a2, a1-a2), c4 = Cross(b2-a2, b1-a2);
 53     return dcmp(c1)*dcmp(c2)<0 && dcmp(c3)*dcmp(c4)<0;
 54 }
 55 Point GetLineIn(Point p, Vector v, Point q, Vector w) {
 56     Vector u = p-q;
 57     double t = Cross(w,u) / Cross(v,w);
 58     return p+v*t;
 59 }
 60 bool OnSeg(Point p, Point a, Point b) {
 61     return dcmp(Cross(a-p, b-p))==0 && dcmp(Dot(a-p, b-p))<0;
 62 }
 63 // 多边形的有向面积
 64 double GetArea(Poly p) {
 65     double area = 0;
 66     int n = p.size();
 67     for(int i = 1; i < n-1; i++) area += Cross(p[i]-p[0], p[(i+1)%n]-p[0]);
 68     return area/2;
 69 }
 70 
 71 struct Edge{
 72     int u, v;
 73     double ang;
 74     Edge(int u=0, int v=0, double ang=0): u(u), v(v), ang(ang){}
 75 };
 76 const int maxe = 10000+10;  // 最大边数
 77 
 78 // 平面直线图(PSGL)实现
 79 struct PSGL{
 80     int n, m, face_cnt;
 81     double x[maxe], y[maxe];
 82     vector<Edge> edges;
 83     vector<int> G[maxe];
 84     int vis[maxe<<1];  // 每条边是否已经访问过
 85     int left[maxe<<1];  // 左面的编号
 86     int pre[maxe<<1];  // 相同起点的上一条边(即顺时针旋转碰到的下一条边)的编号
 87 
 88     vector<Poly> faces;
 89     double area[maxe];
 90 
 91     void init(int n) {
 92         this->n = n;
 93         for(int i = 0; i < n; i++) G[i].clear();
 94         edges.clear();
 95         faces.clear();
 96     }
 97     double getAngle(int u,int v) {
 98         return atan2(y[v]-y[u], x[v]-x[u]);
 99     }
100     void add(int u, int v) {
101         edges.pb(Edge(u,v,getAngle(u,v)));
102         edges.pb(Edge(v,u,getAngle(v,u)));
103         m = edges.size();
104         G[u].pb(m-2);
105         G[v].pb(m-1);
106     }
107     //找出faces并计算面积
108     void build(){
109         for(int  u = 0; u < n; u++) {
110             int d = G[u].size();
111             for(int i = 0; i < d; i++) {
112                 for(int j = i+1; j < d; j++) {
113                     if(edges[G[u][i]].ang > edges[G[u][j]].ang) swap(G[u][i],G[u][j]);
114                 }
115             }
116             for(int i = 0; i < d; i++) pre[G[u][(i+1)%d]] = G[u][i];
117         }
118         memset(vis, 0, sizeof(vis));
119         face_cnt = 0;
120         for(int u = 0; u < n; u++) {
121             for(int i = 0; i < G[u].size(); i++){
122                 int e = G[u][i];
123                 if(!vis[e]){  //逆时针找圈
124                     face_cnt++;
125                     Poly p;
126                     while(1){
127                         vis[e] = 1;
128                         left[e] = face_cnt;
129                         int from = edges[e].u;
130                         p.pb(Point(x[from], y[from]));
131                         e = pre[e^1];
132                         if(e == G[u][i]) break;
133                     }
134                     faces.pb(p);
135                 }
136             }
137         }
138         for(int i = 0; i < faces.size(); i++) {
139             area[i] = GetArea(faces[i]);
140 
141         }
142     }
143 
144 };
145 
146 PSGL g;
147 
148 const int maxv = 110;
149 int n, c;
150 
151 Point p[maxv];
152 Point V[maxv*maxv/2+maxv];
153 
154 int ID(Point p) {
155     return lower_bound(V, V+c, p) - V;
156 }
157 //假定poly没有相邻点重合的情况,只需要删除三点共线的情况
158 Poly simplify(const Poly& p) {
159     Poly temp;
160     int n = p.size();
161     for(int i = 0; i < n; i++) {
162         Point a = p[i];
163         Point b = p[(i+1)%n];
164         Point c = p[(i+2)%n];
165         if(dcmp(Cross(a-b, c-b)) != 0) temp.pb(b);
166     }
167     return temp;
168 }
169 
170 void build_graph(){
171     c = n;
172     for(int i = 0; i < n; i++) V[i] = p[i];
173     vector<double> dis[maxv];    // dist[i][j]是第i条线段上的第j个点离起点(P[i])的距离
174     for(int i = 0; i < n; i++) {
175         for(int j = i+1; j < n; j++) {
176             if(SegProIn(p[i], p[(i+1)%n], p[j], p[(j+1)%n])) {
177                 Point xy = GetLineIn(p[i], p[(i+1)%n]-p[i], p[j], p[(j+1)%n]-p[j]);
178                 V[c++] = xy;
179                 dis[i].pb(Length(xy - p[i]));
180                 dis[j].pb(Length(xy - p[j]));
181             }
182         }
183     }
184     // 为了保证“很接近的点”被看作同一个,这里使用了sort+unique的方法
185   // 必须使用前面提到的“理论上是错误”的小于运算符,否则不能保证“很接近的点”在排序后连续排列
186   // 另一个常见的处理方式是把坐标扩大很多倍(比如100000倍),然后四舍五入变成整点(计算完毕后再还原),用少许的精度损失换来鲁棒性和速度。
187     sort(V, V+c);
188     c = unique(V, V+c) - V;
189 
190     g.init(c);  //c是平面图的点数
191     for(int i = 0; i < c; i++) {
192         g.x[i] = V[i].x;
193         g.y[i] = V[i].y;
194     }
195     for(int  i = 0; i < n; i++) {
196         Vector v = p[(i+1)%n] - p[i];
197         double len = Length(v);
198         dis[i].pb(0);
199         dis[i].pb(len);
200         sort(dis[i].begin(), dis[i].end());
201         int sz = dis[i].size();
202         for(int j = 1; j < sz; j++) {
203             Point a = p[i] + v*(dis[i][j-1]/len);
204             Point b = p[i] + v*(dis[i][j]/len);
205             if(a == b) continue;
206             g.add(ID(a), ID(b));
207         }
208     }
209     g.build();
210 
211     Poly po;
212     for(int i = 0; i < g.faces.size(); i++) {
213         if(g.area[i] < 0) {   // 对于连通图,惟一一个面积小于零的面是无限面
214             po = g.faces[i];
215             reverse(po.begin(),po.end());   // 对于内部区域来说,无限面多边形的各个顶点是顺时针的
216             po = simplify(po);  // 无限面多边形上可能会有相邻共线点
217             break;
218         }
219     }
220     int m = po.size();
221     printf("%d\n", m);
222     int s = 0;
223     for(int i = 0; i < m; i++) if(po[i] < po[s]) s = i;
224     for(int i = s; i < m; i++) printf("%.4lf %.4lf\n", po[i].x, po[i].y);
225     for(int i = 0; i < s; i++) printf("%.4lf %.4lf\n", po[i].x, po[i].y);
226 }
227 
228 
229 int main(){
230     while(scanf("%d", &n)!=EOF) {
231         int x,y;
232         for(int i = 0; i < n; i++) {
233             scanf("%d %d", &x, &y);
234             p[i] = Point(x,y);
235         }
236         build_graph();
237     }
238     return 0;
239 }
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