几何问题。点线向量面积等模版。

<p>首先是我超的模版http://blog.csdn.net/crescent__moon/article/details/9564511</p><p>包括运算符重载，点积叉积等内容。</p><p>后面是补充。</p><p>#include <iostream>  </p>#include <math.h>  #include <iomanip>      #define eps 1e-8  #define zero(x) (((x)>0?(x):-(x))<eps)    #define pi acos(-1.0)        struct point  {      double x, y;  };    struct line  {      point a, b;  };  struct point3  {      double x, y, z;  };  struct line3  {      point3 a, b;  };  struct plane3  {      point3 a, b, c;  };      //计算cross product (P1-P0)x(P2-P0)  double xmult(point p1, point p2, point p0)  {      return (p1.x - p0.x)*(p2.y - p0.y) - (p2.x - p0.x)*(p1.y - p0.y);  }  //计算dot product (P1-P0).(P2-P0)  double dmult(point p1, point p2, point p0)  {      return (p1.x - p0.x)*(p2.x - p0.x) + (p1.y - p0.y)*(p2.y - p0.y);  }  //计算cross product U . V  point3 xmult(point3 u, point3 v)  {      point3 ret;      ret.x = u.y*v.z - v.y*u.z;      ret.y = u.z*v.x - u.x*v.z;      ret.z = u.x*v.y - u.y*v.x;      return ret;  }  //计算dot product U . V  double dmult(point3 u, point3 v)  {      return u.x*v.x + u.y*v.y + u.z*v.z;  }      //两点距离  double distance(point p1, point p2)  {      return sqrt((p1.x - p2.x)*(p1.x - p2.x) + (p1.y - p2.y)*(p1.y - p2.y));  }    //判三点共线  bool dots_inline(point p1, point p2, point p3)  {      return zero(xmult(p1, p2, p3));  }    //判点是否在线段上,包括端点  bool dot_online_in(point p, line l)  {      return zero(xmult(p, l.a, l.b)) && (l.a.x - p.x)*(l.b.x - p.x) < eps && (l.a.y - p.y)*(l.b.y - p.y) < eps;  }    //判点是否在线段上,不包括端点  bool dot_online_ex(point p, line l)  {      return dot_online_in(p, l) && (!zero(p.x - l.a.x) || !zero(p.y - l.a.y)) && (!zero(p.x - l.b.x) || !zero(p.y - l.b.y));  }    //判两点在线段同侧,点在线段上返回0  bool same_side(point p1, point p2, line l)  {      return xmult(l.a, p1, l.b)*xmult(l.a, p2, l.b) > eps;  }    //判两点在线段异侧,点在线段上返回0  bool opposite_side(point p1, point p2, line l)  {      return xmult(l.a, p1, l.b)*xmult(l.a, p2, l.b) < -eps;  }    //判两直线平行  bool parallel(line u, line v)  {      return zero((u.a.x - u.b.x)*(v.a.y - v.b.y) - (v.a.x - v.b.x)*(u.a.y - u.b.y));  }    //判两直线垂直  bool perpendicular(line u, line v)  {      return zero((u.a.x - u.b.x)*(v.a.x - v.b.x) + (u.a.y - u.b.y)*(v.a.y - v.b.y));  }    //判两线段相交,包括端点和部分重合  bool intersect_in(line u, line v)  {      if (!dots_inline(u.a, u.b, v.a) || !dots_inline(u.a, u.b, v.b))          return !same_side(u.a, u.b, v) && !same_side(v.a, v.b, u);      return dot_online_in(u.a, v) || dot_online_in(u.b, v) || dot_online_in(v.a, u) || dot_online_in(v.b, u);  }    //判两线段相交,不包括端点和部分重合  bool intersect_ex(line u, line v)  {      return opposite_side(u.a, u.b, v) && opposite_side(v.a, v.b, u);  }    //计算两直线交点,注意事先判断直线是否平行!  //线段交点请另外判线段相交(同时还是要判断是否平行!)  point intersection(line u, line v)  {      point ret = u.a;      double t = ((u.a.x - v.a.x)*(v.a.y - v.b.y) - (u.a.y - v.a.y)*(v.a.x - v.b.x)) / ((u.a.x - u.b.x)*(v.a.y - v.b.y) - (u.a.y - u.b.y)*(v.a.x - v.b.x));      ret.x += (u.b.x - u.a.x)*t;      ret.y += (u.b.y - u.a.y)*t;      return ret;  }  point intersection(point u1, point u2, point v1, point v2)  {      point ret = u1;      double t = ((u1.x - v1.x)*(v1.y - v2.y) - (u1.y - v1.y)*(v1.x - v2.x)) / ((u1.x - u2.x)*(v1.y - v2.y) - (u1.y - u2.y)*(v1.x - v2.x));      ret.x += (u2.x - u1.x)*t;      ret.y += (u2.y - u1.y)*t;      return ret;  }  //点到直线上的最近点  point ptoline(point p, line l)  {      point t = p;      t.x += l.a.y - l.b.y, t.y += l.b.x - l.a.x;      return intersection(p, t, l.a, l.b);  }    //点到直线距离  double disptoline(point p, line l)  {      return fabs(xmult(p, l.a, l.b)) / distance(l.a, l.b);  }    //点到线段上的最近点  point ptoseg(point p, line l)  {      point t = p;      t.x += l.a.y - l.b.y, t.y += l.b.x - l.a.x;      if (xmult(l.a, t, p)*xmult(l.b, t, p) > eps)          return distance(p, l.a) < distance(p, l.b) ? l.a : l.b;      return intersection(p, t, l.a, l.b);  }    //点到线段距离  double disptoseg(point p, line l)  {      point t = p;      t.x += l.a.y - l.b.y, t.y += l.b.x - l.a.x;      if (xmult(l.a, t, p)*xmult(l.b, t, p) > eps)          return distance(p, l.a) < distance(p, l.b) ? distance(p, l.a) : distance(p, l.b);      return fabs(xmult(p, l.a, l.b)) / distance(l.a, l.b);  }    //矢量V 以P 为顶点逆时针旋转angle 并放大scale 倍  point rotate(point v, point p, double angle, double scale)  {      point ret = p;      v.x -= p.x, v.y -= p.y;      p.x = scale*cos(angle);      p.y = scale*sin(angle);      ret.x += v.x*p.x - v.y*p.y;      ret.y += v.x*p.y + v.y*p.x;      return ret;  }    //计算三角形面积,输入三顶点  double area_triangle(point p1, point p2, point p3)  {      return fabs(xmult(p1, p2, p3)) / 2;  }    //计算三角形面积,输入三边长  double area_triangle(double a, double b, double c)  {      double s = (a + b + c) / 2;      return sqrt(s*(s - a)*(s - b)*(s - c));  }    //计算多边形面积,顶点按顺时针或逆时针给出  double area_polygon(int n, point* p)  {      double s1 = 0, s2 = 0;      int i;      for (i = 0; i < n; i++)          s1 += p[(i + 1)%n].y*p[i].x, s2 += p[(i + 1)%n].y*p[(i + 2)%n].x;      return fabs(s1 - s2) / 2;  }    //计算圆心角lat 表示纬度,-90<=w<=90,lng 表示经度  //返回两点所在大圆劣弧对应圆心角,0<=angle<=pi  double angle(double lng1, double lat1, double lng2, double lat2)  {      double dlng = fabs(lng1 - lng2)*pi / 180;      while (dlng >= pi + pi)          dlng -= pi + pi;      if (dlng > pi)          dlng = pi + pi - dlng;      lat1 *= pi / 180, lat2 *= pi / 180;      return acos(cos(lat1)*cos(lat2)*cos(dlng) + sin(lat1)*sin(lat2));  }    //计算距离,r 为球半径  double line_dist(double r, double lng1, double lat1, double lng2, double lat2)  {      double dlng = fabs(lng1 - lng2)*pi / 180;      while (dlng >= pi + pi)          dlng -= pi + pi;      if (dlng > pi)          dlng = pi + pi - dlng;      lat1 *= pi / 180, lat2 *= pi / 180;      return r*sqrt(2 - 2 * (cos(lat1)*cos(lat2)*cos(dlng) + sin(lat1)*sin(lat2)));  }    //计算球面距离,r 为球半径  inline double sphere_dist(double r, double lng1, double lat1, double lng2, double lat2)  {      return r*angle(lng1, lat1, lng2, lat2);  }    //外心  point circumcenter(point a, point b, point c)  {      line u, v;      u.a.x = (a.x + b.x) / 2;      u.a.y = (a.y + b.y) / 2;      u.b.x = u.a.x - a.y + b.y;      u.b.y = u.a.y + a.x - b.x;      v.a.x = (a.x + c.x) / 2;      v.a.y = (a.y + c.y) / 2;      v.b.x = v.a.x - a.y + c.y;      v.b.y = v.a.y + a.x - c.x;      return intersection(u, v);  }    //内心  point incenter(point a, point b, point c)  {      line u, v;      double m, n;      u.a = a;      m = atan2(b.y - a.y, b.x - a.x);      n = atan2(c.y - a.y, c.x - a.x);      u.b.x = u.a.x + cos((m + n) / 2);      u.b.y = u.a.y + sin((m + n) / 2);      v.a = b;      m = atan2(a.y - b.y, a.x - b.x);      n = atan2(c.y - b.y, c.x - b.x);      v.b.x = v.a.x + cos((m + n) / 2);      v.b.y = v.a.y + sin((m + n) / 2);      return intersection(u, v);  }    //垂心  point perpencenter(point a, point b, point c)  {      line u, v;      u.a = c;      u.b.x = u.a.x - a.y + b.y;      u.b.y = u.a.y + a.x - b.x;      v.a = b;      v.b.x = v.a.x - a.y + c.y;      v.b.y = v.a.y + a.x - c.x;      return intersection(u, v);  }  //重心  //到三角形三顶点距离的平方和最小的点  //三角形内到三边距离之积最大的点  point barycenter(point a, point b, point c)  {      line u, v;      u.a.x = (a.x + b.x) / 2;      u.a.y = (a.y + b.y) / 2;      u.b = c;      v.a.x = (a.x + c.x) / 2;      v.a.y = (a.y + c.y) / 2;      v.b = b;      return intersection(u, v);  }    //费马点  //到三角形三顶点距离之和最小的点  point fermentpoint(point a, point b, point c)  {      point u, v;      double step = fabs(a.x) + fabs(a.y) + fabs(b.x) + fabs(b.y) + fabs(c.x) + fabs(c.y);      int i, j, k;      u.x = (a.x + b.x + c.x) / 3;      u.y = (a.y + b.y + c.y) / 3;      while (step > 1e-10)      {          for (k = 0; k < 10; step /= 2, k++)          {              for (i = -1; i <= 1; i++)              {                  for (j = -1; j <= 1; j++)                  {                      v.x = u.x + step*i;                      v.y = u.y + step*j;                      if (distance(u, a) + distance(u, b) + distance(u, c) > distance(v, a) + distance(v, b) + distance(v, c))                      {                          u = v;                      }                  }              }          }      }      return u;  }    //矢量差 U - V  point3 subt(point3 u, point3 v)  {      point3 ret;      ret.x = u.x - v.x;      ret.y = u.y - v.y;      ret.z = u.z - v.z;      return ret;  }    ///三维///  //取平面法向量  point3 pvec(plane3 s)  {      return xmult(subt(s.a, s.b), subt(s.b, s.c));  }  point3 pvec(point3 s1, point3 s2, point3 s3)  {      return xmult(subt(s1, s2), subt(s2, s3));  }    //两点距离,单参数取向量大小  double distance(point3 p1, point3 p2)  {      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));  }    //向量大小  double vlen(point3 p)  {      return sqrt(p.x*p.x + p.y*p.y + p.z*p.z);  }    //判三点共线  bool dots_inline(point3 p1, point3 p2, point3 p3)  {      return vlen(xmult(subt(p1, p2), subt(p2, p3))) < eps;  }    //判四点共面  bool dots_onplane(point3 a, point3 b, point3 c, point3 d)  {      return zero(dmult(pvec(a, b, c), subt(d, a)));  }    //判点是否在线段上,包括端点和共线  bool dot_online_in(point3 p, line3 l)  {      return zero(vlen(xmult(subt(p, l.a), subt(p, l.b)))) && (l.a.x - p.x)*(l.b.x - p.x) < eps && (l.a.y - p.y)*(l.b.y - p.y) < eps && (l.a.z - p.z)*(l.b.z - p.z) < eps;  }    //判点是否在线段上,不包括端点  bool dot_online_ex(point3 p, line3 l)  {      return dot_online_in(p, l) && (!zero(p.x - l.a.x) || !zero(p.y - l.a.y) || !zero(p.z - l.a.z)) && (!zero(p.x - l.b.x) || !zero(p.y - l.b.y) || !zero(p.z - l.b.z));  }    //判点是否在空间三角形上,包括边界,三点共线无意义  bool dot_inplane_in(point3 p, plane3 s)  {      return zero(vlen(xmult(subt(s.a, s.b), subt(s.a, s.c))) - vlen(xmult(subt(p, s.a), subt(p, s.b))) - vlen(xmult(subt(p, s.b), subt(p, s.c))) - vlen(xmult(subt(p, s.c), subt(p, s.a))));  }    //判点是否在空间三角形上,不包括边界,三点共线无意义  bool dot_inplane_ex(point3 p, plane3 s)  {      return dot_inplane_in(p, s) && vlen(xmult(subt(p, s.a), subt(p, s.b))) > eps && vlen(xmult(subt(p, s.b), subt(p, s.c))) > eps && vlen(xmult(subt(p, s.c), subt(p, s.a))) > eps;  }    //判两点在线段同侧,点在线段上返回0,不共面无意义  bool same_side(point3 p1, point3 p2, line3 l)  {      return dmult(xmult(subt(l.a, l.b), subt(p1, l.b)), xmult(subt(l.a, l.b), subt(p2, l.b))) > eps;  }    //判两点在线段异侧,点在线段上返回0,不共面无意义  bool opposite_side(point3 p1, point3 p2, line3 l)  {      return dmult(xmult(subt(l.a, l.b), subt(p1, l.b)), xmult(subt(l.a, l.b), subt(p2, l.b))) < -eps;  }    //判两点在平面同侧,点在平面上返回0  bool same_side(point3 p1, point3 p2, plane3 s)  {      return dmult(pvec(s), subt(p1, s.a))*dmult(pvec(s), subt(p2, s.a)) > eps;  }  bool same_side(point3 p1, point3 p2, point3 s1, point3 s2, point3 s3)  {      return dmult(pvec(s1, s2, s3), subt(p1, s1))*dmult(pvec(s1, s2, s3), subt(p2, s1)) > eps;  }    //判两点在平面异侧,点在平面上返回0  bool opposite_side(point3 p1, point3 p2, plane3 s)  {      return dmult(pvec(s), subt(p1, s.a))*dmult(pvec(s), subt(p2, s.a)) < -eps;  }  bool opposite_side(point3 p1, point3 p2, point3 s1, point3 s2, point3 s3)  {      return dmult(pvec(s1, s2, s3), subt(p1, s1))*dmult(pvec(s1, s2, s3), subt(p2, s1)) < -eps;  }  //判两直线平行  bool parallel(line3 u, line3 v)  {      return vlen(xmult(subt(u.a, u.b), subt(v.a, v.b))) < eps;  }    //判两平面平行  bool parallel(plane3 u, plane3 v)  {      return vlen(xmult(pvec(u), pvec(v))) < eps;  }    //判直线与平面平行  bool parallel(line3 l, plane3 s)  {      return zero(dmult(subt(l.a, l.b), pvec(s)));  }  bool parallel(point3 l1, point3 l2, point3 s1, point3 s2, point3 s3)  {      return zero(dmult(subt(l1, l2), pvec(s1, s2, s3)));  }    //判两直线垂直  bool perpendicular(line3 u, line3 v)  {      return zero(dmult(subt(u.a, u.b), subt(v.a, v.b)));  }    //判两平面垂直  bool perpendicular(plane3 u, plane3 v)  {      return zero(dmult(pvec(u), pvec(v)));  }    //判直线与平面平行  bool perpendicular(line3 l, plane3 s)  {      return vlen(xmult(subt(l.a, l.b), pvec(s))) < eps;  }    //判两线段相交,包括端点和部分重合  bool intersect_in(line3 u, line3 v)  {      if (!dots_onplane(u.a, u.b, v.a, v.b))          return 0;      if (!dots_inline(u.a, u.b, v.a) || !dots_inline(u.a, u.b, v.b))          return !same_side(u.a, u.b, v) && !same_side(v.a, v.b, u);      return dot_online_in(u.a, v) || dot_online_in(u.b, v) || dot_online_in(v.a, u) || dot_online_in(v.b, u);  }    //判两线段相交,不包括端点和部分重合  bool intersect_ex(line3 u, line3 v)  {      return dots_onplane(u.a, u.b, v.a, v.b) && opposite_side(u.a, u.b, v) && opposite_side(v.a, v.b, u);  }    //判线段与空间三角形相交,包括交于边界和(部分)包含  bool intersect_in(line3 l, plane3 s)  {      return !same_side(l.a, l.b, s) && !same_side(s.a, s.b, l.a, l.b, s.c) && !same_side(s.b, s.c, l.a, l.b, s.a) && !same_side(s.c, s.a, l.a, l.b, s.b);  }    //判线段与空间三角形相交,不包括交于边界和(部分)包含  bool intersect_ex(line3 l, plane3 s)  {      return opposite_side(l.a, l.b, s) && opposite_side(s.a, s.b, l.a, l.b, s.c) && opposite_side(s.b, s.c, l.a, l.b, s.a) && opposite_side(s.c, s.a, l.a, l.b, s.b);  }    //计算两直线交点,注意事先判断直线是否共面和平行!  //线段交点请另外判线段相交(同时还是要判断是否平行!)  point3 intersection(line3 u, line3 v)  {      point3 ret = u.a;      double t = ((u.a.x - v.a.x)*(v.a.y - v.b.y) - (u.a.y - v.a.y)*(v.a.x - v.b.x))          / ((u.a.x - u.b.x)*(v.a.y - v.b.y) - (u.a.y - u.b.y)*(v.a.x - v.b.x));      ret.x += (u.b.x - u.a.x)*t;      ret.y += (u.b.y - u.a.y)*t;      ret.z += (u.b.z - u.a.z)*t;      return ret;  }    //计算直线与平面交点,注意事先判断是否平行,并保证三点不共线!  //线段和空间三角形交点请另外判断  point3 intersection(line3 l, plane3 s)  {      point3 ret = pvec(s);      double t = (ret.x*(s.a.x - l.a.x) + ret.y*(s.a.y - l.a.y) + ret.z*(s.a.z - l.a.z)) / (ret.x*(l.b.x - l.a.x) + ret.y*(l.b.y - l.a.y) + ret.z*(l.b.z - l.a.z));      ret.x = l.a.x + (l.b.x - l.a.x)*t;      ret.y = l.a.y + (l.b.y - l.a.y)*t;      ret.z = l.a.z + (l.b.z - l.a.z)*t;      return ret;  }  point3 intersection(point3 l1, point3 l2, point3 s1, point3 s2, point3 s3)  {      point3 ret = pvec(s1, s2, s3);      double t = (ret.x*(s1.x - l1.x) + ret.y*(s1.y - l1.y) + ret.z*(s1.z - l1.z)) /          (ret.x*(l2.x - l1.x) + ret.y*(l2.y - l1.y) + ret.z*(l2.z - l1.z));      ret.x = l1.x + (l2.x - l1.x)*t;      ret.y = l1.y + (l2.y - l1.y)*t;      ret.z = l1.z + (l2.z - l1.z)*t;      return ret;  }    //计算两平面交线,注意事先判断是否平行,并保证三点不共线!  line3 intersection(plane3 u, plane3 v)  {      line3 ret;      ret.a = parallel(v.a, v.b, u.a, u.b, u.c) ? intersection(v.b, v.c, u.a, u.b, u.c) : intersection(v.a, v.b, u.a, u.b, u.c);      ret.b = parallel(v.c, v.a, u.a, u.b, u.c) ? intersection(v.b, v.c, u.a, u.b, u.c) : intersection(v.c, v.a, u.a, u.b, u.c);      return ret;  }  line3 intersection(point3 u1, point3 u2, point3 u3, point3 v1, point3 v2, point3 v3)  {      line3 ret;      ret.a = parallel(v1, v2, u1, u2, u3) ? intersection(v2, v3, u1, u2, u3) : intersection(v1, v2, u1, u2, u3);      ret.b = parallel(v3, v1, u1, u2, u3) ? intersection(v2, v3, u1, u2, u3) : intersection(v3, v1, u1, u2, u3);      return ret;  }  //点到直线距离  double ptoline(point3 p, line3 l)  {      return vlen(xmult(subt(p, l.a), subt(l.b, l.a))) / distance(l.a, l.b);  }    //点到平面距离  double ptoplane(point3 p, plane3 s)  {      return fabs(dmult(pvec(s), subt(p, s.a))) / vlen(pvec(s));  }    //直线到直线距离  double linetoline(line3 u, line3 v)  {      point3 n = xmult(subt(u.a, u.b), subt(v.a, v.b));      return fabs(dmult(subt(u.a, v.a), n)) / vlen(n);  }    //两直线夹角cos 值  double angle_cos(line3 u, line3 v)  {      return dmult(subt(u.a, u.b), subt(v.a, v.b)) / vlen(subt(u.a, u.b)) / vlen(subt(v.a, v.b));  }    //两平面夹角cos 值  double angle_cos(plane3 u, plane3 v)  {      return dmult(pvec(u), pvec(v)) / vlen(pvec(u)) / vlen(pvec(v));  }    //直线平面夹角sin 值  double angle_sin(line3 l, plane3 s)  {      return dmult(subt(l.a, l.b), pvec(s)) / vlen(subt(l.a, l.b)) / vlen(pvec(s));  }    int main()  {      int t;      std::cin >> t;      std::cout << "INTERSECTING LINES OUTPUT" << std::endl;      while (t--)      {          line a, b;          std::cin >> a.a.x >> a.a.y >> a.b.x >> a.b.y >> b.a.x >> b.a.y >> b.b.x >> b.b.y;            if (xmult(a.a, a.b, b.a) == 0 && xmult(a.a, a.b, b.b) == 0)          {              std::cout << "LINE" << std::endl;          }          else if (parallel(a, b))          {              std::cout << "NONE" << std::endl;          }          else          {              point temp;              temp = intersection(a, b);              std::cout << std::fixed << std::setprecision(2) << "POINT" << ' ' << temp.x << ' ' << temp.y << std::endl;          }      }      std::cout << "END OF OUTPUT" << std::endl;  }