LA 4728 Squares (二维凸包+旋转卡壳)
http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&problem=4199
依然是《训练指南》上面的题。这题题意是,给出一堆正方形区域,问区域中最远点对的距离的平方是多少。
做法是经典的凸包加上旋转卡壳。因为正方形区域距离最大的两个点必然在凸包的顶点上,所以将正方形区域的四个顶点加入点集,然后构造二维位凸包。最后模拟旋转卡壳的做法就可以轻松通过这题了。旋转卡壳,按我的理解是模拟一条紧贴凸包的边的直线,然后找到离这条直线的最远点,这个点就是这条边上的对踵点了。1y~
代码如下:
![](https://images.cnblogs.com/OutliningIndicators/ContractedBlock.gif)
1 #include <cstdio> 2 #include <cstring> 3 #include <cmath> 4 #include <set> 5 #include <vector> 6 #include <iostream> 7 #include <algorithm> 8 9 using namespace std; 10 11 struct Point { 12 double x, y; 13 Point() {} 14 Point(double x, double y) : x(x), y(y) {} 15 } ; 16 template<class T> T sqr(T x) { return x * x;} 17 inline double ptDis(Point a, Point b) { return sqrt(sqr(a.x - b.x) + sqr(a.y - b.y));} 18 19 // basic calculations 20 typedef Point Vec; 21 Vec operator + (Vec a, Vec b) { return Vec(a.x + b.x, a.y + b.y);} 22 Vec operator - (Vec a, Vec b) { return Vec(a.x - b.x, a.y - b.y);} 23 Vec operator * (Vec a, double p) { return Vec(a.x * p, a.y * p);} 24 Vec operator / (Vec a, double p) { return Vec(a.x / p, a.y / p);} 25 26 const double EPS = 5e-13; 27 const double PI = acos(-1.0); 28 inline int sgn(double x) { return fabs(x) < EPS ? 0 : (x < 0 ? -1 : 1);} 29 bool operator < (Point a, Point b) { return a.x < b.x || (a.x == b.x && a.y < b.y);} 30 bool operator == (Point a, Point b) { return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0;} 31 32 inline double dotDet(Vec a, Vec b) { return a.x * b.x + a.y * b.y;} 33 inline double crossDet(Vec a, Vec b) { return a.x * b.y - a.y * b.x;} 34 inline double crossDet(Point o, Point a, Point b) { return crossDet(a - o, b - o);} 35 inline double vecLen(Vec x) { return sqrt(sqr(x.x) + sqr(x.y));} 36 inline double toRad(double deg) { return deg / 180.0 * PI;} 37 inline double angle(Vec v) { return atan2(v.y, v.x);} 38 inline double angle(Vec a, Vec b) { return acos(dotDet(a, b) / vecLen(a) / vecLen(b));} 39 inline double triArea(Point a, Point b, Point c) { return fabs(crossDet(b - a, c - a));} 40 inline Vec vecUnit(Vec x) { return x / vecLen(x);} 41 inline Vec rotate(Vec x, double rad) { return Vec(x.x * cos(rad) - x.y * sin(rad), x.x * sin(rad) + x.y * cos(rad));} 42 Vec normal(Vec x) { 43 double len = vecLen(x); 44 return Vec(- x.y / len, x.x / len); 45 } 46 47 struct Line { 48 Point s, t; 49 Line() {} 50 Line(Point s, Point t) : s(s), t(t) {} 51 Point point(double x) { 52 return s + (t - s) * x; 53 } 54 Line move(double x) { 55 Vec nor = normal(t - s); 56 nor = nor / vecLen(nor) * x; 57 return Line(s + nor, t + nor); 58 } 59 Vec vec() { return t - s;} 60 } ; 61 typedef Line Seg; 62 63 inline bool onSeg(Point x, Point a, Point b) { return sgn(crossDet(a - x, b - x)) == 0 && sgn(dotDet(a - x, b - x)) < 0;} 64 inline bool onSeg(Point x, Seg s) { return onSeg(x, s.s, s.t);} 65 66 // 0 : not intersect 67 // 1 : proper intersect 68 // 2 : improper intersect 69 int segIntersect(Point a, Point c, Point b, Point d) { 70 Vec v1 = b - a, v2 = c - b, v3 = d - c, v4 = a - d; 71 int a_bc = sgn(crossDet(v1, v2)); 72 int b_cd = sgn(crossDet(v2, v3)); 73 int c_da = sgn(crossDet(v3, v4)); 74 int d_ab = sgn(crossDet(v4, v1)); 75 if (a_bc * c_da > 0 && b_cd * d_ab > 0) return 1; 76 if (onSeg(b, a, c) && c_da) return 2; 77 if (onSeg(c, b, d) && d_ab) return 2; 78 if (onSeg(d, c, a) && a_bc) return 2; 79 if (onSeg(a, d, b) && b_cd) return 2; 80 return 0; 81 } 82 inline int segIntersect(Seg a, Seg b) { return segIntersect(a.s, a.t, b.s, b.t);} 83 84 // point of the intersection of 2 lines 85 Point lineIntersect(Point P, Vec v, Point Q, Vec w) { 86 Vec u = P - Q; 87 double t = crossDet(w, u) / crossDet(v, w); 88 return P + v * t; 89 } 90 inline Point lineIntersect(Line a, Line b) { return lineIntersect(a.s, a.t - a.s, b.s, b.t - b.s);} 91 92 // Warning: This is a DIRECTED Distance!!! 93 double pt2Line(Point x, Point a, Point b) { 94 Vec v1 = b - a, v2 = x - a; 95 return crossDet(v1, v2) / vecLen(v1); 96 } 97 inline double pt2Line(Point x, Line L) { return pt2Line(x, L.s, L.t);} 98 99 double pt2Seg(Point x, Point a, Point b) { 100 if (a == b) return vecLen(x - a); 101 Vec v1 = b - a, v2 = x - a, v3 = x - b; 102 if (sgn(dotDet(v1, v2)) < 0) return vecLen(v2); 103 if (sgn(dotDet(v1, v3)) > 0) return vecLen(v3); 104 return fabs(crossDet(v1, v2)) / vecLen(v1); 105 } 106 inline double pt2Seg(Point x, Seg s) { return pt2Seg(x, s.s, s.t);} 107 108 struct Poly { 109 vector<Point> pt; 110 Poly() { pt.clear();} 111 Poly(vector<Point> pt) : pt(pt) {} 112 Point operator [] (int x) const { return pt[x];} 113 int size() { return pt.size();} 114 double area() { 115 double ret = 0.0; 116 int sz = pt.size(); 117 for (int i = 1; i < sz; i++) { 118 ret += crossDet(pt[i], pt[i - 1]); 119 } 120 return fabs(ret / 2.0); 121 } 122 } ; 123 124 struct Circle { 125 Point c; 126 double r; 127 Circle() {} 128 Circle(Point c, double r) : c(c), r(r) {} 129 Point point(double a) { 130 return Point(c.x + cos(a) * r, c.y + sin(a) * r); 131 } 132 } ; 133 134 int lineCircleIntersect(Line L, Circle C, double &t1, double &t2, vector<Point> &sol) { 135 double a = L.s.x, b = L.t.x - C.c.x, c = L.s.y, d = L.t.y - C.c.y; 136 double e = sqr(a) + sqr(c), f = 2 * (a * b + c * d), g = sqr(b) + sqr(d) - sqr(C.r); 137 double delta = sqr(f) - 4.0 * e * g; 138 if (sgn(delta) < 0) return 0; 139 if (sgn(delta) == 0) { 140 t1 = t2 = -f / (2.0 * e); 141 sol.push_back(L.point(t1)); 142 return 1; 143 } 144 t1 = (-f - sqrt(delta)) / (2.0 * e); 145 sol.push_back(L.point(t1)); 146 t2 = (-f + sqrt(delta)) / (2.0 * e); 147 sol.push_back(L.point(t2)); 148 return 2; 149 } 150 151 int lineCircleIntersect(Line L, Circle C, vector<Point> &sol) { 152 Vec dir = L.t - L.s, nor = normal(dir); 153 Point mid = lineIntersect(C.c, nor, L.s, dir); 154 double len = sqr(C.r) - sqr(ptDis(C.c, mid)); 155 if (sgn(len) < 0) return 0; 156 if (sgn(len) == 0) { 157 sol.push_back(mid); 158 return 1; 159 } 160 Vec dis = vecUnit(dir); 161 len = sqrt(len); 162 sol.push_back(mid + dis * len); 163 sol.push_back(mid - dis * len); 164 return 2; 165 } 166 167 // -1 : coincide 168 int circleCircleIntersect(Circle C1, Circle C2, vector<Point> &sol) { 169 double d = vecLen(C1.c - C2.c); 170 if (sgn(d) == 0) { 171 if (sgn(C1.r - C2.r) == 0) { 172 return -1; 173 } 174 return 0; 175 } 176 if (sgn(C1.r + C2.r - d) < 0) return 0; 177 if (sgn(fabs(C1.r - C2.r) - d) > 0) return 0; 178 double a = angle(C2.c - C1.c); 179 double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2.0 * C1.r * d)); 180 Point p1 = C1.point(a - da), p2 = C1.point(a + da); 181 sol.push_back(p1); 182 if (p1 == p2) return 1; 183 sol.push_back(p2); 184 return 2; 185 } 186 187 void circleCircleIntersect(Circle C1, Circle C2, vector<double> &sol) { 188 double d = vecLen(C1.c - C2.c); 189 if (sgn(d) == 0) return ; 190 if (sgn(C1.r + C2.r - d) < 0) return ; 191 if (sgn(fabs(C1.r - C2.r) - d) > 0) return ; 192 double a = angle(C2.c - C1.c); 193 double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2.0 * C1.r * d)); 194 sol.push_back(a - da); 195 sol.push_back(a + da); 196 } 197 198 int tangent(Point p, Circle C, vector<Vec> &sol) { 199 Vec u = C.c - p; 200 double dist = vecLen(u); 201 if (dist < C.r) return 0; 202 if (sgn(dist - C.r) == 0) { 203 sol.push_back(rotate(u, PI / 2.0)); 204 return 1; 205 } 206 double ang = asin(C.r / dist); 207 sol.push_back(rotate(u, -ang)); 208 sol.push_back(rotate(u, ang)); 209 return 2; 210 } 211 212 // ptA : points of tangency on circle A 213 // ptB : points of tangency on circle B 214 int tangent(Circle A, Circle B, vector<Point> &ptA, vector<Point> &ptB) { 215 if (A.r < B.r) { 216 swap(A, B); 217 swap(ptA, ptB); 218 } 219 int d2 = sqr(A.c.x - B.c.x) + sqr(A.c.y - B.c.y); 220 int rdiff = A.r - B.r, rsum = A.r + B.r; 221 if (d2 < sqr(rdiff)) return 0; 222 double base = atan2(B.c.y - A.c.y, B.c.x - A.c.x); 223 if (d2 == 0 && A.r == B.r) return -1; 224 if (d2 == sqr(rdiff)) { 225 ptA.push_back(A.point(base)); 226 ptB.push_back(B.point(base)); 227 return 1; 228 } 229 double ang = acos((A.r - B.r) / sqrt(d2)); 230 ptA.push_back(A.point(base + ang)); 231 ptB.push_back(B.point(base + ang)); 232 ptA.push_back(A.point(base - ang)); 233 ptB.push_back(B.point(base - ang)); 234 if (d2 == sqr(rsum)) { 235 ptA.push_back(A.point(base)); 236 ptB.push_back(B.point(PI + base)); 237 } else if (d2 > sqr(rsum)) { 238 ang = acos((A.r + B.r) / sqrt(d2)); 239 ptA.push_back(A.point(base + ang)); 240 ptB.push_back(B.point(PI + base + ang)); 241 ptA.push_back(A.point(base - ang)); 242 ptB.push_back(B.point(PI + base - ang)); 243 } 244 return (int) ptA.size(); 245 } 246 247 void getCoor(double R, double lat, double lng, double &x, double &y, double &z) { 248 lat = toRad(lat); 249 lng = toRad(lng); 250 x = R * cos(lat) * cos(lng); 251 y = R * cos(lat) * sin(lng); 252 z = R * sin(lat); 253 } 254 255 // -1 : onside 256 // 0 : outside 257 // 1 : inside 258 int ptInPoly(Point p, Poly poly) { 259 int wn = 0, sz = poly.size(); 260 for (int i = 0; i < sz; i++) { 261 if (onSeg(p, poly[i], poly[(i + 1) % sz])) return -1; 262 int k = sgn(crossDet(poly[(i + 1) % sz] - poly[i], p - poly[i])); 263 int d1 = sgn(poly[i].y - p.y); 264 int d2 = sgn(poly[(i + 1) % sz].y - p.y); 265 if (k > 0 && d1 <= 0 && d2 > 0) wn++; 266 if (k < 0 && d2 <= 0 && d1 > 0) wn--; 267 } 268 if (wn != 0) return 1; 269 return 0; 270 } 271 272 // if DO NOT need a high precision 273 /* 274 int ptInPoly(Point p, Poly poly) { 275 int sz = poly.size(); 276 double ang = 0.0, tmp; 277 for (int i = 0; i < sz; i++) { 278 if (onSeg(p, poly[i], poly[(i + 1) % sz])) return -1; 279 tmp = angle(poly[i] - p) - angle(poly[(i + 1) % sz] - p) + PI; 280 ang += tmp - floor(tmp / (2.0 * PI)) * 2.0 * PI - PI; 281 } 282 if (sgn(ang - PI) == 0) return -1; 283 if (sgn(ang) == 0) return 0; 284 return 1; 285 } 286 */ 287 288 // convex hull algorithms 289 // return the number of points in convex hull 290 291 // andwer's algorithm 292 // if DO NOT want the points on the side of convex hull, change all "<" into "<=" 293 int andrew(Point *pt, int n, Point *ch) { 294 sort(pt, pt + n); 295 int m = 0; 296 for (int i = 0; i < n; i++) { 297 while (m > 1 && crossDet(ch[m - 1] - ch[m - 2], pt[i] - ch[m - 2]) <= 0) m--; 298 ch[m++] = pt[i]; 299 } 300 int k = m; 301 for (int i = n - 2; i >= 0; i--) { 302 while (m > k && crossDet(ch[m - 1] - ch[m - 2], pt[i] - ch[m - 2]) <= 0) m--; 303 ch[m++] = pt[i]; 304 } 305 if (n > 1) m--; 306 return m; 307 } 308 309 // graham's algorithm 310 // if DO NOT want the points on the side of convex hull, change all "<=" into "<" 311 Point origin; 312 inline bool cmpAng(Point p1, Point p2) { return crossDet(origin, p1, p2) > 0;} 313 inline bool cmpDis(Point p1, Point p2) { return ptDis(p1, origin) > ptDis(p2, origin);} 314 315 void removePt(Point *pt, int &n) { 316 int idx = 1; 317 for (int i = 2; i < n; i++) { 318 if (sgn(crossDet(origin, pt[i], pt[idx]))) pt[++idx] = pt[i]; 319 else if (cmpDis(pt[i], pt[idx])) pt[idx] = pt[i]; 320 } 321 n = idx + 1; 322 } 323 324 int graham(Point *pt, int n, Point *ch) { 325 int top = -1; 326 for (int i = 1; i < n; i++) { 327 if (pt[i].y < pt[0].y || (pt[i].y == pt[0].y && pt[i].x < pt[0].x)) swap(pt[i], pt[0]); 328 } 329 origin = pt[0]; 330 sort(pt + 1, pt + n, cmpAng); 331 removePt(pt, n); 332 for (int i = 0; i < n; i++) { 333 if (i >= 2) { 334 while (!(crossDet(ch[top - 1], pt[i], ch[top]) <= 0)) top--; 335 } 336 ch[++top] = pt[i]; 337 } 338 return top + 1; 339 } 340 341 /****************** template above *******************/ 342 343 #define REP(i, n) for (int i = 0; i < (n); i++) 344 #define REP_1(i, n) for (int i = 1; i <= (n); i++) 345 346 inline int nextID(int x, int n) { return (x + 1) % n;} 347 348 double diameter(Point *pt, int n) { 349 double ret = 0.0; 350 int top = 0; 351 REP(i, n) { 352 while (pt2Line(pt[top], pt[i], pt[i + 1]) <= pt2Line(pt[nextID(top, n)], pt[i], pt[i + 1])) { 353 ret = max(ret, max(ptDis(pt[top], pt[i]), ptDis(pt[top], pt[i + 1]))); 354 top = nextID(top, n); 355 } 356 ret = max(ret, max(ptDis(pt[top], pt[i]), ptDis(pt[top], pt[i + 1]))); 357 } 358 return ret; 359 } 360 361 const int N = 444444; 362 Point pt[N], con[N]; 363 const int dir[4][2] = { {0, 0}, {1, 0}, {0, 1}, {1, 1}}; 364 365 int main() { 366 // freopen("in", "r", stdin); 367 int n, T; 368 scanf("%d", &T); 369 while (T-- && ~scanf("%d", &n)) { 370 double x, y, w; 371 REP(i, n) { 372 scanf("%lf%lf%lf", &x, &y, &w); 373 REP(j, 4) { 374 pt[(i << 2) + j] = Point(x + w * dir[j][0], y + w * dir[j][1]); 375 } 376 } 377 n = andrew(pt, n << 2, con); 378 con[n] = con[0]; 379 printf("%.0f\n", sqr(diameter(con, n))); 380 } 381 return 0; 382 }
稍微精简点的代码,总觉得会有bug,不过最后还是过了:
![](https://images.cnblogs.com/OutliningIndicators/ContractedBlock.gif)
1 #include <cstdio> 2 #include <cstring> 3 #include <cmath> 4 #include <set> 5 #include <vector> 6 #include <iostream> 7 #include <algorithm> 8 9 using namespace std; 10 11 struct Point { 12 double x, y; 13 Point() {} 14 Point(double x, double y) : x(x), y(y) {} 15 } ; 16 template<class T> T sqr(T x) { return x * x;} 17 inline double ptDis(Point a, Point b) { return sqrt(sqr(a.x - b.x) + sqr(a.y - b.y));} 18 19 // basic calculations 20 typedef Point Vec; 21 Vec operator + (Vec a, Vec b) { return Vec(a.x + b.x, a.y + b.y);} 22 Vec operator - (Vec a, Vec b) { return Vec(a.x - b.x, a.y - b.y);} 23 Vec operator * (Vec a, double p) { return Vec(a.x * p, a.y * p);} 24 Vec operator / (Vec a, double p) { return Vec(a.x / p, a.y / p);} 25 26 const double EPS = 5e-13; 27 const double PI = acos(-1.0); 28 inline int sgn(double x) { return fabs(x) < EPS ? 0 : (x < 0 ? -1 : 1);} 29 bool operator < (Point a, Point b) { return a.x < b.x || (a.x == b.x && a.y < b.y);} 30 bool operator == (Point a, Point b) { return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0;} 31 32 inline double dotDet(Vec a, Vec b) { return a.x * b.x + a.y * b.y;} 33 inline double crossDet(Vec a, Vec b) { return a.x * b.y - a.y * b.x;} 34 inline double crossDet(Point o, Point a, Point b) { return crossDet(a - o, b - o);} 35 inline double vecLen(Vec x) { return sqrt(sqr(x.x) + sqr(x.y));} 36 inline double toRad(double deg) { return deg / 180.0 * PI;} 37 inline double angle(Vec v) { return atan2(v.y, v.x);} 38 inline double angle(Vec a, Vec b) { return acos(dotDet(a, b) / vecLen(a) / vecLen(b));} 39 inline double triArea(Point a, Point b, Point c) { return fabs(crossDet(b - a, c - a));} 40 inline Vec vecUnit(Vec x) { return x / vecLen(x);} 41 inline Vec rotate(Vec x, double rad) { return Vec(x.x * cos(rad) - x.y * sin(rad), x.x * sin(rad) + x.y * cos(rad));} 42 Vec normal(Vec x) { 43 double len = vecLen(x); 44 return Vec(- x.y / len, x.x / len); 45 } 46 47 struct Line { 48 Point s, t; 49 Line() {} 50 Line(Point s, Point t) : s(s), t(t) {} 51 Point point(double x) { 52 return s + (t - s) * x; 53 } 54 Line move(double x) { 55 Vec nor = normal(t - s); 56 nor = nor / vecLen(nor) * x; 57 return Line(s + nor, t + nor); 58 } 59 Vec vec() { return t - s;} 60 } ; 61 typedef Line Seg; 62 63 inline bool onSeg(Point x, Point a, Point b) { return sgn(crossDet(a - x, b - x)) == 0 && sgn(dotDet(a - x, b - x)) < 0;} 64 inline bool onSeg(Point x, Seg s) { return onSeg(x, s.s, s.t);} 65 66 // 0 : not intersect 67 // 1 : proper intersect 68 // 2 : improper intersect 69 int segIntersect(Point a, Point c, Point b, Point d) { 70 Vec v1 = b - a, v2 = c - b, v3 = d - c, v4 = a - d; 71 int a_bc = sgn(crossDet(v1, v2)); 72 int b_cd = sgn(crossDet(v2, v3)); 73 int c_da = sgn(crossDet(v3, v4)); 74 int d_ab = sgn(crossDet(v4, v1)); 75 if (a_bc * c_da > 0 && b_cd * d_ab > 0) return 1; 76 if (onSeg(b, a, c) && c_da) return 2; 77 if (onSeg(c, b, d) && d_ab) return 2; 78 if (onSeg(d, c, a) && a_bc) return 2; 79 if (onSeg(a, d, b) && b_cd) return 2; 80 return 0; 81 } 82 inline int segIntersect(Seg a, Seg b) { return segIntersect(a.s, a.t, b.s, b.t);} 83 84 // point of the intersection of 2 lines 85 Point lineIntersect(Point P, Vec v, Point Q, Vec w) { 86 Vec u = P - Q; 87 double t = crossDet(w, u) / crossDet(v, w); 88 return P + v * t; 89 } 90 inline Point lineIntersect(Line a, Line b) { return lineIntersect(a.s, a.t - a.s, b.s, b.t - b.s);} 91 92 // Warning: This is a DIRECTED Distance!!! 93 double pt2Line(Point x, Point a, Point b) { 94 Vec v1 = b - a, v2 = x - a; 95 return crossDet(v1, v2) / vecLen(v1); 96 } 97 inline double pt2Line(Point x, Line L) { return pt2Line(x, L.s, L.t);} 98 99 double pt2Seg(Point x, Point a, Point b) { 100 if (a == b) return vecLen(x - a); 101 Vec v1 = b - a, v2 = x - a, v3 = x - b; 102 if (sgn(dotDet(v1, v2)) < 0) return vecLen(v2); 103 if (sgn(dotDet(v1, v3)) > 0) return vecLen(v3); 104 return fabs(crossDet(v1, v2)) / vecLen(v1); 105 } 106 inline double pt2Seg(Point x, Seg s) { return pt2Seg(x, s.s, s.t);} 107 108 struct Poly { 109 vector<Point> pt; 110 Poly() { pt.clear();} 111 Poly(vector<Point> pt) : pt(pt) {} 112 Point operator [] (int x) const { return pt[x];} 113 int size() { return pt.size();} 114 double area() { 115 double ret = 0.0; 116 int sz = pt.size(); 117 for (int i = 1; i < sz; i++) { 118 ret += crossDet(pt[i], pt[i - 1]); 119 } 120 return fabs(ret / 2.0); 121 } 122 } ; 123 124 struct Circle { 125 Point c; 126 double r; 127 Circle() {} 128 Circle(Point c, double r) : c(c), r(r) {} 129 Point point(double a) { 130 return Point(c.x + cos(a) * r, c.y + sin(a) * r); 131 } 132 } ; 133 134 int lineCircleIntersect(Line L, Circle C, double &t1, double &t2, vector<Point> &sol) { 135 double a = L.s.x, b = L.t.x - C.c.x, c = L.s.y, d = L.t.y - C.c.y; 136 double e = sqr(a) + sqr(c), f = 2 * (a * b + c * d), g = sqr(b) + sqr(d) - sqr(C.r); 137 double delta = sqr(f) - 4.0 * e * g; 138 if (sgn(delta) < 0) return 0; 139 if (sgn(delta) == 0) { 140 t1 = t2 = -f / (2.0 * e); 141 sol.push_back(L.point(t1)); 142 return 1; 143 } 144 t1 = (-f - sqrt(delta)) / (2.0 * e); 145 sol.push_back(L.point(t1)); 146 t2 = (-f + sqrt(delta)) / (2.0 * e); 147 sol.push_back(L.point(t2)); 148 return 2; 149 } 150 151 int lineCircleIntersect(Line L, Circle C, vector<Point> &sol) { 152 Vec dir = L.t - L.s, nor = normal(dir); 153 Point mid = lineIntersect(C.c, nor, L.s, dir); 154 double len = sqr(C.r) - sqr(ptDis(C.c, mid)); 155 if (sgn(len) < 0) return 0; 156 if (sgn(len) == 0) { 157 sol.push_back(mid); 158 return 1; 159 } 160 Vec dis = vecUnit(dir); 161 len = sqrt(len); 162 sol.push_back(mid + dis * len); 163 sol.push_back(mid - dis * len); 164 return 2; 165 } 166 167 // -1 : coincide 168 int circleCircleIntersect(Circle C1, Circle C2, vector<Point> &sol) { 169 double d = vecLen(C1.c - C2.c); 170 if (sgn(d) == 0) { 171 if (sgn(C1.r - C2.r) == 0) { 172 return -1; 173 } 174 return 0; 175 } 176 if (sgn(C1.r + C2.r - d) < 0) return 0; 177 if (sgn(fabs(C1.r - C2.r) - d) > 0) return 0; 178 double a = angle(C2.c - C1.c); 179 double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2.0 * C1.r * d)); 180 Point p1 = C1.point(a - da), p2 = C1.point(a + da); 181 sol.push_back(p1); 182 if (p1 == p2) return 1; 183 sol.push_back(p2); 184 return 2; 185 } 186 187 void circleCircleIntersect(Circle C1, Circle C2, vector<double> &sol) { 188 double d = vecLen(C1.c - C2.c); 189 if (sgn(d) == 0) return ; 190 if (sgn(C1.r + C2.r - d) < 0) return ; 191 if (sgn(fabs(C1.r - C2.r) - d) > 0) return ; 192 double a = angle(C2.c - C1.c); 193 double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2.0 * C1.r * d)); 194 sol.push_back(a - da); 195 sol.push_back(a + da); 196 } 197 198 int tangent(Point p, Circle C, vector<Vec> &sol) { 199 Vec u = C.c - p; 200 double dist = vecLen(u); 201 if (dist < C.r) return 0; 202 if (sgn(dist - C.r) == 0) { 203 sol.push_back(rotate(u, PI / 2.0)); 204 return 1; 205 } 206 double ang = asin(C.r / dist); 207 sol.push_back(rotate(u, -ang)); 208 sol.push_back(rotate(u, ang)); 209 return 2; 210 } 211 212 // ptA : points of tangency on circle A 213 // ptB : points of tangency on circle B 214 int tangent(Circle A, Circle B, vector<Point> &ptA, vector<Point> &ptB) { 215 if (A.r < B.r) { 216 swap(A, B); 217 swap(ptA, ptB); 218 } 219 int d2 = sqr(A.c.x - B.c.x) + sqr(A.c.y - B.c.y); 220 int rdiff = A.r - B.r, rsum = A.r + B.r; 221 if (d2 < sqr(rdiff)) return 0; 222 double base = atan2(B.c.y - A.c.y, B.c.x - A.c.x); 223 if (d2 == 0 && A.r == B.r) return -1; 224 if (d2 == sqr(rdiff)) { 225 ptA.push_back(A.point(base)); 226 ptB.push_back(B.point(base)); 227 return 1; 228 } 229 double ang = acos((A.r - B.r) / sqrt(d2)); 230 ptA.push_back(A.point(base + ang)); 231 ptB.push_back(B.point(base + ang)); 232 ptA.push_back(A.point(base - ang)); 233 ptB.push_back(B.point(base - ang)); 234 if (d2 == sqr(rsum)) { 235 ptA.push_back(A.point(base)); 236 ptB.push_back(B.point(PI + base)); 237 } else if (d2 > sqr(rsum)) { 238 ang = acos((A.r + B.r) / sqrt(d2)); 239 ptA.push_back(A.point(base + ang)); 240 ptB.push_back(B.point(PI + base + ang)); 241 ptA.push_back(A.point(base - ang)); 242 ptB.push_back(B.point(PI + base - ang)); 243 } 244 return (int) ptA.size(); 245 } 246 247 void getCoor(double R, double lat, double lng, double &x, double &y, double &z) { 248 lat = toRad(lat); 249 lng = toRad(lng); 250 x = R * cos(lat) * cos(lng); 251 y = R * cos(lat) * sin(lng); 252 z = R * sin(lat); 253 } 254 255 // -1 : onside 256 // 0 : outside 257 // 1 : inside 258 int ptInPoly(Point p, Poly poly) { 259 int wn = 0, sz = poly.size(); 260 for (int i = 0; i < sz; i++) { 261 if (onSeg(p, poly[i], poly[(i + 1) % sz])) return -1; 262 int k = sgn(crossDet(poly[(i + 1) % sz] - poly[i], p - poly[i])); 263 int d1 = sgn(poly[i].y - p.y); 264 int d2 = sgn(poly[(i + 1) % sz].y - p.y); 265 if (k > 0 && d1 <= 0 && d2 > 0) wn++; 266 if (k < 0 && d2 <= 0 && d1 > 0) wn--; 267 } 268 if (wn != 0) return 1; 269 return 0; 270 } 271 272 // if DO NOT need a high precision 273 /* 274 int ptInPoly(Point p, Poly poly) { 275 int sz = poly.size(); 276 double ang = 0.0, tmp; 277 for (int i = 0; i < sz; i++) { 278 if (onSeg(p, poly[i], poly[(i + 1) % sz])) return -1; 279 tmp = angle(poly[i] - p) - angle(poly[(i + 1) % sz] - p) + PI; 280 ang += tmp - floor(tmp / (2.0 * PI)) * 2.0 * PI - PI; 281 } 282 if (sgn(ang - PI) == 0) return -1; 283 if (sgn(ang) == 0) return 0; 284 return 1; 285 } 286 */ 287 288 // convex hull algorithms 289 // return the number of points in convex hull 290 291 // andwer's algorithm 292 // if DO NOT want the points on the side of convex hull, change all "<" into "<=" 293 int andrew(Point *pt, int n, Point *ch) { 294 sort(pt, pt + n); 295 int m = 0; 296 for (int i = 0; i < n; i++) { 297 while (m > 1 && crossDet(ch[m - 1] - ch[m - 2], pt[i] - ch[m - 2]) <= 0) m--; 298 ch[m++] = pt[i]; 299 } 300 int k = m; 301 for (int i = n - 2; i >= 0; i--) { 302 while (m > k && crossDet(ch[m - 1] - ch[m - 2], pt[i] - ch[m - 2]) <= 0) m--; 303 ch[m++] = pt[i]; 304 } 305 if (n > 1) m--; 306 return m; 307 } 308 309 // graham's algorithm 310 // if DO NOT want the points on the side of convex hull, change all "<=" into "<" 311 Point origin; 312 inline bool cmpAng(Point p1, Point p2) { return crossDet(origin, p1, p2) > 0;} 313 inline bool cmpDis(Point p1, Point p2) { return ptDis(p1, origin) > ptDis(p2, origin);} 314 315 void removePt(Point *pt, int &n) { 316 int idx = 1; 317 for (int i = 2; i < n; i++) { 318 if (sgn(crossDet(origin, pt[i], pt[idx]))) pt[++idx] = pt[i]; 319 else if (cmpDis(pt[i], pt[idx])) pt[idx] = pt[i]; 320 } 321 n = idx + 1; 322 } 323 324 int graham(Point *pt, int n, Point *ch) { 325 int top = -1; 326 for (int i = 1; i < n; i++) { 327 if (pt[i].y < pt[0].y || (pt[i].y == pt[0].y && pt[i].x < pt[0].x)) swap(pt[i], pt[0]); 328 } 329 origin = pt[0]; 330 sort(pt + 1, pt + n, cmpAng); 331 removePt(pt, n); 332 for (int i = 0; i < n; i++) { 333 if (i >= 2) { 334 while (!(crossDet(ch[top - 1], pt[i], ch[top]) <= 0)) top--; 335 } 336 ch[++top] = pt[i]; 337 } 338 return top + 1; 339 } 340 341 /****************** template above *******************/ 342 343 #define REP(i, n) for (int i = 0; i < (n); i++) 344 #define REP_1(i, n) for (int i = 1; i <= (n); i++) 345 346 inline int nextID(int x, int n) { return (x + 1) % n;} 347 348 double diameter(Point *pt, int n) { 349 double ret = 0.0; 350 int top = 0; 351 REP(i, n) { 352 while (pt2Line(pt[top], pt[i], pt[i + 1]) <= pt2Line(pt[nextID(top, n)], pt[i], pt[i + 1])) { 353 // ret = max(ret, max(ptDis(pt[top], pt[i]), ptDis(pt[top], pt[i + 1]))); 354 top = nextID(top, n); 355 } 356 ret = max(ret, max(ptDis(pt[top], pt[i]), ptDis(pt[top], pt[i + 1]))); 357 } 358 return ret; 359 } 360 361 const int N = 444444; 362 Point pt[N], con[N]; 363 const int dir[4][2] = { {0, 0}, {1, 0}, {0, 1}, {1, 1}}; 364 365 int main() { 366 // freopen("in", "r", stdin); 367 int n, T; 368 scanf("%d", &T); 369 while (T-- && ~scanf("%d", &n)) { 370 double x, y, w; 371 REP(i, n) { 372 scanf("%lf%lf%lf", &x, &y, &w); 373 REP(j, 4) { 374 pt[(i << 2) + j] = Point(x + w * dir[j][0], y + w * dir[j][1]); 375 } 376 } 377 n = andrew(pt, n << 2, con); 378 con[n] = con[0]; 379 printf("%.0f\n", sqr(diameter(con, n))); 380 } 381 return 0; 382 }
——written by Lyon