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;
}

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