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https://github.com/Kiritow/OJ-Problems-Source.git
synced 2024-03-22 13:11:29 +08:00
1394 lines
43 KiB
C++
1394 lines
43 KiB
C++
//随机化求最小圆覆盖
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//圆心为O, 半径为R
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double R, EPS = 1e-9;
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struct Point { double x, y; } a[N], O;
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inline double dis(Point x, Point y) { return sqrt((x.x - y.x) * (x.x - y.x) + (x.y - y.y) * (x.y - y.y)); }
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Point center(Point x, Point y, Point z) {
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double a1 = y.x - x.x, a2 = z.x - x.x, b1 = y.y - x.y, b2 = z.y - x.y,
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c1 = (a1 * a1 + b1 * b1) / 2, c2 = (a2 * a2 + b2 * b2) / 2,
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d = a1 * b2 - a2 * b1;
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return (Point) {x.x + (c1 * b2 - c2 * b1) / d, x.y + (a1 * c2 - a2 * c1) / d};
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}
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void cal(int n, Point b[]) {
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O = a[0]; R = 0;
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for (int i = 0; i < n; i++) { a[i] = b[i]; }
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for (int i = 0; i < n; i++) { swap(a[rand() % n], a[i]); }
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for (int i = 1; i < n; i++) {
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if (dis(a[i], O) > R + EPS) {
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O = a[i]; R = 0;
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for (int j = 0; j < i; j++) {
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if (dis(a[j], O) > R + EPS) {
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O = (Point) {(a[i].x + a[j].x) / 2, (a[i].y + a[j].y) / 2}, R = dis(O, a[i]);
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for (int k = 0; k < j; k++) {
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if (dis(a[k], O) > R + EPS) { O = center(a[k], a[j], a[i]), R = dis(O, a[i]); }
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}
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}
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}
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}
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}
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}
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//kuangbin
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//1、基本函数
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//1.1 Point 定义
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const double PI = acos(-1.0);
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const double EPS = 1e-8;
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inline int sgn(double x) {
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return (fabs(x) < EPS ? 0 : (x < 0 ? -1 : 1));
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}
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struct Point {
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double x, y;
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Point() {}
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Point(double _x, double _y): x(_x), y(_y) {}
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Point operator-(const Point &b)const {
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return Point(x - b.x, y - b.y);
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}
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//叉积
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double operator^(const Point &b)const {
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return x * b.y - y * b.x;
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}
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//点积
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double operator*(const Point &b)const {
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return x * b.x + y * b.y;
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}
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//绕原点旋转角度B(弧度值), 后x, y的变化
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void transXY(double B) {
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double tx = x, ty = y;
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x = tx * cos(B) - ty * sin(B);
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y = tx * sin(B) + ty * cos(B);
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}
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};
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//1.2 Line 定义
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struct Line {
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Point s, e;
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Line() {}
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Line(Point _s, Point _e): s(_s), e(_e) {}
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//两直线相交求交点
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//第一个值为0表示直线重合, 为1表示平行, 为0表示相交, 为2是相交
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//只有第一个值为2时, 交点才有意义
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pair<int, Point> operator&(const Line &b)const {
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Point res = s;
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if (sgn((s - e) ^ (b.s - b.e)) == 0) {
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if (sgn((s - b.e) ^ (b.s - b.e)) == 0) { return make_pair(0, res); } //重合
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else { return make_pair(1, res); } //平行
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}
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double t = ((s - b.s) ^ (b.s - b.e)) / ((s - e) ^ (b.s - b.e));
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res.x += (e.x - s.x) * t;
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res.y += (e.y - s.y) * t;
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return make_pair(2, res);
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}
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};
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//1.3 两点间距离
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//*两点间距离
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double dist(const Point &a, const Point &b) {
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return sqrt((a - b) * (a - b));
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}
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//1.4 判断: 线段相交
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//*判断线段相交
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bool inter(const Line &l1, const Line &l2) {
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return max(l1.s.x, l1.e.x) >= min(l2.s.x, l2.e.x)
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&& max(l2.s.x, l2.e.x) >= min(l1.s.x, l1.e.x)
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&& max(l1.s.y, l1.e.y) >= min(l2.s.y, l2.e.y)
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&& max(l2.s.y, l2.e.y) >= min(l1.s.y, l1.e.y)
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&& sgn((l2.s - l1.e) ^ (l1.s - l1.e)) * sgn((l2.e-l1.e) ^ (l1.s - l1.e)) <= 0
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&& sgn((l1.s - l2.e) ^ (l2.s - l2.e)) * sgn((l1.e-l2.e) ^ (l2.s - l2.e)) <= 0;
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}
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//1.5 判断：直线和线段相交
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//判断直线和线段相交
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bool Seg_inter_line(const Line &l1, const Line &l2) { //判断直线l1和线段l2是否相交
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return sgn((l2.s - l1.e) ^ (l1.s - l1.e)) * sgn((l2.e-l1.e) ^ (l1.s - l1.e)) <= 0;
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}
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//1.6 点到直线距离
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//点到直线距离
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//返回为result, 是点到直线最近的点
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Point PointToLine(const Point &P, const Line &L) {
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double t = ((P - L.s) * (L.e-L.s)) / ((L.e - L.s) * (L.e - L.s));
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return Point(L.s.x + (L.e.x - L.s.x) * t, L.s.y + (L.e.y - L.s.y) * t);
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}
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//1.7 点到线段距离
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//点到线段的距离
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//返回点到线段最近的点
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Point NearestPointToLineSeg(const Point &P, const Line &L) {
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double t = ((P - L.s) * (L.e-L.s)) / ((L.e-L.s) * (L.e-L.s));
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Point result;
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if (t >= 0 && t <= 1) {
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result.x = L.s.x + (L.e.x - L.s.x) * t;
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result.y = L.s.y + (L.e.y - L.s.y) * t;
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} else {
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if (dist(P, L.s) < dist(P, L.e)) { result = L.s; }
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else { result = L.e; }
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}
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return result;
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}
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//1.8 计算多边形面积
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//计算多边形面积
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//点的编号从0 ~ n - 1
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double CalcArea(Point p[], int n) {
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double res = 0;
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for (int i = 0; i < n; i++) {
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res += (p[i] ^ p[(i + 1) % n]) * 0.5;
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}
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return fabs(res);
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}
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//1.9 判断点在线段上
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//*判断点在线段上
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bool OnSeg(const Point &P, const Line &L) {
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return sgn((L.s - P) ^ (L.e-P)) == 0
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&& sgn((P.x - L.s.x) * (P.x - L.e.x)) <= 0
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&& sgn((P.y - L.s.y) * (P.y - L.e.y)) <= 0;
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}
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//1.10 判断点在凸多边形内
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//*判断点在凸多边形内
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//点形成一个凸包, 而且按逆时针排序(如果是顺时针把里面的<0改为>0)
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//点的编号:0 ~ n - 1
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//返回值:
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//-1:点在凸多边形外
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//0:点在凸多边形边界上
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//1:点在凸多边形内
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int inConvexPoly(const Point &a, Point p[], int n) {
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for (int i = 0; i < n; i++) {
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if (sgn((p[i] - a) ^ (p[(i + 1) % n] - a)) < 0) { return -1; }
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else if (OnSeg(a, Line(p[i], p[(i + 1) % n]))) { return 0; }
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}
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return 1;
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}
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//1.11 判断点在任意多边形内
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//*判断点在任意多边形内
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//射线法, poly[]的顶点数要大于等于3,点的编号0~n-1
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//返回值
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//-1:点在凸多边形外
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//0:点在凸多边形边界上
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//1:点在凸多边形内
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int inPoly(const Point &p, Point poly[], int n) {
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int cnt = 0;
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Line ray(p, Point(-100000000000.0, p.y)), side; //-INF,注意取值防止越界
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for (int i = 0; i < n; i++) {
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side.s = poly[i];
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side.e = poly[(i + 1) % n];
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if (OnSeg(p, side)) { return 0; }
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//如果平行轴则不考虑
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if (sgn(side.s.y - side.e.y) == 0) { continue; }
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if (OnSeg(side.s, ray)) {
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if (sgn(side.s.y - side.e.y) > 0) { cnt++; }
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} else if (OnSeg(side.e, ray)) {
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if (sgn(side.e.y - side.s.y) > 0) { cnt++; }
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} else if (inter(ray, side)) {
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cnt++;
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}
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}
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if (cnt & 1) { return 1; }
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else { return -1; }
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}
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//1.12 判断凸多边形
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//判断凸多边形
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//允许共线边
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//点可以是顺时针给出也可以是逆时针给出
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//点的编号1~n-1
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bool isconvex(Point poly[], int n) {
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bool s[3] = {0};
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for (int i = 0; i < n; i++) {
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s[sgn((poly[(i + 1) % n] - poly[i]) ^ (poly[(i + 2) % n] - poly[i])) + 1] = true;
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if (s[0] && s[2]) { return false; }
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}
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return true;
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}
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//2、凸包
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/*
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* 求凸包, Graham算法
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* 点的编号0 ~ n - 1
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* 返回凸包结果Stack[0 ~ top - 1]为凸包的编号
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*/
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const int N = 1010;
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Point lst[N];
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int Stack[N], top;
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//相对于lst[0]的极角排序
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bool _cmp(const Point &p1, const Point &p2) {
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double tmp = (p1 - lst[0]) ^ (p2 - lst[0]);
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if (sgn(tmp) > 0) { return true; }
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else if (sgn(tmp) == 0 && sgn(dist(p1, lst[0]) - dist(p2, lst[0])) <= 0) { return true; }
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else { return false; }
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}
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void Graham(int n) {
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Point p0 = lst[0];
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int k = 0;
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//找最下边的一个点
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for (int i = 1; i < n; i++) {
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if ((p0.y > lst[i].y) || (p0.y == lst[i].y && p0.x > lst[i].x)) {
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p0 = lst[i]; k = i;
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}
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}
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swap(lst[k], lst[0]);
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sort(lst + 1, lst + n, _cmp);
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if (n == 1) {
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top = 1;
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Stack[0] = 0;
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return;
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}
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if (n == 2) {
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top = 2;
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Stack[0] = 0;
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Stack[1] = 1;
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return ;
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}
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top = 2;
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Stack[0] = 0;
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Stack[1] = 1;
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for (int i = 2; i < n; i++) {
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while (top > 1 && sgn((lst[Stack[top - 1]] - lst[Stack[top - 2]]) ^ (lst[i] - lst[Stack[top - 2]])) <= 0) {
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top--;
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}
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Stack[top++] = i;
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}
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}
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//3、平面最近点对(HDU 1007)
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const double EPS = 1e-6;
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const double INF = 1e20;
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const int N = 100005;
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struct Point {
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double x, y;
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} p[N], tmpt[N];
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double dist(const Point &a, const Point &b) {
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return sqrt((a.x - b.x) * (a.x - b.x) + (a.y - b.y) * (a.y - b.y));
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}
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bool cmpxy(const Point &a, const Point &b) {
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return a.x < b.x || (a.x == b.x && a.y < b.y);
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}
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bool cmpy(Point a, Point b) {
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return a.y < b.y;
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}
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double Closest_Pair(int left, int right) {
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double d = INF;
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if (left == right) { return d; }
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if (left + 1 == right) { return dist(p[left], p[right]); }
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int mid = (left + right) / 2;
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double d1 = Closest_Pair(left, mid);
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double d2 = Closest_Pair(mid + 1, right);
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d = min(d1, d2);
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int k = 0;
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for (int i = left; i <= right; i++) {
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if (fabs(p[mid].x - p[i].x) <= d) {
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tmpt[k++] = p[i];
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}
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}
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sort(tmpt, tmpt + k, cmpy);
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for (int i = 0; i < k; i++) {
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for (int j = i + 1; j < k && tmpt[j].y - tmpt[i].y < d; j++) {
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d = min(d, dist(tmpt[i], tmpt[j]));
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}
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}
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return d;
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}
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int main() {
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int n;
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while (scanf("%d", &n) == 1 && n) {
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for (int i = 0; i < n; i++) {
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scanf("%lf%lf", &p[i].x, &p[i].y);
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}
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sort(p, p + n, cmpxy);
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printf("%.2lf\n", Closest_Pair(0, n - 1) / 2);
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}
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}
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//4、旋转卡壳
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//4.1 求解平面最远点对(POJ 2187 Beauty Contest)
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struct Point {
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int x, y;
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Point(int _x = 0, int _y = 0): x(_x), y(_y) {}
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Point operator-(const Point &b)const {
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return Point(x - b.x, y - b.y);
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}
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int operator^(const Point &b)const {
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return x * b.y - y * b.x;
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}
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int operator*(const Point &b)const {
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return x * b.x + y * b.y;
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}
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void input() {
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scanf("%d%d", &x, &y);
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}
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};
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//距离的平方
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int dist2(const Point &a, const Point &b) {
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return (a - b) * (a - b);
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}
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//******二维凸包, int***********
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const int N = 50005;
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Point lst[N], p[N];
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int Stack[N], top;
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bool _cmp(const Point &p1, const Point &p2) {
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int tmp = (p1 - lst[0]) ^ (p2 - lst[0]);
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if (tmp > 0) { return true; }
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else if (tmp == 0 && dist2(p1, lst[0]) <= dist2(p2, lst[0])) { return true; }
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else { return false; }
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}
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void Graham(int n) {
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Point p0 = lst[0];
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int k = 0;
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for (int i = 1; i < n; i++) {
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if (p0.y > lst[i].y || (p0.y == lst[i].y && p0.x > lst[i].x)) {
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p0 = lst[i];
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k = i;
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}
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}
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swap(lst[k], lst[0]);
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sort(lst + 1, lst + n, _cmp);
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if (n == 1) {
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top = 1;
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Stack[0] = 0;
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return;
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}
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if (n == 2) {
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top = 2;
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Stack[0] = 0; Stack[1] = 1;
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return;
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}
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top = 2;
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Stack[0] = 0;
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Stack[1] = 1;
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for (int i = 2; i < n; i++) {
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while (top > 1 && ((lst[Stack[top - 1]] - lst[Stack[top - 2]]) ^ (lst[i] - lst[Stack[top - 2]])) <= 0) {
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top--;
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}
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Stack[top++] = i;
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}
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}
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//旋转卡壳, 求两点间距离平方的最大值
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int rotating_calipers(Point p[], int n) {
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int ans = 0, cur = 1;
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Point v;
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for (int i = 0; i < n; i++) {
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v = p[i] - p[(i + 1) % n];
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while ((v ^ (p[(cur + 1) % n] - p[cur])) < 0) {
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cur = (cur + 1) % n;
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}
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ans = max(ans, max(dist2(p[i], p[cur]), dist2(p[(i + 1) % n], p[(cur + 1) % n])));
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}
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return ans;
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}
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int main() {
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int n;
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while (~scanf("%d", &n)) {
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for (int i = 0; i < n; i++) { lst[i].input(); }
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Graham(n);
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for (int i = 0; i < top; i++) { p[i] = lst[Stack[i]]; }
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printf("%d\n", rotating_calipers(p, top));
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}
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}
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//4.2 求解平面点集最大三角形
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//旋转卡壳计算平面点集最大三角形面积
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Point lst[N], p[N];
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int rotating_calipers(Point p[], int n) {
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int ans = 0;
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Point v;
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for (int i = 0; i < n; i++) {
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int j = (i + 1) % n, k = (j + 1) % n;
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while (j != i && k != i) {
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ans = max(ans, abs((p[j] - p[i]) ^ (p[k] - p[i])));
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while (((p[i] - p[j]) ^ (p[(k + 1) % n] - p[k])) < 0) {
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k = (k + 1) % n;
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}
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j = (j + 1) % n;
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}
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}
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return ans;
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}
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int main() {
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int n;
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while (scanf("%d", &n) == 1) {
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if (n == -1) { break; }
|
||
for (int i = 0; i < n; i++) { lst[i].input(); }
|
||
Graham(n);
|
||
for (int i = 0; i < top; i++) { p[i] = lst[Stack[i]]; }
|
||
printf("%.2f\n", rotating_calipers(p, top) / 2.0);
|
||
}
|
||
}
|
||
//4.3 求解两凸包最小距离(POJ 3608)
|
||
const double EPS = 1e-8;
|
||
inline int sgn(double x) {
|
||
return (fabs(x) < EPS ? 0 : (x < 0 ? -1 : 1));
|
||
}
|
||
struct Point {
|
||
double x, y;
|
||
Point() {}
|
||
Point(double _x = 0, double _y = 0): x(_x), y(_y) {}
|
||
Point operator-(const Point &b)const {
|
||
return Point(x - b.x, y - b.y);
|
||
}
|
||
//叉积
|
||
double operator^(const Point &b)const {
|
||
return x * b.y - y * b.x;
|
||
}
|
||
//点积
|
||
double operator*(const Point &b)const {
|
||
return x * b.x + y * b.y;
|
||
}
|
||
void input() {
|
||
scanf("%lf%lf", &x, &y);
|
||
}
|
||
};
|
||
struct Line {
|
||
Point s, e;
|
||
Line() {}
|
||
Line(Point _s, Point _e): s(_s), e(_e) {}
|
||
};
|
||
//两点间距离
|
||
double dist(const Point &a, const Point &b) {
|
||
return sqrt((a - b) * (a - b));
|
||
}
|
||
//点到线段的距离, 返回点到线段最近的点
|
||
Point NearestPointToLineSeg(const Point &P, const Line &L) {
|
||
double t = ((P - L.s) * (L.e-L.s)) / ((L.e-L.s) * (L.e-L.s));
|
||
Point result;
|
||
if (t >= 0 && t <= 1) {
|
||
result.x = L.s.x + (L.e.x - L.s.x) * t;
|
||
result.y = L.s.y + (L.e.y - L.s.y) * t;
|
||
} else {
|
||
if (dist(P, L.s) < dist(P, L.e)) { result = L.s; }
|
||
else { result = L.e; }
|
||
}
|
||
return result;
|
||
}
|
||
/*
|
||
* 求凸包, Graham算法
|
||
* 点的编号0~n-1
|
||
* 返回凸包结果Stack[0~top-1]为凸包的编号
|
||
*/
|
||
const int N = 10005;
|
||
Point lst[N];
|
||
int Stack[N], top;
|
||
//相对于list[0]的极角排序
|
||
bool _cmp(const Point &p1, const Point &p2) {
|
||
double tmp = (p1 - lst[0]) ^ (p2 - lst[0]);
|
||
if (sgn(tmp) > 0) { return true; }
|
||
else if (sgn(tmp) == 0 && sgn(dist(p1, lst[0]) - dist(p2, lst[0])) <= 0) { return true; }
|
||
else { return false; }
|
||
}
|
||
void Graham(int n) {
|
||
Point p0 = lst[0];
|
||
int k = 0;
|
||
//找最下边的一个点
|
||
for (int i = 1; i < n; i++) {
|
||
if ((p0.y > lst[i].y) || (p0.y == lst[i].y && p0.x > lst[i].x)) {
|
||
p0 = lst[i];
|
||
k = i;
|
||
}
|
||
}
|
||
swap(lst[k], lst[0]);
|
||
sort(lst + 1, lst + n, _cmp);
|
||
if (n == 1) {
|
||
top = 1;
|
||
Stack[0] = 0;
|
||
return;
|
||
}
|
||
if (n == 2) {
|
||
top = 2;
|
||
Stack[0] = 0;
|
||
Stack[1] = 1;
|
||
return ;
|
||
}
|
||
top = 2;
|
||
Stack[0] = 0;
|
||
Stack[1] = 1;
|
||
for (int i = 2; i < n; i++) {
|
||
while (top > 1 && sgn((lst[Stack[top - 1]] - lst[Stack[top - 2]]) ^ (lst[i] - lst[Stack[top - 2]])) <= 0) {
|
||
top--;
|
||
}
|
||
Stack[top++] = i;
|
||
}
|
||
}
|
||
//点p0到线段p1p2的距离
|
||
double pointtoseg(const Point &p0, const Point &p1, const Point &p2) {
|
||
return dist(p0, NearestPointToLineSeg(p0, Line(p1, p2)));
|
||
}
|
||
//平行线段p0p1和p2p3的距离
|
||
double dispallseg(const Point &p0, const Point &p1, const Point &p2, const Point &p3) {
|
||
double ans1 = min(pointtoseg(p0, p2, p3), pointtoseg(p1, p2, p3));
|
||
double ans2 = min(pointtoseg(p2, p0, p1), pointtoseg(p3, p0, p1));
|
||
return min(ans1, ans2);
|
||
}
|
||
//得到向量a1a2和b1b2的位置关系
|
||
double Get_angle(const Point &a1, const Point &a2, const Point &b1, const Point &b2) {
|
||
return (a2 - a1) ^ (b1 - b2);
|
||
}
|
||
double rotating_calipers(Point p[], int np, Point q[], int nq) {
|
||
int sp = 0, sq = 0;
|
||
for (int i = 0; i < np; i++) {
|
||
if (sgn(p[i].y - p[sp].y) < 0) { sp = i; }
|
||
}
|
||
for (int i = 0; i < nq; i++) {
|
||
if (sgn(q[i].y - q[sq].y) > 0) { sq = i; }
|
||
}
|
||
double ans = dist(p[sp], q[sq]), tmp;
|
||
for (int i = 0; i < np; i++) {
|
||
while (sgn(tmp = Get_angle(p[sp], p[(sp + 1) % np], q[sq], q[(sq + 1) % nq])) < 0) {
|
||
sq = (sq + 1) % nq;
|
||
}
|
||
if (sgn(tmp) == 0) {
|
||
ans = min(ans, dispallseg(p[sp], p[(sp + 1) % np], q[sq], q[(sq + 1) % nq]));
|
||
} else { ans = min(ans, pointtoseg(q[sq], p[sp], p[(sp + 1) % np])); }
|
||
sp = (sp + 1) % np;
|
||
}
|
||
return ans;
|
||
}
|
||
double solve(Point p[], int n, Point q[], int m) {
|
||
return min(rotating_calipers(p, n, q, m), rotating_calipers(q, m, p, n));
|
||
}
|
||
Point p[N], q[N];
|
||
int main() {
|
||
int n, m;
|
||
while (scanf("%d%d", &n, &m) == 2) {
|
||
if (n == 0 && m == 0) { break; }
|
||
for (int i = 0; i < n; i++) { lst[i].input(); }
|
||
Graham(n);
|
||
for (int i = 0; i < top; i++) { p[i] = lst[i]; }
|
||
n = top;
|
||
for (int i = 0; i < m; i++) { lst[i].input(); }
|
||
Graham(m);
|
||
for (int i = 0; i < top; i++) { q[i] = lst[i]; }
|
||
m = top;
|
||
printf("%.4f\n", solve(p, n, q, m));
|
||
}
|
||
}
|
||
//5、半平面交
|
||
//5.1 半平面交模板(from UESTC)
|
||
const double PI = acos(-1.0);
|
||
const double EPS = 1e-8;
|
||
const int N = 105;
|
||
inline int sgn(double x) {
|
||
return (fabs(x) < EPS ? 0 : (x < 0 ? -1 : 1));
|
||
}
|
||
struct Point {
|
||
double x, y;
|
||
Point() {}
|
||
Point(double _x, double _y): x(_x), y(_y) {}
|
||
Point operator-(const Point &b)const {
|
||
return Point(x - b.x, y - b.y);
|
||
}
|
||
double operator^(const Point &b)const {
|
||
return x * b.y - y * b.x;
|
||
}
|
||
double operator*(const Point &b)const {
|
||
return x * b.x + y * b.y;
|
||
}
|
||
};
|
||
struct Line {
|
||
Point s, e;
|
||
double k;
|
||
Line() {}
|
||
Line(Point _s, Point _e): s(_s), e(_e), k(atan2(e.y - s.y, e.x - s.x)) {}
|
||
Point operator&(const Line &b)const {
|
||
Point res = s;
|
||
double t = ((s - b.s) ^ (b.s - b.e)) / ((s - e) ^ (b.s - b.e));
|
||
res.x += (e.x - s.x) * t;
|
||
res.y += (e.y - s.y) * t;
|
||
return res;
|
||
}
|
||
};
|
||
//半平面交, 直线的左边代表有效区域
|
||
bool HPIcmp(const Line &a, const Line &b) {
|
||
if (fabs(a.k - b.k) > EPS) { return a.k < b.k; }
|
||
return ((a.s - b.s) ^ (b.e - b.s)) < 0;
|
||
}
|
||
Line Q[N];
|
||
void HPI(Line line[], int n, Point res[], int &resn) {
|
||
int tot = 1;
|
||
sort(line, line + n, HPIcmp);
|
||
for (int i = 1; i < n; i++) {
|
||
if (fabs(line[i].k - line[i - 1].k) > EPS) { line[tot++] = line[i]; }
|
||
}
|
||
int head = 0, tail = 1;
|
||
Q[0] = line[0]; Q[1] = line[1];
|
||
resn = 0;
|
||
for (int i = 2; i < tot; i++) {
|
||
if (fabs((Q[tail].e-Q[tail].s) ^ (Q[tail - 1].e-Q[tail - 1].s)) < EPS
|
||
|| fabs((Q[head].e-Q[head].s) ^ (Q[head + 1].e-Q[head + 1].s)) < EPS) {
|
||
return;
|
||
}
|
||
while (head < tail
|
||
&& (((Q[tail]&Q[tail - 1]) - line[i].s) ^ (line[i].e-line[i].s)) > EPS) {
|
||
tail--;
|
||
}
|
||
while (head < tail
|
||
&& (((Q[head]&Q[head + 1]) - line[i].s) ^ (line[i].e-line[i].s)) > EPS) {
|
||
head++;
|
||
}
|
||
Q[++tail] = line[i];
|
||
}
|
||
while (head < tail
|
||
&& (((Q[tail]&Q[tail - 1]) - Q[head].s) ^ (Q[head].e-Q[head].s)) > EPS) {
|
||
tail--;
|
||
}
|
||
while (head < tail
|
||
&& (((Q[head]&Q[head - 1]) - Q[tail].s) ^ (Q[tail].e-Q[tail].e)) > EPS) {
|
||
head++;
|
||
}
|
||
if (tail <= head + 1) { return; }
|
||
for (int i = head; i < tail; i++) {
|
||
res[resn++] = Q[i] & Q[i + 1];
|
||
}
|
||
if (head < tail - 1) {
|
||
res[resn++] = Q[head] & Q[tail];
|
||
}
|
||
}
|
||
//5.2 普通半平面交写法
|
||
//POJ 1750
|
||
const double EPS = 1e-18;
|
||
const double INF = 100000000000.0;
|
||
const int N = 105;
|
||
inline int sgn(double x) {
|
||
return (fabs(x) < EPS ? 0 : (x < 0 ? -1 : 1));
|
||
}
|
||
struct Point {
|
||
double x, y;
|
||
Point() {}
|
||
Point(double _x, double _y): x(_x), y(_y) {}
|
||
Point operator-(const Point &b)const {
|
||
return Point(x - b.x, y - b.y);
|
||
}
|
||
double operator^(const Point &b)const {
|
||
return x * b.y - y * b.x;
|
||
}
|
||
double operator*(const Point &b)const {
|
||
return x * b.x + y * b.y;
|
||
}
|
||
};
|
||
//计算多边形面积
|
||
double CalcArea(Point p[], int n) {
|
||
double res = 0;
|
||
for (int i = 0; i < n; i++) {
|
||
res += (p[i] ^ p[(i + 1) % n]);
|
||
}
|
||
return fabs(res / 2);
|
||
}
|
||
//通过两点, 确定直线方程
|
||
void Get_equation(const Point &p1, const Point &p2, double &a, double &b, double &c) {
|
||
a = p2.y - p1.y;
|
||
b = p1.x - p2.x;
|
||
c = p2.x * p1.y - p1.x * p2.y;
|
||
}
|
||
//求交点
|
||
Point Intersection(const Point &p1, const Point &p2, double a, double b, double c) {
|
||
double u = fabs(a * p1.x + b * p1.y + c);
|
||
double v = fabs(a * p2.x + b * p2.y + c);
|
||
return Point((p1.x * v + p2.x * u) / (u + v), (p1.y * v + p2.y * u) / (u + v));
|
||
}
|
||
Point tp[N];
|
||
void Cut(double a, double b, double c, Point p[], int &cnt) {
|
||
int tmp = 0;
|
||
for (int i = 1; i <= cnt; i++) {
|
||
//当前点在左侧, 逆时针的点
|
||
if (a * p[i].x + b * p[i].y + c < EPS) { tp[++tmp] = p[i]; }
|
||
else {
|
||
if (a * p[i - 1].x + b * p[i - 1].y + c < -EPS) {
|
||
tp[++tmp] = Intersection(p[i - 1], p[i], a, b, c);
|
||
}
|
||
if (a * p[i + 1].x + b * p[i + 1].y + c < -EPS) {
|
||
tp[++tmp] = Intersection(p[i], p[i + 1], a, b, c);
|
||
}
|
||
}
|
||
}
|
||
for (int i = 1; i <= tmp; i++) { p[i] = tp[i]; }
|
||
p[0] = p[tmp];
|
||
p[tmp + 1] = p[1];
|
||
cnt = tmp;
|
||
}
|
||
double V[N], U[N], W[N];
|
||
int n;
|
||
Point p[N];
|
||
bool solve(int id) {
|
||
p[1] = Point(0, 0);
|
||
p[2] = Point(INF, 0);
|
||
p[3] = Point(INF, INF);
|
||
p[4] = Point(0, INF);
|
||
p[0] = p[4];
|
||
p[5] = p[1];
|
||
int cnt = 4;
|
||
for (int i = 0; i < n; i++) {
|
||
if (i != id) {
|
||
double a = (V[i] - V[id]) / (V[i] * V[id]);
|
||
double b = (U[i] - U[id]) / (U[i] * U[id]);
|
||
double c = (W[i] - W[id]) / (W[i] * W[id]);
|
||
if (sgn(a) == 0 && sgn(b) == 0) {
|
||
if (sgn(c) >= 0) { return false; }
|
||
else { continue; }
|
||
}
|
||
Cut(a, b, c, p, cnt);
|
||
}
|
||
}
|
||
if (sgn(CalcArea(p, cnt)) == 0) { return false; }
|
||
else { return true; }
|
||
}
|
||
int main() {
|
||
while (scanf("%d", &n) == 1) {
|
||
for (int i = 0; i < n; i++) {
|
||
scanf("%lf%lf%lf", &V[i], &U[i], &W[i]);
|
||
}
|
||
for (int i = 0; i < n; i++) {
|
||
if (solve(i)) { printf("Yes\n"); }
|
||
else { printf("No\n"); }
|
||
}
|
||
}
|
||
}
|
||
//6、三点求圆心坐标(三角形外心)
|
||
//过三点求圆心坐标
|
||
Point waixin(const Point &a, const Point &b, const Point &c) {
|
||
double a1 = b.x - a.x, b1 = b.y - a.y, c1 = (a1 * a1 + b1 * b1) / 2;
|
||
double a2 = c.x - a.x, b2 = c.y - a.y, c2 = (a2 * a2 + b2 * b2) / 2;
|
||
double d = a1 * b2 - a2 * b1;
|
||
return Point(a.x + (c1 * b2 - c2 * b1) / d, a.y + (a1 * c2 - a2 * c1) / d);
|
||
}
|
||
//7、求两圆相交的面积
|
||
//两个圆的公共部分面积
|
||
double Area_of_overlap(const Point &c1, double r1, const Point &c2, double r2) {
|
||
double d = dist(c1, c2);
|
||
if (r1 + r2 < d + EPS) { return 0; }
|
||
if (d < fabs(r1 - r2) + EPS) {
|
||
double r = min(r1, r2);
|
||
return PI * r * r;
|
||
}
|
||
double x = (d * d + r1 * r1 - r2 * r2) / (2 * d);
|
||
double t1 = acos(x / r1);
|
||
double t2 = acos((d - x) / r2);
|
||
return r1 * r1 * t1 + r2 * r2 * t2 - d * r1 * sin(t1);
|
||
}
|
||
//8、Pick 公式
|
||
//顶点坐标均是整点的简单多边形: 面积 = 内部格点数目 + 边上格点数目 / 2 - 1
|
||
|
||
|
||
|
||
//http://www.cppblog.com/abilitytao/archive/2009/08/04/92171.html
|
||
|
||
//常量区
|
||
const double M_PI = acos(-1.0); //PI
|
||
const double INF = 1e10; //无穷大
|
||
const double EPS = 1e-8; //计算精度
|
||
const int LEFT = 0; //点在直线左边
|
||
const int RIGHT = 1; //点在直线右边
|
||
const int ONLINE = 2; //点在直线上
|
||
const int CROSS = 0; //两直线相交
|
||
const int COLINE = 1; //两直线共线
|
||
const int PARALLEL = 2; //两直线平行
|
||
const int NOTCOPLANAR = 3; //两直线不共面
|
||
const int INSIDE = 1; //点在图形内部
|
||
const int OUTSIDE = 2; //点在图形外部
|
||
const int BORDER = 3; //点在图形边界
|
||
const int BAOHAN = 1; //大圆包含小圆
|
||
const int NEIQIE = 2; //内切
|
||
const int XIANJIAO = 3; //相交
|
||
const int WAIQIE = 4; //外切
|
||
const int XIANLI = 5; //相离
|
||
///////////////////////////////////////////////////////////////////
|
||
//类型定义区
|
||
struct Point { //二维点或矢量
|
||
double x, y;
|
||
double angle, dis;
|
||
Point(): x(0), y(0), angle(0), dis(0) {}
|
||
Point(double x0, double y0): x(x0), y(y0), angle(0), dis(0) {}
|
||
};
|
||
struct Point3D { //三维点或矢量
|
||
double x, y, z;
|
||
Point3D(): x(0), y(0), z(0) {}
|
||
Point3D(double x0, double y0, double z0): x(x0), y(y0), z(z0) {}
|
||
};
|
||
struct Line { //二维的直线或线段
|
||
Point p1, p2;
|
||
Line(): p1(), p2() {}
|
||
Line(Point p10, Point p20): p1(p10), p2(p20) {}
|
||
};
|
||
struct Line3D { //三维的直线或线段
|
||
Point3D p1, p2;
|
||
Line3D(): p1(), p2() {}
|
||
Line3D(Point3D p10, Point3D p20): p1(p10), p2(p20) {}
|
||
};
|
||
struct Rect { //用长宽表示矩形的方法 w, h分别表示宽度和高度
|
||
double w, h;
|
||
Rect(): w(0), h(0) {}
|
||
Rect(double _w, double _h) : w(_w), h(_h) {}
|
||
};
|
||
struct Rect_2 { //表示矩形, 左下角坐标是(xl, yl), 右上角坐标是(xh, yh)
|
||
double xl, yl, xh, yh;
|
||
Rect_2(): xl(0), yl(0), xh(0), yh(0) {}
|
||
Rect_2(double _xl, double _yl, double _xh, double _yh) : xl(_xl), yl(_yl), xh(_xh), yh(_yh) {}
|
||
};
|
||
struct Circle { //圆
|
||
Point c;
|
||
double r;
|
||
Circle(): c(), r(0) {}
|
||
Circle(Point _c, double _r) : c(_c), r(_r) {}
|
||
};
|
||
typedef vector<Point> Polygon; //二维多边形
|
||
typedef vector<Point> Points; //二维点集
|
||
typedef vector<Point3D> Points3D; //三维点集
|
||
///////////////////////////////////////////////////////////////////
|
||
//基本函数区
|
||
inline bool ZERO(double x) { //x == 0
|
||
return (fabs(x) < EPS);
|
||
}
|
||
inline bool ZERO(Point p) { //p == 0
|
||
return (ZERO(p.x) && ZERO(p.y));
|
||
}
|
||
inline bool ZERO(Point3D p) { //p == 0
|
||
return (ZERO(p.x) && ZERO(p.y) && ZERO(p.z));
|
||
}
|
||
inline bool EQ(double x, double y) { //eqaul, x == y
|
||
return (fabs(x - y) < EPS);
|
||
}
|
||
inline bool NEQ(double x, double y) { //not equal, x != y
|
||
return (fabs(x - y) >= EPS);
|
||
}
|
||
inline bool LT(double x, double y) { //less than, x < y
|
||
return (NEQ(x, y) && (x < y));
|
||
}
|
||
inline bool GT(double x, double y) { //greater than, x > y
|
||
return (NEQ(x, y) && (x > y));
|
||
}
|
||
inline bool LEQ(double x, double y) { //less equal, x <= y
|
||
return (EQ(x, y) || (x < y));
|
||
}
|
||
inline bool GEQ(double x, double y) { //greater equal, x >= y
|
||
return (EQ(x, y) || (x > y));
|
||
}
|
||
//注意！！！
|
||
//如果是一个很小的负的浮点数
|
||
//保留有效位数输出的时候会出现-0.000这样的形式,
|
||
//前面多了一个负号
|
||
//这就会导致错误！！！！！！
|
||
//因此在输出浮点数之前, 一定要调用次函数进行修正！
|
||
inline double FIX(double x) {
|
||
return (fabs(x) < EPS) ? 0 : x;
|
||
}
|
||
/////////////////////////////////////////////////////////////////////////////////////
|
||
//二维矢量运算
|
||
bool operator==(Point p1, Point p2) {
|
||
return (EQ(p1.x, p2.x) && EQ(p1.y, p2.y));
|
||
}
|
||
bool operator!=(Point p1, Point p2) {
|
||
return (NEQ(p1.x, p2.x) || NEQ(p1.y, p2.y));
|
||
}
|
||
bool operator<(Point p1, Point p2) {
|
||
if (NEQ(p1.x, p2.x)) { return (p1.x < p2.x); }
|
||
else { return (p1.y < p2.y); }
|
||
}
|
||
Point operator+(Point p1, Point p2) {
|
||
return Point(p1.x + p2.x, p1.y + p2.y);
|
||
}
|
||
Point operator-(Point p1, Point p2) {
|
||
return Point(p1.x - p2.x, p1.y - p2.y);
|
||
}
|
||
double operator*(Point p1, Point p2) { //计算叉乘 p1 × p2
|
||
return (p1.x * p2.y - p2.x * p1.y);
|
||
}
|
||
double operator&(Point p1, Point p2) { //计算点积 p1·p2
|
||
return (p1.x * p2.x + p1.y * p2.y);
|
||
}
|
||
double Norm(Point p) { //计算矢量p的模
|
||
return sqrt(p.x * p.x + p.y * p.y);
|
||
}
|
||
//把矢量p旋转角度angle (弧度表示)
|
||
//angle > 0表示逆时针旋转
|
||
//angle < 0表示顺时针旋转
|
||
Point Rotate(Point p, double angle) {
|
||
Point result;
|
||
result.x = p.x * cos(angle) - p.y * sin(angle);
|
||
result.y = p.x * sin(angle) + p.y * cos(angle);
|
||
return result;
|
||
}
|
||
//////////////////////////////////////////////////////////////////////////////////////
|
||
//三维矢量运算
|
||
bool operator==(Point3D p1, Point3D p2) {
|
||
return (EQ(p1.x, p2.x) && EQ(p1.y, p2.y) && EQ(p1.z, p2.z));
|
||
}
|
||
bool operator<(Point3D p1, Point3D p2) {
|
||
if (NEQ(p1.x, p2.x)) { return (p1.x < p2.x); }
|
||
else if (NEQ(p1.y, p2.y)) { return (p1.y < p2.y); }
|
||
else { return (p1.z < p2.z); }
|
||
}
|
||
Point3D operator+(Point3D p1, Point3D p2) {
|
||
return Point3D(p1.x + p2.x, p1.y + p2.y, p1.z + p2.z);
|
||
}
|
||
Point3D operator-(Point3D p1, Point3D p2) {
|
||
return Point3D(p1.x - p2.x, p1.y - p2.y, p1.z - p2.z);
|
||
}
|
||
Point3D operator*(Point3D p1, Point3D p2) { //计算叉乘 p1 x p2
|
||
return Point3D(p1.y * p2.z - p1.z * p2.y, p1.z * p2.x - p1.x * p2.z, p1.x * p2.y - p1.y * p2.x);
|
||
}
|
||
double operator&(Point3D p1, Point3D p2) { //计算点积 p1·p2
|
||
return (p1.x * p2.x + p1.y * p2.y + p1.z * p2.z);
|
||
}
|
||
double Norm(Point3D p) { //计算矢量p的模
|
||
return sqrt(p.x * p.x + p.y * p.y + p.z * p.z);
|
||
}
|
||
/////////////////////////////////////////////////////////////////////////////////////
|
||
//几何题面积计算
|
||
//
|
||
//根据三个顶点坐标计算三角形面积
|
||
//面积的正负按照右手旋规则确定
|
||
double Area(Point A, Point B, Point C) { //三角形面积
|
||
return ((B - A) * (C - A) / 2.0);
|
||
}
|
||
//根据三条边长计算三角形面积
|
||
double Area(double a, double b, double c) { //三角形面积
|
||
double s = (a + b + c) / 2.0;
|
||
return sqrt(s * (s - a) * (s - b) * (s - c));
|
||
}
|
||
double Area(const Circle &C) {
|
||
return M_PI * C.r * C.r;
|
||
}
|
||
//计算多边形面积
|
||
//面积的正负按照右手旋规则确定
|
||
double Area(const Polygon &poly) { //多边形面积
|
||
double res = 0;
|
||
int n = poly.size();
|
||
if (n < 3) { return 0; }
|
||
for (int i = 0; i < n; i++) {
|
||
res += poly[i].x * poly[(i + 1) % n].y;
|
||
res -= poly[i].y * poly[(i + 1) % n].x;
|
||
}
|
||
return (res / 2.0);
|
||
}
|
||
/////////////////////////////////////////////////////////////////////////////////////
|
||
//点.线段.直线问题
|
||
//
|
||
double Distance(Point p1, Point p2) { //2点间的距离
|
||
return sqrt((p1.x - p2.x) * (p1.x - p2.x) + (p1.y - p2.y) * (p1.y - p2.y));
|
||
}
|
||
double Distance(Point3D p1, Point3D p2) { //2点间的距离,三维
|
||
return sqrt((p1.x - p2.x) * (p1.x - p2.x) + (p1.y - p2.y) * (p1.y - p2.y) + (p1.z - p2.z) * (p1.z - p2.z));
|
||
}
|
||
double Distance(Point p, Line L) { //求二维平面上点到直线的距离
|
||
return (fabs((p - L.p1) * (L.p2 - L.p1)) / Norm(L.p2 - L.p1));
|
||
}
|
||
double Distance(Point3D p, Line3D L) { //求三维空间中点到直线的距离
|
||
return (Norm((p - L.p1) * (L.p2 - L.p1)) / Norm(L.p2 - L.p1));
|
||
}
|
||
bool OnLine(Point p, Line L) { //判断二维平面上点p是否在直线L上
|
||
return ZERO((p - L.p1) * (L.p2 - L.p1));
|
||
}
|
||
bool OnLine(Point3D p, Line3D L) { //判断三维空间中点p是否在直线L上
|
||
return ZERO((p - L.p1) * (L.p2 - L.p1));
|
||
}
|
||
int Relation(Point p, Line L) { //计算点p与直线L的相对关系 ,返回ONLINE,LEFT,RIGHT
|
||
double res = (L.p2 - L.p1) * (p - L.p1);
|
||
if (EQ(res, 0)) { return ONLINE; }
|
||
else if (res > 0) { return LEFT; }
|
||
else { return RIGHT; }
|
||
}
|
||
bool SameSide(Point p1, Point p2, Line L) { //判断点p1, p2是否在直线L的同侧
|
||
double m1 = (p1 - L.p1) * (L.p2 - L.p1);
|
||
double m2 = (p2 - L.p1) * (L.p2 - L.p1);
|
||
return GT(m1 * m2, 0);
|
||
}
|
||
bool OnLineSeg(Point p, Line L) { //判断二维平面上点p是否在线段l上
|
||
return (ZERO((L.p1 - p) * (L.p2 - p)) && LEQ((p.x - L.p1.x) * (p.x - L.p2.x), 0)
|
||
&& LEQ((p.y - L.p1.y) * (p.y - L.p2.y), 0));
|
||
}
|
||
bool OnLineSeg(Point3D p, Line3D L) { //判断三维空间中点p是否在线段l上
|
||
return (ZERO((L.p1 - p) * (L.p2 - p)) && EQ(Norm(p - L.p1) + Norm(p - L.p2), Norm(L.p2 - L.p1)));
|
||
}
|
||
Point SymPoint(Point p, Line L) { //求二维平面上点p关于直线L的对称点
|
||
Point result;
|
||
double a = L.p2.x - L.p1.x;
|
||
double b = L.p2.y - L.p1.y;
|
||
double t = ((p.x - L.p1.x) * a + (p.y - L.p1.y) * b) / (a * a + b * b);
|
||
result.x = 2 * L.p1.x + 2 * a * t - p.x;
|
||
result.y = 2 * L.p1.y + 2 * b * t - p.y;
|
||
return result;
|
||
}
|
||
bool Coplanar(Points3D points) { //判断一个点集中的点是否全部共面
|
||
Point3D p;
|
||
if (points.size() < 4) { return true; }
|
||
p = (points[2] - points[0]) * (points[1] - points[0]);
|
||
for (size_t i = 3; i < points.size(); i++) {
|
||
if (!ZERO(p & (points[i] - points[0]))) { return false; }
|
||
}
|
||
return true;
|
||
}
|
||
bool LineIntersect(Line L1, Line L2) { //判断二维的两直线是否相交
|
||
return (!ZERO((L1.p1 - L1.p2) * (L2.p1 - L2.p2))); //是否平行
|
||
}
|
||
bool LineIntersect(Line3D L1, Line3D L2) { //判断三维的两直线是否相交
|
||
Point3D p1 = L1.p1 - L1.p2;
|
||
Point3D p2 = L2.p1 - L2.p2;
|
||
Point3D p = p1 * p2;
|
||
if (ZERO(p)) { return false; } //是否平行
|
||
p = (L2.p1 - L1.p2) * (L1.p1 - L1.p2);
|
||
return ZERO(p & L2.p2); //是否共面
|
||
}
|
||
bool LineSegIntersect(Line L1, Line L2) { //判断二维的两条线段是否相交
|
||
return (GEQ(max(L1.p1.x, L1.p2.x), min(L2.p1.x, L2.p2.x)) &&
|
||
GEQ(max(L2.p1.x, L2.p2.x), min(L1.p1.x, L1.p2.x)) &&
|
||
GEQ(max(L1.p1.y, L1.p2.y), min(L2.p1.y, L2.p2.y)) &&
|
||
GEQ(max(L2.p1.y, L2.p2.y), min(L1.p1.y, L1.p2.y)) &&
|
||
LEQ(((L2.p1 - L1.p1) * (L1.p2 - L1.p1)) * ((L2.p2 - L1.p1) * (L1.p2 - L1.p1)), 0) &&
|
||
LEQ(((L1.p1 - L2.p1) * (L2.p2 - L2.p1)) * ((L1.p2 - L2.p1) * (L2.p2 - L2.p1)), 0));
|
||
}
|
||
bool LineSegIntersect(Line3D L1, Line3D L2) { //判断三维的两条线段是否相交
|
||
//todo
|
||
return true;
|
||
}
|
||
//计算两条二维直线的交点, 结果在参数P中返回
|
||
//返回值说明了两条直线的位置关系: COLINE -- 共线 PARALLEL -- 平行 CROSS -- 相交
|
||
int CalCrossPoint(Line L1, Line L2, Point &P) {
|
||
double A1, B1, C1, A2, B2, C2;
|
||
A1 = L1.p2.y - L1.p1.y;
|
||
B1 = L1.p1.x - L1.p2.x;
|
||
C1 = L1.p2.x * L1.p1.y - L1.p1.x * L1.p2.y;
|
||
A2 = L2.p2.y - L2.p1.y;
|
||
B2 = L2.p1.x - L2.p2.x;
|
||
C2 = L2.p2.x * L2.p1.y - L2.p1.x * L2.p2.y;
|
||
if (EQ(A1 * B2, B1 * A2)) {
|
||
if (EQ((A1 + B1) * C2, (A2 + B2) * C1)) { return COLINE; }
|
||
else { return PARALLEL; }
|
||
} else {
|
||
P.x = (B2 * C1 - B1 * C2) / (A2 * B1 - A1 * B2);
|
||
P.y = (A1 * C2 - A2 * C1) / (A2 * B1 - A1 * B2);
|
||
return CROSS;
|
||
}
|
||
}
|
||
//计算两条三维直线的交点, 结果在参数P中返回
|
||
//返回值说明了两条直线的位置关系 COLINE -- 共线 PARALLEL -- 平行 CROSS -- 相交 NONCOPLANAR -- 不公面
|
||
int CalCrossPoint(Line3D L1, Line3D L2, Point3D &P) {
|
||
//todo
|
||
return 0;
|
||
}
|
||
//计算点P到直线L的最近点
|
||
Point NearestPointToLine(Point P, Line L) {
|
||
Point result;
|
||
double a, b, t;
|
||
a = L.p2.x - L.p1.x;
|
||
b = L.p2.y - L.p1.y;
|
||
t = ((P.x - L.p1.x) * a + (P.y - L.p1.y) * b) / (a * a + b * b);
|
||
result.x = L.p1.x + a * t;
|
||
result.y = L.p1.y + b * t;
|
||
return result;
|
||
}
|
||
//计算点P到线段L的最近点
|
||
Point NearestPointToLineSeg(Point P, Line L) {
|
||
Point result;
|
||
double a, b, t;
|
||
a = L.p2.x - L.p1.x;
|
||
b = L.p2.y - L.p1.y;
|
||
t = ((P.x - L.p1.x) * a + (P.y - L.p1.y) * b) / (a * a + b * b);
|
||
if (GEQ(t, 0) && LEQ(t, 1)) {
|
||
result.x = L.p1.x + a * t;
|
||
result.y = L.p1.y + b * t;
|
||
} else {
|
||
if (Norm(P - L.p1) < Norm(P - L.p2)) { result = L.p1; }
|
||
else { result = L.p2; }
|
||
}
|
||
return result;
|
||
}
|
||
//计算险段L1到线段L2的最短距离
|
||
double MinDistance(Line L1, Line L2) {
|
||
double d1, d2, d3, d4;
|
||
if (LineSegIntersect(L1, L2)) {
|
||
return 0;
|
||
} else {
|
||
d1 = Norm(NearestPointToLineSeg(L1.p1, L2) - L1.p1);
|
||
d2 = Norm(NearestPointToLineSeg(L1.p2, L2) - L1.p2);
|
||
d3 = Norm(NearestPointToLineSeg(L2.p1, L1) - L2.p1);
|
||
d4 = Norm(NearestPointToLineSeg(L2.p2, L1) - L2.p2);
|
||
return min(min(d1, d2), min(d3, d4));
|
||
}
|
||
}
|
||
//求二维两直线的夹角,
|
||
//返回值是0~Pi之间的弧度
|
||
double Inclination(Line L1, Line L2) {
|
||
Point u = L1.p2 - L1.p1;
|
||
Point v = L2.p2 - L2.p1;
|
||
return acos((u & v) / (Norm(u) * Norm(v)));
|
||
}
|
||
//求三维两直线的夹角,
|
||
//返回值是0~Pi之间的弧度
|
||
double Inclination(Line3D L1, Line3D L2) {
|
||
Point3D u = L1.p2 - L1.p1;
|
||
Point3D v = L2.p2 - L2.p1;
|
||
return acos((u & v) / (Norm(u) * Norm(v)));
|
||
}
|
||
/////////////////////////////////////////////////////////////////////////////
|
||
//多边行问题:
|
||
//判断点p是否在凸多边形poly内
|
||
//poly的顶点数目要大于等于3
|
||
//返回值为：
|
||
//INSIDE -- 点在poly内
|
||
//BORDER -- 点在poly边界上
|
||
//OUTSIDE -- 点在poly外
|
||
int InsideConvex(Point p, const Polygon &poly) { //判断点p是否在凸多边形poly内
|
||
Point q(0, 0);
|
||
Line side;
|
||
int i, n = poly.size();
|
||
for (i = 0; i < n; i++) {
|
||
q.x += poly[i].x; q.y += poly[i].y;
|
||
}
|
||
q.x /= n; q.y /= n;
|
||
for (i = 0; i < n; i++) {
|
||
side.p1 = poly[i];
|
||
side.p2 = poly[(i + 1) % n];
|
||
if (OnLineSeg(p, side)) { return BORDER; }
|
||
else if (!SameSide(p, q, side)) { return OUTSIDE; }
|
||
}
|
||
return INSIDE;
|
||
}
|
||
//判断多边形poly是否是凸的
|
||
bool IsConvex(const Polygon &poly) {
|
||
int i, n, rel;
|
||
Line side;
|
||
n = poly.size();
|
||
if (n < 3) { return false; }
|
||
side.p1 = poly[0];
|
||
side.p2 = poly[1];
|
||
rel = Relation(poly[2], side);
|
||
for (i = 1; i < n; i++) {
|
||
side.p1 = poly[i];
|
||
side.p2 = poly[(i + 1) % n];
|
||
if (Relation(poly[(i + 2) % n], side) != rel) { return false; }
|
||
}
|
||
return true;
|
||
}
|
||
//判断点p是否在简单多边形poly内, 多边形可以是凸的或凹的
|
||
//poly的顶点数目要大于等于3
|
||
//返回值为：
|
||
//INSIDE -- 点在poly内
|
||
//BORDER -- 点在poly边界上
|
||
//OUTSIDE -- 点在poly外
|
||
int InsidePolygon(const Polygon &poly, Point p) {
|
||
int i, n, count;
|
||
Line ray, side;
|
||
n = poly.size();
|
||
count = 0;
|
||
ray.p1 = p;
|
||
ray.p2.y = p.y;
|
||
ray.p2.x = - INF;
|
||
for (i = 0; i < n; i++) {
|
||
side.p1 = poly[i];
|
||
side.p2 = poly[(i + 1) % n];
|
||
if (OnLineSeg(p, side)) {
|
||
return BORDER;
|
||
}
|
||
//如果side平行x轴则不作考虑
|
||
if (EQ(side.p1.y, side.p2.y)) { continue; }
|
||
if (OnLineSeg(side.p1, ray)) {
|
||
if (GT(side.p1.y, side.p2.y)) { count++; }
|
||
} else if (OnLineSeg(side.p2, ray)) {
|
||
if (GT(side.p2.y, side.p1.y)) { count++; }
|
||
} else if (LineSegIntersect(ray, side)) {
|
||
count++;
|
||
}
|
||
}
|
||
return ((count % 2 == 1) ? INSIDE : OUTSIDE);
|
||
}
|
||
//判断线段是否在多边形内 (线段的点可能在多边形上)
|
||
//多边形可以是任意简单多边形
|
||
bool InsidePolygon(const Polygon &poly, Line L) {
|
||
bool result;
|
||
int n;
|
||
Points points;
|
||
Point p;
|
||
Line side;
|
||
result = ((InsidePolygon(poly, L.p1) != OUTSIDE) && (InsidePolygon(poly, L.p2) != OUTSIDE));
|
||
if (!result) { return false; }
|
||
n = poly.size();
|
||
for (int i = 0; i < n; i++) {
|
||
side.p1 = poly[i];
|
||
side.p2 = poly[(i + 1) % n];
|
||
if (OnLineSeg(L.p1, side)) {
|
||
points.push_back(L.p1);
|
||
} else if (OnLineSeg(L.p2, side)) {
|
||
points.push_back(L.p2);
|
||
} else if (OnLineSeg(side.p1, L)) {
|
||
points.push_back(side.p1);
|
||
} else if (OnLineSeg(side.p2, L)) {
|
||
points.push_back(side.p2);
|
||
} else if (LineSegIntersect(side, L)) {
|
||
return false;
|
||
}
|
||
}
|
||
//对交点进行排序
|
||
sort(points.begin(), points.end());
|
||
for (size_t i = 1; i < points.size(); i++) {
|
||
if (points[i - 1] != points[i]) {
|
||
p.x = (points[i - 1].x + points[i].x) / 2.0;
|
||
p.y = (points[i - 1].y + points[i].y) / 2.0;
|
||
if (InsidePolygon(poly, p) == OUTSIDE) {
|
||
return false;
|
||
}
|
||
}
|
||
}
|
||
return true;
|
||
}
|
||
//寻找凸包 graham 扫描法
|
||
//生成的多边形顶点按逆时针方向排列
|
||
bool GrahamComp(const Point &left, const Point &right) {
|
||
if (EQ(left.angle, right.angle)) {
|
||
return (left.dis < right.dis);
|
||
} else {
|
||
return (left.angle < right.angle);
|
||
}
|
||
}
|
||
void GrahamScan(Points &points, Polygon &result) {
|
||
int i, k, n;
|
||
Point p;
|
||
n = points.size();
|
||
result.clear();
|
||
if (n < 3) { return; }
|
||
//选取points中y坐标最小的点points[k],
|
||
//如果这样的点有多个, 则取最左边的一个
|
||
k = 0;
|
||
for (i = 1; i < n; i++) {
|
||
if (EQ(points[i].y, points[k].y)) {
|
||
if (points[i].x <= points[k].x) { k = i; }
|
||
} else if (points[i].y < points[k].y) {
|
||
k = i;
|
||
}
|
||
}
|
||
swap(points[0], points[k]);
|
||
//现在points中y坐标最小的点在points[0]
|
||
//计算每个点相对于points[0]的极角和距离
|
||
for (i = 1; i < n; i++) {
|
||
points[i].angle = atan2(points[i].y - points[0].y, points[i].x - points[0].x);
|
||
points[i].dis = Norm(points[i] - points[0]);
|
||
}
|
||
//对顶点按照相对points[0]的极角从小到大进行排序
|
||
//对于极角相同的按照距points[0]的距离从小到大排序
|
||
sort(points.begin() + 1, points.end(), GrahamComp);
|
||
//下面计算凸包
|
||
result.push_back(points[0]);
|
||
for (i = 1; i < n; i++) {
|
||
//如果有极角相同的点, 只取相对于points[0]最远的一个
|
||
if ((i + 1 < n) && EQ(points[i].angle, points[i + 1].angle)) { continue; }
|
||
if (result.size() >= 3) {
|
||
p = result[result.size() - 2];
|
||
while (GEQ((points[i] - p) * (result.back() - p), 0)) {
|
||
result.pop_back();
|
||
p = result[result.size() - 2];
|
||
}
|
||
}
|
||
result.push_back(points[i]);
|
||
}
|
||
}
|
||
//用有向直线line切割凸多边形,
|
||
//result[LEFT]和result[RIGHT]分别保存被切割后line的左边和右边部分
|
||
//result[ONLINE]没有用到, 只是用来作为辅助空间
|
||
//返回值是切割多边形的切口的长度,
|
||
//如果返回值是0 则说明未作切割。
|
||
//当未作切割时, 如果多边形在该直线的右侧, 则result[RIGHT]等于该多边形, 否则result[LEFT]等于该多边形
|
||
//注意：被切割的多边形一定要是凸多边形, 顶点按照逆时针排列
|
||
//可利用这个函数来求多边形的核, 初始的核设为一个很大的矩形, 然后依次用多边形的每条边去割
|
||
double CutConvex(const Polygon &poly, const Line &line, Polygon result[3]) {
|
||
vector<Point> points;
|
||
Line side;
|
||
Point p;
|
||
int i, n, cur, pre;
|
||
result[LEFT].clear();
|
||
result[RIGHT].clear();
|
||
result[ONLINE].clear();
|
||
n = poly.size();
|
||
if (n == 0) { return 0; }
|
||
pre = cur = Relation(poly[0], line);
|
||
for (i = 0; i < n; i++) {
|
||
cur = Relation(poly[(i + 1) % n], line);
|
||
if (cur == pre) {
|
||
result[cur].push_back(poly[(i + 1) % n]);
|
||
} else {
|
||
side.p1 = poly[i];
|
||
side.p2 = poly[(i + 1) % n];
|
||
CalCrossPoint(side, line, p);
|
||
points.push_back(p);
|
||
result[pre].push_back(p);
|
||
result[cur].push_back(p);
|
||
result[cur].push_back(poly[(i + 1) % n]);
|
||
pre = cur;
|
||
}
|
||
}
|
||
sort(points.begin(), points.end());
|
||
if (points.size() < 2) { return 0; }
|
||
else { return Norm(points.front() - points.back()); }
|
||
}
|
||
//求多边形的重心, 适用于凸的或凹的简单多边形
|
||
//该算法可以一边读入多边性的顶点一边计算重心
|
||
Point CenterOfPolygon(const Polygon &poly) {
|
||
Point p, p0, p1, p2, p3;
|
||
double m, m0;
|
||
p1 = poly[0];
|
||
p2 = poly[1];
|
||
p.x = p.y = m = 0;
|
||
for (size_t i = 2; i < poly.size(); i++) {
|
||
p3 = poly[i];
|
||
p0.x = (p1.x + p2.x + p3.x) / 3.0;
|
||
p0.y = (p1.y + p2.y + p3.y) / 3.0;
|
||
m0 = p1.x * p2.y + p2.x * p3.y + p3.x * p1.y - p1.y * p2.x - p2.y * p3.x - p3.y * p1.x;
|
||
if (ZERO(m + m0)) { m0 += EPS; } //为了防止除0溢出, 对m0做一点点修正
|
||
p.x = (m * p.x + m0 * p0.x) / (m + m0);
|
||
p.y = (m * p.y + m0 * p0.y) / (m + m0);
|
||
m = m + m0;
|
||
p2 = p3;
|
||
}
|
||
return p;
|
||
}
|
||
//判断两个矩形是否相交
|
||
//如果相邻不算相交
|
||
bool Intersect(Rect_2 r1, Rect_2 r2) {
|
||
return (max(r1.xl, r2.xl) < min(r1.xh, r2.xh) && max(r1.yl, r2.yl) < min(r1.yh, r2.yh));
|
||
}
|
||
//判断矩形r2是否可以放置在矩形r1内
|
||
//r2可以任意地旋转
|
||
//发现原来的给出的方法过不了OJ上的无归之室这题,
|
||
//所以用了自己的代码
|
||
bool IsContain(Rect r1, Rect r2) { //矩形的w > h
|
||
if (r1.w > r2.w && r1.h > r2.h) { return true; }
|
||
else {
|
||
double r = sqrt(r2.w * r2.w + r2.h * r2.h) / 2.0;
|
||
double alpha = atan2(r2.h, r2.w);
|
||
double sita = asin((r1.h / 2.0) / r);
|
||
double x = r * cos(sita - 2 * alpha);
|
||
double y = r * sin(sita - 2 * alpha);
|
||
if (x < r1.w / 2.0 && y < r1.h / 2.0 && x > 0 && y > -r1.h / 2.0) { return true; }
|
||
else { return false; }
|
||
}
|
||
}
|
||
////////////////////////////////////////////////////////////////////////
|
||
//圆
|
||
Point Center(const Circle &C) { //圆心
|
||
return C.c;
|
||
}
|
||
//两个圆的公共面积
|
||
double CommonArea(const Circle &A, const Circle &B) {
|
||
double area = 0.0;
|
||
const Circle &M = (A.r > B.r) ? A : B;
|
||
const Circle &N = (A.r > B.r) ? B : A;
|
||
double D = Distance(Center(M), Center(N));
|
||
if ((D < M.r + N.r) && (D > M.r - N.r)) {
|
||
double cosM = (M.r * M.r + D * D - N.r * N.r) / (2.0 * M.r * D);
|
||
double cosN = (N.r * N.r + D * D - M.r * M.r) / (2.0 * N.r * D);
|
||
double alpha = 2.0 * acos(cosM);
|
||
double beta = 2.0 * acos(cosN);
|
||
double TM = 0.5 * M.r * M.r * sin(alpha);
|
||
double TN = 0.5 * N.r * N.r * sin(beta);
|
||
double FM = (alpha / 360.0) * Area(M);
|
||
double FN = (beta / 360.0) * Area(N);
|
||
area = FM + FN - TM - TN;
|
||
} else if (D <= M.r - N.r) {
|
||
area = Area(N);
|
||
}
|
||
return area;
|
||
}
|
||
//判断圆是否在矩形内(不允许相切)
|
||
bool IsInCircle(const Circle &C, const Rect_2 &rect) {
|
||
return (GT(C.c.x - C.r, rect.xl) && LT(C.c.x + C.r, rect.xh) && GT(C.c.y - C.r, rect.yl)
|
||
&& LT(C.c.y + C.r, rect.yh));
|
||
}
|
||
//判断2圆的位置关系
|
||
//返回:
|
||
//BAOHAN = 1; //大圆包含小圆
|
||
//NEIQIE = 2; //内切
|
||
//XIANJIAO = 3; //相交
|
||
//WAIQIE = 4; //外切
|
||
//XIANLI = 5; //相离
|
||
int CirCir(const Circle &c1, const Circle &c2) {
|
||
double dis = Distance(c1.c, c2.c);
|
||
if (LT(dis, fabs(c1.r - c2.r))) { return BAOHAN; }
|
||
if (EQ(dis, fabs(c1.r - c2.r))) { return NEIQIE; }
|
||
if (LT(dis, c1.r + c2.r) && GT(dis, fabs(c1.r - c2.r))) { return XIANJIAO; }
|
||
if (EQ(dis, c1.r + c2.r)) { return WAIQIE; }
|
||
return XIANLI;
|
||
}
|