/***********基础*************/struct Point3 { double x, y, z; Point3(double x=0, double y=0, double z=0):x(x),y(y),z(z) { }};typedef Point3 Vector3;Vector3 operator + (const Vector3& A, const Vector3& B) { return Vector3(A.x+B.x, A.y+B.y, A.z+B.z); }Vector3 operator - (const Point3& A, const Point3& B) { return Vector3(A.x-B.x, A.y-B.y, A.z-B.z); }Vector3 operator * (const Vector3& A, double p) { return Vector3(A.x*p, A.y*p, A.z*p); }Vector3 operator / (const Vector3& A, double p) { return Vector3(A.x/p, A.y/p, A.z/p); }double Dot(const Vector3& A, const Vector3& B) { return A.x*B.x + A.y*B.y + A.z*B.z; }double Length(const Vector3& A) { return sqrt(Dot(A, A)); }double Angle(const Vector3& A, const Vector3& B) { return acos(Dot(A, B) / Length(A) / Length(B)); }Vector3 Cross(const Vector3& A, const Vector3& B) { return Vector3(A.y*B.z - A.z*B.y, A.z*B.x - A.x*B.z, A.x*B.y - A.y*B.x); }double Area2(const Point3& A, const Point3& B, const Point3& C) { return Length(Cross(B-A, C-A)); }double Volume6(const Point3& A, const Point3& B, const Point3& C, const Point3& D) { return Dot(D-A, Cross(B-A, C-A)); }// 四面体的重心Point3 Centroid(const Point3& A, const Point3& B, const Point3& C, const Point3& D) { return (A + B + C + D)/4.0; }/************点线面*************/// 点p到平面p0-n的距离。n必须为单位向量double DistanceToPlane(const Point3& p, const Point3& p0, const Vector3& n) { return fabs(Dot(p-p0, n)); // 如果不取绝对值，得到的是有向距离}// 点p在平面p0-n上的投影。n必须为单位向量Point3 GetPlaneProjection(const Point3& p, const Point3& p0, const Vector3& n) { return p-n*Dot(p-p0, n);}//直线p1-p2 与平面p0-n的交点Point3 LinePlaneIntersection(Point3 p1, Point3 p2, Point3 p0, Vector3 n){ vector3 = p2-p1; double t = (Dot(n, p0-p1) / Dot(n, p2-p1));//分母为0，直线与平面平行或在平面上 return p1 + v*t; //如果是线段 判断t是否在0~1之间}// 点P到直线AB的距离double DistanceToLine(const Point3& P, const Point3& A, const Point3& B) { Vector3 v1 = B - A, v2 = P - A; return Length(Cross(v1, v2)) / Length(v1);}//点到线段的距离double DistanceToSeg(Point3 p, Point3 a, Point3 b){ if(a == b) return Length(p-a); Vector3 v1 = b-a, v2 = p-a, v3 = p-b; if(dcmp(Dot(v1, v2)) < 0) return Length(v2); else if(dcmp(Dot(v1, v3)) > 0) return Length(v3); else return Length(Cross(v1, v2)) / Length(v1); }//求异面直线 p1+s*u与p2+t*v的公垂线对应的s 如果平行|重合，返回falsebool LineDistance3D(Point3 p1, Vector3 u, Point3 p2, Vector3 v, double& s){ double b = Dot(u, u) * Dot(v, v) - Dot(u, v) * Dot(u, v); if(dcmp(b) == 0) return false; double a = Dot(u, v) * Dot(v, p1-p2) - Dot(v, v) * Dot(u, p1-p2); s = a/b; return true;}// p1和p2是否在线段a-b的同侧bool SameSide(const Point3& p1, const Point3& p2, const Point3& a, const Point3& b) { return dcmp(Dot(Cross(b-a, p1-a), Cross(b-a, p2-a))) >= 0;}// 点P在三角形P0, P1, P2中bool PointInTri(const Point3& P, const Point3& P0, const Point3& P1, const Point3& P2) { return SameSide(P, P0, P1, P2) && SameSide(P, P1, P0, P2) && SameSide(P, P2, P0, P1);}// 三角形P0P1P2是否和线段AB相交bool TriSegIntersection(const Point3& P0, const Point3& P1, const Point3& P2, const Point3& A, const Point3& B, Point3& P) { Vector3 n = Cross(P1-P0, P2-P0); if(dcmp(Dot(n, B-A)) == 0) return false; // 线段A-B和平面P0P1P2平行或共面 else { // 平面A和直线P1-P2有惟一交点 double t = Dot(n, P0-A) / Dot(n, B-A); if(dcmp(t) < 0 || dcmp(t-1) > 0) return false; // 不在线段AB上 P = A + (B-A)*t; // 交点 return PointInTri(P, P0, P1, P2); }}//空间两三角形是否相交bool TriTriIntersection(Point3* T1, Point3* T2) { Point3 P; for(int i = 0; i < 3; i++) { if(TriSegIntersection(T1[0], T1[1], T1[2], T2[i], T2[(i+1)%3], P)) return true; if(TriSegIntersection(T2[0], T2[1], T2[2], T1[i], T1[(i+1)%3], P)) return true; } return false;}//空间两直线上最近点对 返回最近距离 点对保存在ans1 ans2中double SegSegDistance(Point3 a1, Point3 b1, Point3 a2, Point b2){ Vector v1 = (a1-b1), v2 = (a2-b2); Vector N = Cross(v1, v2); Vector ab = (a1-a2); double ans = Dot(N, ab) / Length(N); Point p1 = a1, p2 = a2; Vector d1 = b1-a1, d2 = b2-a2; double t1, t2; t1 = Dot((Cross(p2-p1, d2)), Cross(d1, d2)); t2 = Dot((Cross(p2-p1, d1)), Cross(d1, d2)); double dd = Length((Cross(d1, d2))); t1 /= dd*dd; t2 /= dd*dd; ans1 = (a1 + (b1-a1)*t1); ans2 = (a2 + (b2-a2)*t2); return fabs(ans);}// 判断P是否在三角形A, B, C中，并且到三条边的距离都至少为mindist。保证P, A, B, C共面bool InsideWithMinDistance(const Point3& P, const Point3& A, const Point3& B, const Point3& C, double mindist) { if(!PointInTri(P, A, B, C)) return false; if(DistanceToLine(P, A, B) < mindist) return false; if(DistanceToLine(P, B, C) < mindist) return false; if(DistanceToLine(P, C, A) < mindist) return false; return true;}// 判断P是否在凸四边形ABCD（顺时针或逆时针）中，并且到四条边的距离都至少为mindist。保证P, A, B, C, D共面bool InsideWithMinDistance(const Point3& P, const Point3& A, const Point3& B, const Point3& C, const Point3& D, double mindist) { if(!PointInTri(P, A, B, C)) return false; if(!PointInTri(P, C, D, A)) return false; if(DistanceToLine(P, A, B) < mindist) return false; if(DistanceToLine(P, B, C) < mindist) return false; if(DistanceToLine(P, C, D) < mindist) return false; if(DistanceToLine(P, D, A) < mindist) return false; return true;}/*************凸包相关问题*******************///加干扰double rand01() { return rand() / (double)RAND_MAX; }double randeps() { return (rand01() - 0.5) * eps; }Point3 add_noise(const Point3& p) { return Point3(p.x + randeps(), p.y + randeps(), p.z + randeps());}struct Face { int v[3]; Face(int a, int b, int c) { v[0] = a; v[1] = b; v[2] = c; } Vector3 Normal(const vector<Point3>& P) const { return Cross(P[v[1]]-P[v[0]], P[v[2]]-P[v[0]]); } // f是否能看见P[i] int CanSee(const vector<Point3>& P, int i) const { return Dot(P[i]-P[v[0]], Normal(P)) > 0; }};// 增量法求三维凸包// 注意：没有考虑各种特殊情况（如四点共面）。实践中，请在调用前对输入点进行微小扰动vector<Face> CH3D(const vector<Point3>& P) { int n = P.size(); vector<vector<int> > vis(n); for(int i = 0; i < n; i++) vis[i].resize(n); vector<Face> cur; cur.push_back(Face(0, 1, 2)); // 由于已经进行扰动，前三个点不共线 cur.push_back(Face(2, 1, 0)); for(int i = 3; i < n; i++) { vector<Face> next; // 计算每条边的“左面”的可见性 for(int j = 0; j < cur.size(); j++) { Face& f = cur[j]; int res = f.CanSee(P, i); if(!res) next.push_back(f); for(int k = 0; k < 3; k++) vis[f.v[k]][f.v[(k+1)%3]] = res; } for(int j = 0; j < cur.size(); j++) for(int k = 0; k < 3; k++) { int a = cur[j].v[k], b = cur[j].v[(k+1)%3]; if(vis[a][b] != vis[b][a] && vis[a][b]) // (a,b)是分界线，左边对P[i]可见 next.push_back(Face(a, b, i)); } cur = next; } return cur;}struct ConvexPolyhedron { int n; vector<Point3> P, P2; vector<Face> faces; bool read() { if(scanf("%d", &n) != 1) return false; P.resize(n); P2.resize(n); for(int i = 0; i < n; i++) { P[i] = read_point3(); P2[i] = add_noise(P[i]); } faces = CH3D(P2); return true; } //三维凸包重心 Point3 centroid() { Point3 C = P[0]; double totv = 0; Point3 tot(0,0,0); for(int i = 0; i < faces.size(); i++) { Point3 p1 = P[faces[i].v[0]], p2 = P[faces[i].v[1]], p3 = P[faces[i].v[2]]; double v = -Volume6(p1, p2, p3, C); totv += v; tot = tot + Centroid(p1, p2, p3, C)*v; } return tot / totv; } //凸包重心到表面最近距离 double mindist(Point3 C) { double ans = 1e30; for(int i = 0; i < faces.size(); i++) { Point3 p1 = P[faces[i].v[0]], p2 = P[faces[i].v[1]], p3 = P[faces[i].v[2]]; ans = min(ans, fabs(-Volume6(p1, p2, p3, C) / Area2(p1, p2, p3))); } return ans; }};//三维凸包struct Point { double x, y, z; Point(double x = 0, double y = 0, double z = 0) : x(x), y(y), z(z) {} inline void read() { scanf("%lf%lf%lf", &x, &y, &z); } //两向量之差 inline Point operator- (Point p) { return Point(x - p.x, y - p.y, z - p.z); } //两向量之和 inline Point operator+ (Point p) { return Point(x + p.x, y + p.y, z + p.z); } //叉乘 inline Point operator* (Point p) { return Point(y * p.z - z * p.y, z * p.x - x * p.z, x * p.y - y * p.x); } inline Point operator* (double d) { return Point(x * d, y * d, z * d); } inline Point operator/ (double d) { return Point(x / d, y / d, z / d); } //点乘 inline double operator^ (Point p) { return (x * p.x + y * p.y + z * p.z); } }; struct CH3D { struct face { //表示凸包一个面上的三个点的编号 int a,b,c; //表示该面是否属于最终凸包上的面 bool ok; }; //初始顶点数 int n; //初始顶点 Point P[MAXN]; //凸包表面的三角形数 int num; //凸包表面的三角形 face F[8*MAXN]; //凸包表面的三角形 int g[MAXN][MAXN]; //向量长度 inline double Length(Point a) { return sqrt(a.x * a.x + a.y * a.y + a.z * a.z); } //叉乘 inline Point cross(Point a, Point b, Point c) { return Point((b.y - a.y) * (c.z - a.z) - (b.z - a.z) * (c.y - a.y), (b.z - a.z) * (c.x - a.x) - (b.x - a.x) * (c.z - a.z), (b.x - a.x) * (c.y - a.y) - (b.y - a.y) * (c.x - a.x) ); } //三角形面积*2 inline double area(Point a, Point b, Point c) { return Length((b - a) * (c - a)); } //四面体有向体积*6 inline double volume(Point a, Point b, Point c, Point d) { return (b - a) * (c - a) ^ (d - a); } //正：点在面同向 inline double dblcmp(Point p, face f) { Point m = P[f.b] - P[f.a]; Point n = P[f.c] - P[f.a]; Point t = p - P[f.a]; return (m * n) ^ t; } void deal(int p, int a, int b) { int f = g[a][b];//搜索与该边相邻的另一个平面 face add; if(F[f].ok) { if(dblcmp(P[p],F[f])>eps) dfs(p,f); else { add.a = b; add.b = a; add.c = p;//这里注意顺序，要成右手系 add.ok = true; g[p][b] = g[a][p] = g[b][a] = num; F[num++] = add; } } } void dfs(int p, int now)//递归搜索所有应该从凸包内删除的面 { F[now].ok = 0; deal(p,F[now].b, F[now].a); deal(p,F[now].c, F[now].b); deal(p,F[now].a, F[now].c); } bool same(int s, int t) { Point &a = P[F[s].a]; Point &b = P[F[s].b]; Point &c = P[F[s].c]; return fabs(volume(a, b, c, P[F[t].a])) < eps && fabs(volume(a, b, c, P[F[t].b])) < eps && fabs(volume(a, b, c, P[F[t].c])) < eps; } //构建三维凸包 void create() { face add; num = 0; if(n < 4) return; //********************************************** //此段是为了保证前四个点不共面 bool flag = true; FF(i, 1, n) { if(Length(P[0] - P[i]) > eps) { swap(P[1], P[i]); flag=false; break; } } if(flag) return; flag = true; //使前三个点不共线 FF(i, 2, n) { if(Length((P[0] - P[1]) * (P[1] - P[i])) > eps) { swap(P[2], P[i]); flag = false; break; } } if(flag) return; flag = true; //使前四个点不共面 FF(i, 3, n) { if(fabs((P[0] - P[1]) * (P[1] - P[2]) ^ (P[0] - P[i])) > eps) { swap(P[3], P[i]); flag = false; break; } } if(flag) return; //***************************************** REP(i, 4) { add.a = (i + 1) % 4; add.b = (i + 2) % 4; add.c = (i + 3) % 4; add.ok = true; if(dblcmp(P[i], add) > 0) swap(add.b, add.c); g[add.a][add.b] = g[add.b][add.c] = g[add.c][add.a] = num; F[num++] = add; } FF(i, 4, n) { REP(j, num) { if(F[j].ok && dblcmp(P[i],F[j]) > eps) { dfs(i, j); break; } } } int tmp = num; num = 0; REP(i, tmp) if(F[i].ok) F[num++] = F[i]; } //表面积 double area() { double res = 0; if(n == 3) { Point p = cross(P[0], P[1], P[2]); res = Length(p) / 2.0; return res; } REP(i, num) res += area(P[F[i].a], P[F[i].b], P[F[i].c]); return res / 2.0; } double volume() { double res = 0; Point tmp(0, 0, 0); REP(i, num) res += volume(tmp, P[F[i].a], P[F[i].b], P[F[i].c]); return fabs(res / 6.0); } //表面三角形个数 inline int triangle() { return num; } //表面多边形个数 int polygon() { int res = 0, flag; REP(i, num) { flag = 1; REP(j, i) if(same(i, j)) { flag = 0; break; } res += flag; } return res; } //三维凸包重心 Point barycenter() { Point ans(0,0,0), t(0,0,0); double all = 0, vol; REP(i, num) { vol = volume(t, P[F[i].a], P[F[i].b], P[F[i].c]); ans = ans + (t + P[F[i].a] + P[F[i].b] + P[F[i].c]) / 4.0 * vol; all += vol; } return ans / all; } //点到面的距离 inline double ptoface(Point p, int i) { double Len = Length((P[F[i].b] - P[F[i].a]) * (P[F[i].c] - P[F[i].a])); return fabs(volume(P[F[i].a], P[F[i].b], P[F[i].c],p) / Len); } } hull;
