分析：计算几何、凸包、旋转卡壳。最小面积矩形一定存在一条边与凸包共线。（不共线时，可以通过旋转变得更小）
根据以上结论，直接枚举底边，然后利用旋转卡壳确定其余三个顶点即可。
如图：L1为底边，初始化确定四个边界点，然后利用向量，控制L2，L3，L4与L1的角度即可。
注意：1.输入点可以小于3个，可能是点或者线段；
2.精度问题，之前用点到直线距离公式WA了N久，换成向量计算就AC了。
#include#include #include #include #include #define esp 1e-6 using namespace std; //点结构 typedef struct pnode { double x,y,d; pnode( double a, double b ){x = a;y = b;} pnode(){} }point; point P[ 1005 ]; //直线结构 typedef struct lnode { double x,y,dx,dy; lnode( point a, point b ){x = a.x;y = a.y;dx = b.x-a.x;dy = b.y-a.y;} lnode(){} }line; //叉乘ab*ac double crossproduct( point a, point b, point c ) { return (b.x-a.x)*(c.y-a.y)-(c.x-a.x)*(b.y-a.y); } //点到点距离 double dist( point a, point b ) { return sqrt((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y)); } //点到直线距离 double dist( point p, point a, point b ) { return fabs(crossproduct( p, a, b )/dist( a, b )); } //坐标排序 bool cmp1( point a, point b ) { return (a.x == b.x) (a.y < b.y):(a.x < b.x); } //级角排序 bool cmp2( point a, point b ) { double cp = crossproduct( P[0], a, b ); if ( cp == 0 ) return a.d < b.d; else return cp > 0; } //叉乘判断平行 double judge1( point a, point b, point c, point d ) { return crossproduct( a, b, point( a.x+d.x-c.x, a.y+d.y-c.y ) ); } //点乘判断垂直 double judge2( point a, point b, point c, point d ) { return ((b.x-a.x)*(d.x-c.x)+(b.y-a.y)*(d.y-c.y)); } double Graham( int n ) { if ( n < 3 ) return 0.0; sort( P+0, P+n, cmp1 ); for ( int i = 1 ; i < n ; ++ i ) P[i].d = dist( P[0], P[i] ); sort( P+1, P+n, cmp2 ); //计算凸包 int top = 1; for ( int i = 2 ; i < n ; ++ i ) { while ( top > 0 && crossproduct( P[top-1], P[top], P[i] ) < esp ) -- top; P[++ top] = P[i]; } P[++ top] = P[0]; if ( top < 3 ) return 0.0; //旋转卡壳 int L2 = 0,L3 = 0,L4 = 0; for ( int i = 0 ; i < top ; ++ i ) { if ( P[i].y <= P[L2].y ) L2 = i; if ( P[i].x >= P[L3].x ) L3 = i; if ( P[i].y >= P[L4].y ) L4 = i; } double Min = (P[0].x-P[L3].x)*(P[L2].y-P[L4].y); for ( int L1 = 0 ; L1 < top ; ++ L1 ) { //旋转平行边 while ( judge1( P[L1], P[L1+1], P[L3], P[L3+1] ) > esp ) L3 = (L3+1)%top; //旋转垂直边 while ( L2 != L3 && judge2( P[L1], P[L1+1], P[L2], P[L2+1] ) > esp ) L2 = (L2+1)%top; while ( L4 != L1 && judge2( P[L1+1], P[L1], P[L4], P[L4+1] ) > esp ) L4 = (L4+1)%top; double D = dist( P[L3], P[L1], P[L1+1] ); double L = dist( P[L2], P[L4], point( P[L4].x-P[L1+1].y+P[L1].y, P[L4].y+P[L1+1].x-P[L1].x ) ); if ( Min > D*L ) Min = D*L; } return Min; } int main() { int n; while ( ~scanf("%d",&n) && n ) { for ( int i = 0 ; i < n ; ++ i ) scanf("%lf%lf",&P[i].x,&P[i].y); printf("%.4lf\n",Graham( n )); } return 0; }