LA 4973异面线段

题目大意：给两条线段求他们间的最小距离的平方（以分数形式输出）。

贴个模版吧！太抽象了。

 

#include<cstdio>
#include<cmath>
#include<algorithm>
using namespace std;

struct Point3 {
  int x, y, z;
  Point3(int x=0, int y=0, int 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, int p) { return Vector3(A.x*p, A.y*p, A.z*p); }

bool operator == (const Point3& a, const Point3& b) {
  return a.x==b.x && a.y==b.y && a.z==b.z;
}

Point3 read_point3() {
  Point3 p;
  scanf("%d%d%d", &p.x, &p.y, &p.z);
  return p;
}

int Dot(const Vector3& A, const Vector3& B) { return A.x*B.x + A.y*B.y + A.z*B.z; }
int Length2(const Vector3& A) { return Dot(A, A); }
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); }

typedef long long LL;

LL gcd(LL a, LL b) { return b ? gcd(b, a%b) : a; }
LL lcm(LL a, LL b) { return a / gcd(a,b) * b; }

struct Rat {
  LL a, b;
  Rat(LL a=0):a(a),b(1) { }
  Rat(LL x, LL y):a(x),b(y) {
    if(b < 0) a = -a, b = -b;
    LL d = gcd(a, b); if(d < 0) d = -d;
    a /= d; b /= d;
  }
};

Rat operator + (const Rat& A, const Rat& B) {
  LL x = lcm(A.b, B.b);
  return Rat(A.a*(x/A.b)+B.a*(x/B.b), x);
}

Rat operator - (const Rat& A, const Rat& B) { return A + Rat(-B.a, B.b); }
Rat operator * (const Rat& A, const Rat& B) { return Rat(A.a*B.a, A.b*B.b); }

void updatemin(Rat& A, const Rat& B) {
  if(A.a*B.b > B.a*A.b) A.a = B.a, A.b = B.b;
}

// 点P到线段AB的距离的平方
Rat Rat_Distance2ToSegment(const Point3& P, const Point3& A, const Point3& B) {
  if(A == B) return Length2(P-A);
  Vector3 v1 = B - A, v2 = P - A, v3 = P - B;
  if(Dot(v1, v2) < 0) return Length2(v2);
  else if(Dot(v1, v3) > 0) return Length2(v3);
  else return Rat(Length2(Cross(v1, v2)), Length2(v1));
}

// 求异面直线p1+su和p2+tv的公垂线对应的s。如果平行/重合，返回false
bool Rat_LineDistance3D(const Point3& p1, const Vector3& u, const Point3& p2, const Vector3& v, Rat& s) {
  LL b = (LL)Dot(u,u)*Dot(v,v) - (LL)Dot(u,v)*Dot(u,v);
  if(b == 0) return false;
  LL a = (LL)Dot(u,v)*Dot(v,p1-p2) - (LL)Dot(v,v)*Dot(u,p1-p2);
  s = Rat(a, b);
  return true;
}

void Rat_GetPointOnLine(const Point3& A, const Point3& B, const Rat& t, Rat& x, Rat& y, Rat& z) {
  x = Rat(A.x) + Rat(B.x-A.x) * t;
  y = Rat(A.y) + Rat(B.y-A.y) * t;
  z = Rat(A.z) + Rat(B.z-A.z) * t;
}

Rat Rat_Distance2(const Rat& x1, const Rat& y1, const Rat& z1, const Rat& x2, const Rat& y2, const Rat& z2) {
  return (x1-x2)*(x1-x2)+(y1-y2)*(y1-y2)+(z1-z2)*(z1-z2);
}

int main() {
  int T;
  scanf("%d", &T);
  LL maxx = 0;
  while(T--) {
    Point3 A = read_point3();
    Point3 B = read_point3();
    Point3 C = read_point3();
    Point3 D = read_point3();
    Rat s, t;
    bool ok = false;
    Rat ans = Rat(1000000000);
    if(Rat_LineDistance3D(A, B-A, C, D-C, s))
      if(s.a > 0 && s.a < s.b && Rat_LineDistance3D(C, D-C, A, B-A, t))
        if(t.a > 0 && t.a < t.b) {
          ok = true; // 异面直线/相交直线
          Rat x1, y1, z1, x2, y2, z2;
          Rat_GetPointOnLine(A, B, s, x1, y1, z1);
          Rat_GetPointOnLine(C, D, t, x2, y2, z2);
          ans = Rat_Distance2(x1, y1, z1, x2, y2, z2);
    }
    if(!ok) { // 平行直线/重合直线
      updatemin(ans, Rat_Distance2ToSegment(A, C, D));
      updatemin(ans, Rat_Distance2ToSegment(B, C, D));
      updatemin(ans, Rat_Distance2ToSegment(C, A, B));
      updatemin(ans, Rat_Distance2ToSegment(D, A, B));
    }
    printf("%lld %lld\n", ans.a, ans.b);
  }
  return 0;
}