//随机化求最小圆覆盖 //圆心为O, 半径为R double R, EPS = 1e-9; struct Point { double x, y; } a[N], O; 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)); } Point center(Point x, Point y, Point z) { double a1 = y.x - x.x, a2 = z.x - x.x, b1 = y.y - x.y, b2 = z.y - x.y, c1 = (a1 * a1 + b1 * b1) / 2, c2 = (a2 * a2 + b2 * b2) / 2, d = a1 * b2 - a2 * b1; return (Point) {x.x + (c1 * b2 - c2 * b1) / d, x.y + (a1 * c2 - a2 * c1) / d}; } void cal(int n, Point b[]) { O = a[0]; R = 0; for (int i = 0; i < n; i++) { a[i] = b[i]; } for (int i = 0; i < n; i++) { swap(a[rand() % n], a[i]); } for (int i = 1; i < n; i++) { if (dis(a[i], O) > R + EPS) { O = a[i]; R = 0; for (int j = 0; j < i; j++) { if (dis(a[j], O) > R + EPS) { O = (Point) {(a[i].x + a[j].x) / 2, (a[i].y + a[j].y) / 2}, R = dis(O, a[i]); for (int k = 0; k < j; k++) { if (dis(a[k], O) > R + EPS) { O = center(a[k], a[j], a[i]), R = dis(O, a[i]); } } } } } } } //kuangbin //1、基本函数 //1.1 Point 定义 const double PI = acos(-1.0); 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, double _y): 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; } //绕原点旋转角度B(弧度值), 后x, y的变化 void transXY(double B) { double tx = x, ty = y; x = tx * cos(B) - ty * sin(B); y = tx * sin(B) + ty * cos(B); } }; //1.2 Line 定义 struct Line { Point s, e; Line() {} Line(Point _s, Point _e): s(_s), e(_e) {} //两直线相交求交点 //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 //只有第一个值为2时, 交点才有意义 pair operator&(const Line &b)const { Point res = s; if (sgn((s - e) ^ (b.s - b.e)) == 0) { if (sgn((s - b.e) ^ (b.s - b.e)) == 0) { return make_pair(0, res); } //重合 else { return make_pair(1, res); } //平行 } 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 make_pair(2, res); } }; //1.3 两点间距离 //*两点间距离 double dist(const Point &a, const Point &b) { return sqrt((a - b) * (a - b)); } //1.4 判断: 线段相交 //*判断线段相交 bool inter(const Line &l1, const Line &l2) { return max(l1.s.x, l1.e.x) >= min(l2.s.x, l2.e.x) && max(l2.s.x, l2.e.x) >= min(l1.s.x, l1.e.x) && max(l1.s.y, l1.e.y) >= min(l2.s.y, l2.e.y) && max(l2.s.y, l2.e.y) >= min(l1.s.y, l1.e.y) && sgn((l2.s - l1.e) ^ (l1.s - l1.e)) * sgn((l2.e-l1.e) ^ (l1.s - l1.e)) <= 0 && sgn((l1.s - l2.e) ^ (l2.s - l2.e)) * sgn((l1.e-l2.e) ^ (l2.s - l2.e)) <= 0; } //1.5 判断：直线和线段相交 //判断直线和线段相交 bool Seg_inter_line(const Line &l1, const Line &l2) { //判断直线l1和线段l2是否相交 return sgn((l2.s - l1.e) ^ (l1.s - l1.e)) * sgn((l2.e-l1.e) ^ (l1.s - l1.e)) <= 0; } //1.6 点到直线距离 //点到直线距离 //返回为result, 是点到直线最近的点 Point PointToLine(const Point &P, const Line &L) { double t = ((P - L.s) * (L.e-L.s)) / ((L.e - L.s) * (L.e - L.s)); return Point(L.s.x + (L.e.x - L.s.x) * t, L.s.y + (L.e.y - L.s.y) * t); } //1.7 点到线段距离 //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; } //1.8 计算多边形面积 //计算多边形面积 //点的编号从0 ~ n - 1 double CalcArea(Point p[], int n) { double res = 0; for (int i = 0; i < n; i++) { res += (p[i] ^ p[(i + 1) % n]) * 0.5; } return fabs(res); } //1.9 判断点在线段上 //*判断点在线段上 bool OnSeg(const Point &P, const Line &L) { return sgn((L.s - P) ^ (L.e-P)) == 0 && sgn((P.x - L.s.x) * (P.x - L.e.x)) <= 0 && sgn((P.y - L.s.y) * (P.y - L.e.y)) <= 0; } //1.10 判断点在凸多边形内 //*判断点在凸多边形内 //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) //点的编号:0 ~ n - 1 //返回值: //-1:点在凸多边形外 //0:点在凸多边形边界上 //1:点在凸多边形内 int inConvexPoly(const Point &a, Point p[], int n) { for (int i = 0; i < n; i++) { if (sgn((p[i] - a) ^ (p[(i + 1) % n] - a)) < 0) { return -1; } else if (OnSeg(a, Line(p[i], p[(i + 1) % n]))) { return 0; } } return 1; } //1.11 判断点在任意多边形内 //*判断点在任意多边形内 //射线法, poly[]的顶点数要大于等于3,点的编号0~n-1 //返回值 //-1:点在凸多边形外 //0:点在凸多边形边界上 //1:点在凸多边形内 int inPoly(const Point &p, Point poly[], int n) { int cnt = 0; Line ray(p, Point(-100000000000.0, p.y)), side; //-INF,注意取值防止越界 for (int i = 0; i < n; i++) { side.s = poly[i]; side.e = poly[(i + 1) % n]; if (OnSeg(p, side)) { return 0; } //如果平行轴则不考虑 if (sgn(side.s.y - side.e.y) == 0) { continue; } if (OnSeg(side.s, ray)) { if (sgn(side.s.y - side.e.y) > 0) { cnt++; } } else if (OnSeg(side.e, ray)) { if (sgn(side.e.y - side.s.y) > 0) { cnt++; } } else if (inter(ray, side)) { cnt++; } } if (cnt & 1) { return 1; } else { return -1; } } //1.12 判断凸多边形 //判断凸多边形 //允许共线边 //点可以是顺时针给出也可以是逆时针给出 //点的编号1~n-1 bool isconvex(Point poly[], int n) { bool s[3] = {0}; for (int i = 0; i < n; i++) { s[sgn((poly[(i + 1) % n] - poly[i]) ^ (poly[(i + 2) % n] - poly[i])) + 1] = true; if (s[0] && s[2]) { return false; } } return true; } //2、凸包 /* * 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 = 1010; Point lst[N]; int Stack[N], top; //相对于lst[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; } } //3、平面最近点对(HDU 1007) const double EPS = 1e-6; const double INF = 1e20; const int N = 100005; struct Point { double x, y; } p[N], tmpt[N]; double dist(const Point &a, const Point &b) { return sqrt((a.x - b.x) * (a.x - b.x) + (a.y - b.y) * (a.y - b.y)); } bool cmpxy(const Point &a, const Point &b) { return a.x < b.x || (a.x == b.x && a.y < b.y); } bool cmpy(Point a, Point b) { return a.y < b.y; } double Closest_Pair(int left, int right) { double d = INF; if (left == right) { return d; } if (left + 1 == right) { return dist(p[left], p[right]); } int mid = (left + right) / 2; double d1 = Closest_Pair(left, mid); double d2 = Closest_Pair(mid + 1, right); d = min(d1, d2); int k = 0; for (int i = left; i <= right; i++) { if (fabs(p[mid].x - p[i].x) <= d) { tmpt[k++] = p[i]; } } sort(tmpt, tmpt + k, cmpy); for (int i = 0; i < k; i++) { for (int j = i + 1; j < k && tmpt[j].y - tmpt[i].y < d; j++) { d = min(d, dist(tmpt[i], tmpt[j])); } } return d; } int main() { int n; while (scanf("%d", &n) == 1 && n) { for (int i = 0; i < n; i++) { scanf("%lf%lf", &p[i].x, &p[i].y); } sort(p, p + n, cmpxy); printf("%.2lf\n", Closest_Pair(0, n - 1) / 2); } } //4、旋转卡壳 //4.1 求解平面最远点对(POJ 2187 Beauty Contest) struct Point { int x, y; Point(int _x = 0, int _y = 0): x(_x), y(_y) {} Point operator-(const Point &b)const { return Point(x - b.x, y - b.y); } int operator^(const Point &b)const { return x * b.y - y * b.x; } int operator*(const Point &b)const { return x * b.x + y * b.y; } void input() { scanf("%d%d", &x, &y); } }; //距离的平方 int dist2(const Point &a, const Point &b) { return (a - b) * (a - b); } //******二维凸包, int*********** const int N = 50005; Point lst[N], p[N]; int Stack[N], top; bool _cmp(const Point &p1, const Point &p2) { int tmp = (p1 - lst[0]) ^ (p2 - lst[0]); if (tmp > 0) { return true; } else if (tmp == 0 && dist2(p1, lst[0]) <= dist2(p2, lst[0])) { return true; } else { return false; } } void Graham(int n) { Point p0 = lst[0]; int k = 0; 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 && ((lst[Stack[top - 1]] - lst[Stack[top - 2]]) ^ (lst[i] - lst[Stack[top - 2]])) <= 0) { top--; } Stack[top++] = i; } } //旋转卡壳, 求两点间距离平方的最大值 int rotating_calipers(Point p[], int n) { int ans = 0, cur = 1; Point v; for (int i = 0; i < n; i++) { v = p[i] - p[(i + 1) % n]; while ((v ^ (p[(cur + 1) % n] - p[cur])) < 0) { cur = (cur + 1) % n; } ans = max(ans, max(dist2(p[i], p[cur]), dist2(p[(i + 1) % n], p[(cur + 1) % n]))); } return ans; } int main() { int n; while (~scanf("%d", &n)) { 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("%d\n", rotating_calipers(p, top)); } } //4.2 求解平面点集最大三角形 //旋转卡壳计算平面点集最大三角形面积 Point lst[N], p[N]; int rotating_calipers(Point p[], int n) { int ans = 0; Point v; for (int i = 0; i < n; i++) { int j = (i + 1) % n, k = (j + 1) % n; while (j != i && k != i) { ans = max(ans, abs((p[j] - p[i]) ^ (p[k] - p[i]))); while (((p[i] - p[j]) ^ (p[(k + 1) % n] - p[k])) < 0) { k = (k + 1) % n; } j = (j + 1) % n; } } return ans; } int main() { int n; while (scanf("%d", &n) == 1) { 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 Polygon; //二维多边形 typedef vector Points; //二维点集 typedef vector 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 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; }