uvalive 3218 Find the Border

  1 #include <cstdio>
  2 #include <cstring>
  3 #include <cmath>
  4 #include <algorithm>
  5 using namespace std;
  6 const double PI = acos(-1);
  7 const double eps = 1e-9;
  8 struct Point
  9 {
 10     double x, y;
 11     Point(double x = 0, double y = 0): x(x), y(y) {}
 12 };
 13 typedef Point Vector;
 14 double dcmp(double x)
 15 {
 16     if (fabs(x) < eps)
 17         return 0;
 18     return x < 0 ? -1 : 1;
 19 }
 20 Vector operator + (const Point& A, const Point& B)
 21 {
 22     return Vector(A.x + B.x, A.y + B.y);
 23 }
 24 Vector operator - (const Point& A, const Point& B)
 25 {
 26     return Vector(A.x - B.x, A.y - B.y);
 27 }
 28 Vector operator * (const Point& A, double a)
 29 {
 30     return Vector(A.x * a, A.y * a);
 31 }
 32 Vector operator / (const Point& A, double a)
 33 {
 34     return Vector(A.x / a, A.y / a);
 35 }
 36 double Cross(const Vector& A, const Vector& B)
 37 {
 38     return A.x * B.y - A.y * B.x;
 39 }
 40 double Dot(const Vector& A, const Vector& B)
 41 {
 42     return A.x * B.x + A.y * B.y;
 43 }
 44 double Length(const Vector& A)
 45 {
 46     return sqrt(Dot(A, A));
 47 }
 48 bool operator < (const Point& A, const Point& B)
 49 {
 50     return A.x < B.x || (A.x == B.x && A.y < B.y);
 51 }
 52 bool operator == (const Point& A, const Point& B)
 53 {
 54     return A.x == B.x && A.y == B.y;
 55 }
 56 Point GetLineIntersection(Point P, Point v, Point Q, Point w)
 57 {
 58     Point u = P - Q;
 59     double t = Cross(w, u) / Cross(v, w);
 60     return P + v * t;
 61 }
 62 bool SegmentProperIntersection(const Point& a1, const Point& a2, const Point& b1, const Point& b2)
 63 {
 64     double c1 = Cross(a2 - a1, b1 - a1);
 65     double c2 = Cross(a2 - a1, b2 - a1);
 66     double c3 = Cross(b2 - b1, a1 - b1);
 67     double c4 = Cross(b2 - b1, a2 - b1);
 68     return dcmp(c1) * dcmp(c2) < 0 && dcmp(c3) * dcmp(c4) < 0;
 69 }
 70 bool OnSegment(const Point& p, const Point& a1, const Point& a2)
 71 {
 72     return dcmp(Cross(a1 - p, a2 - p)) == 0 && dcmp(Dot(a1 - p, a2 - p)) < 0;
 73 }
 74 
 75 const int N = 10005;
 76 const int M = 205;
 77 Point P[M], res[N], temp[M];
 78 double ang[M], s;
 79 int n, cnt;
 80 
 81 void add(Point a, Point b)
 82 {
 83     temp[cnt] = a;
 84     ang[cnt] = atan2(temp[cnt].y - b.y, temp[cnt].x - b.x) - s;
 85     while (dcmp(ang[cnt]) <= 0)
 86         ang[cnt] += 2 * PI;
 87     cnt++;
 88 }
 89 
 90 int main()
 91 {
 92     while (scanf("%d", &n) == 1)
 93     {
 94         int minid = 0;
 95         for (int i = 0; i < n; i++)
 96         {
 97             scanf("%lf%lf", &P[i].x, &P[i].y);
 98             if (P[i] < P[minid])
 99                 minid = i;
100         }
101         res[0] = P[minid];
102         int num = 1;
103         s = -PI;
104         while (1)
105         {
106             cnt = 0;
107             for (int i = 0; i < n; i++)
108             {
109                 if (res[num - 1] == P[i])
110                 {
111                     add(P[(i + 1) % n], res[num - 1]);
112                     add(P[(i + n - 1) % n], res[num - 1]);
113                     break;
114                 }
115             }
116             for (int i = 0; i < n; i++)
117             {
118                 if (OnSegment(res[num - 1], P[i], P[(i + 1) % n]))
119                 {
120                     add(P[(i + 1) % n], res[num - 1]);
121                     add(P[i], res[num - 1]);
122                 }
123             }
124             int id = 0;
125             for (int i = 0; i < cnt; i++)
126                 if (ang[i] < ang[id])
127                     id = i;
128             double minlen = 1e9;
129             Point RP = temp[id], its;
130             for (int i = 0; i < n; i++)
131             {
132                 if (SegmentProperIntersection(temp[id], res[num - 1], P[i], P[(i + 1) % n]))
133                 {
134                     its = GetLineIntersection(temp[id], temp[id] - res[num - 1], P[i], P[i] - P[(i + 1) % n]);
135                     if (Length(its - res[num - 1]) < minlen)
136                     {
137                         minlen = Length(its - res[num - 1]);
138                         RP = its;
139                     }
140                 }
141             }
142             res[num] = RP;
143             s = atan2(res[num - 1].y - res[num].y, res[num - 1].x - res[num].x);
144             num++;
145             if (res[num - 1] == res[0])
146                 break;
147         }
148         printf("%d\n", num - 1);
149         for (int i = 0; i < num - 1; i++)
150             printf("%.4lf %.4lf\n", res[i].x, res[i].y);
151     }
152     return 0;
153 }

  1 #include<cstdio>
  2 #include<vector>
  3 #include<cmath>
  4 #include<algorithm>
  5 #include<cstring>
  6 #include<cassert>
  7 using namespace std;
  8 
  9 const double eps = 1e-8;
 10 double dcmp(double x)
 11 {
 12     if(fabs(x) < eps) return 0;
 13     else return x < 0 ? -1 : 1;
 14 }
 15 
 16 struct Point
 17 {
 18     double x, y;
 19     Point(double x=0, double y=0):x(x),y(y) { }
 20 };
 21 
 22 typedef Point Vector;
 23 
 24 Vector operator + (Vector A, Vector B)
 25 {
 26     return Vector(A.x+B.x, A.y+B.y);
 27 }
 28 
 29 Vector operator - (Point A, Point B)
 30 {
 31     return Vector(A.x-B.x, A.y-B.y);
 32 }
 33 
 34 Vector operator * (Vector A, double p)
 35 {
 36     return Vector(A.x*p, A.y*p);
 37 }
 38 
 39 // 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
 40 // 所以使用这种“小于”运算符的前提是能排除上述情况
 41 bool operator < (const Point& a, const Point& b)
 42 {
 43     return dcmp(a.x - b.x) < 0 || (dcmp(a.x - b.x) == 0 && dcmp(a.y - b.y) < 0);
 44 }
 45 
 46 bool operator == (const Point& a, const Point &b)
 47 {
 48     return dcmp(a.x-b.x) == 0 && dcmp(a.y-b.y) == 0;
 49 }
 50 
 51 double Dot(Vector A, Vector B)
 52 {
 53     return A.x*B.x + A.y*B.y;
 54 }
 55 double Cross(Vector A, Vector B)
 56 {
 57     return A.x*B.y - A.y*B.x;
 58 }
 59 double Length(Vector A)
 60 {
 61     return sqrt(Dot(A, A));
 62 }
 63 
 64 typedef vector<Point> Polygon;
 65 
 66 Point GetLineIntersection(const Point& P, const Vector& v, const Point& Q, const Vector& w)
 67 {
 68     Vector u = P-Q;
 69     double t = Cross(w, u) / Cross(v, w);
 70     return P+v*t;
 71 }
 72 
 73 bool SegmentProperIntersection(const Point& a1, const Point& a2, const Point& b1, const Point& b2)
 74 {
 75     double c1 = Cross(a2-a1,b1-a1), c2 = Cross(a2-a1,b2-a1),
 76            c3 = Cross(b2-b1,a1-b1), c4=Cross(b2-b1,a2-b1);
 77     return dcmp(c1)*dcmp(c2)<0 && dcmp(c3)*dcmp(c4)<0;
 78 }
 79 
 80 bool OnSegment(Point p, Point a1, Point a2)
 81 {
 82     return dcmp(Cross(a1-p, a2-p)) == 0 && dcmp(Dot(a1-p, a2-p)) < 0;
 83 }
 84 
 85 // 多边形的有向面积
 86 double PolygonArea(Polygon poly)
 87 {
 88     double area = 0;
 89     int n = poly.size();
 90     for(int i = 1; i < n-1; i++)
 91         area += Cross(poly[i]-poly[0], poly[(i+1)%n]-poly[0]);
 92     return area/2;
 93 }
 94 
 95 struct Edge
 96 {
 97     int from, to; // 起点，终点，左边的面编号
 98     double ang;
 99 };
100 
101 const int maxn = 10000 + 10; // 最大边数
102 
103 // 平面直线图（PSGL）实现
104 struct PSLG
105 {
106     int n, m, face_cnt;
107     double x[maxn], y[maxn];
108     vector<Edge> edges;
109     vector<int> G[maxn];
110     int vis[maxn*2];  // 每条边是否已经访问过
111     int left[maxn*2]; // 左面的编号
112     int prev[maxn*2]; // 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
113 
114     vector<Polygon> faces;
115     double area[maxn]; // 每个polygon的面积
116 
117     void init(int n)
118     {
119         this->n = n;
120         for(int i = 0; i < n; i++) G[i].clear();
121         edges.clear();
122         faces.clear();
123     }
124 
125     // 有向线段from->to的极角
126     double getAngle(int from, int to)
127     {
128         return atan2(y[to]-y[from], x[to]-x[from]);
129     }
130 
131     void AddEdge(int from, int to)
132     {
133         edges.push_back((Edge)
134         {
135             from, to, getAngle(from, to)
136         });
137         edges.push_back((Edge)
138         {
139             to, from, getAngle(to, from)
140         });
141         m = edges.size();
142         G[from].push_back(m-2);
143         G[to].push_back(m-1);
144     }
145 
146     // 找出faces并计算面积
147     void Build()
148     {
149         for(int u = 0; u < n; u++)
150         {
151             // 给从u出发的各条边按极角排序
152             int d = G[u].size();
153             for(int i = 0; i < d; i++)
154                 for(int j = i+1; j < d; j++) // 这里偷个懒，假设从每个点出发的线段不会太多
155                     if(edges[G[u][i]].ang > edges[G[u][j]].ang) swap(G[u][i], G[u][j]);
156             for(int i = 0; i < d; i++)
157                 prev[G[u][(i+1)%d]] = G[u][i];
158         }
159 
160         memset(vis, 0, sizeof(vis));
161         face_cnt = 0;
162         for(int u = 0; u < n; u++)
163             for(int i = 0; i < G[u].size(); i++)
164             {
165                 int e = G[u][i];
166                 if(!vis[e])   // 逆时针找圈
167                 {
168                     face_cnt++;
169                     Polygon poly;
170                     for(;;)
171                     {
172                         vis[e] = 1;
173                         left[e] = face_cnt;
174                         int from = edges[e].from;
175                         poly.push_back(Point(x[from], y[from]));
176                         e = prev[e^1];
177                         if(e == G[u][i]) break;
178                         assert(vis[e] == 0);
179                     }
180                     faces.push_back(poly);
181                 }
182             }
183 
184         for(int i = 0; i < faces.size(); i++)
185         {
186             area[i] = PolygonArea(faces[i]);
187         }
188     }
189 };
190 
191 PSLG g;
192 
193 const int maxp = 100 + 5;
194 int n, c;
195 Point P[maxp];
196 
197 Point V[maxp*(maxp-1)/2+maxp];
198 
199 // 在V数组里找到点p
200 int ID(Point p)
201 {
202     return lower_bound(V, V+c, p) - V;
203 }
204 
205 // 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
206 Polygon simplify(const Polygon& poly)
207 {
208     Polygon ans;
209     int n = poly.size();
210     for(int i = 0; i < n; i++)
211     {
212         Point a = poly[i];
213         Point b = poly[(i+1)%n];
214         Point c = poly[(i+2)%n];
215         if(dcmp(Cross(a-b, c-b)) != 0) ans.push_back(b);
216     }
217     return ans;
218 }
219 
220 void build_graph()
221 {
222     c = n;
223     for(int i = 0; i < n; i++)
224         V[i] = P[i];
225 
226     vector<double> dist[maxp]; // dist[i][j]是第i条线段上的第j个点离起点（P[i]）的距离
227     for(int i = 0; i < n; i++)
228         for(int j = i+1; j < n; j++)
229             if(SegmentProperIntersection(P[i], P[(i+1)%n], P[j], P[(j+1)%n]))
230             {
231                 Point p = GetLineIntersection(P[i], P[(i+1)%n]-P[i], P[j], P[(j+1)%n]-P[j]);
232                 V[c++] = p;
233                 dist[i].push_back(Length(p - P[i]));
234                 dist[j].push_back(Length(p - P[j]));
235             }
236 
237     // 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
238     // 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
239     // 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。
240     sort(V, V+c);
241     c = unique(V, V+c) - V;
242 
243     g.init(c); // c是平面图的点数
244     for(int i = 0; i < c; i++)
245     {
246         g.x[i] = V[i].x;
247         g.y[i] = V[i].y;
248     }
249     for(int i = 0; i < n; i++)
250     {
251         Vector v = P[(i+1)%n] - P[i];
252         double len = Length(v);
253         dist[i].push_back(0);
254         dist[i].push_back(len);
255         sort(dist[i].begin(), dist[i].end());
256         int sz = dist[i].size();
257         for(int j = 1; j < sz; j++)
258         {
259             Point a = P[i] + v * (dist[i][j-1] / len);
260             Point b = P[i] + v * (dist[i][j] / len);
261             if(a == b) continue;
262             g.AddEdge(ID(a), ID(b));
263         }
264     }
265 
266     g.Build();
267 
268     Polygon poly;
269     for(int i = 0; i < g.faces.size(); i++)
270         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
271         {
272             poly = g.faces[i];
273             reverse(poly.begin(), poly.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
274             poly = simplify(poly); // 无限面多边形上可能会有相邻共线点
275             break;
276         }
277 
278     int m = poly.size();
279     printf("%d\n", m);
280 
281     // 挑选坐标最小的点作为输出的起点
282     int start = 0;
283     for(int i = 0; i < m; i++)
284         if(poly[i] < poly[start]) start = i;
285     for(int i = start; i < m; i++)
286         printf("%.4lf %.4lf\n", poly[i].x, poly[i].y);
287     for(int i = 0; i < start; i++)
288         printf("%.4lf %.4lf\n", poly[i].x, poly[i].y);
289 }
290 
291 int main()
292 {
293     while(scanf("%d", &n) == 1 && n)
294     {
295         for(int i = 0; i < n; i++)
296         {
297             int x, y;
298             scanf("%d%d", &x, &y);
299             P[i] = Point(x, y);
300         }
301         build_graph();
302     }
303     return 0;
304 }