UVa 10256 - The Great Divide 判断凸包相交
模板敲错了于是WA了好几遍……
判断由红点和蓝点分别组成的两个凸包是否相离,是输出Yes,否输出No。
训练指南上的分析:
1.任取红凸包上的一条线段和蓝凸包上的一条线段,判断二者是否相交。如果相交(不一定是规范相交,有公共点就算相交),则无解
2.任取一个红点,判断是否在蓝凸包内。如果是,则无解。蓝点红凸包同理。
其中任何一个凸包退化成点或者线段时需要特判。
其实只需要把上面两个判断顺序颠倒一下,就可以不需要特判。
先判断点是否在凸包内,因为这个考虑了点在凸包边界上的情况,所以后面判凸包线段是否相交时直接用规范相交判断即可。
此时特殊情况包含在了上两种情况中,因此不需要特判。
1 #include <cstdio> 2 #include <cmath> 3 #include <algorithm> 4 5 using namespace std; 6 7 const int MAXN = 540; 8 9 const double eps = 1e-10; 10 11 struct Point 12 { 13 double x, y; 14 Point( double x = 0, double y = 0 ):x(x), y(y) { } 15 }; 16 17 typedef Point Vector; 18 19 Vector operator+( Vector A, Vector B ) //向量加 20 { 21 return Vector( A.x + B.x, A.y + B.y ); 22 } 23 24 Vector operator-( Vector A, Vector B ) //向量减 25 { 26 return Vector( A.x - B.x, A.y - B.y ); 27 } 28 29 Vector operator*( Vector A, double p ) //向量数乘 30 { 31 return Vector( A.x * p, A.y * p ); 32 } 33 34 Vector operator/( Vector A, double p ) //向量数除 35 { 36 return Vector( A.x / p, A.y / p ); 37 } 38 39 bool operator<( const Point& A, const Point& B ) //两点比较 40 { 41 return A.x < B.x || ( A.x == B.x && A.y < B.y ); 42 } 43 44 int dcmp( double x ) //控制精度 45 { 46 if ( fabs(x) < eps ) return 0; 47 else return x < 0 ? -1 : 1; 48 } 49 50 bool operator==( const Point& a, const Point& b ) //两点相等 51 { 52 return dcmp( a.x - b.x ) == 0 && dcmp( a.y - b.y ) == 0; 53 } 54 55 double Dot( Vector A, Vector B ) //向量点乘 56 { 57 return A.x * B.x + A.y * B.y; 58 } 59 60 double Length( Vector A ) //向量模 61 { 62 return sqrt( Dot( A, A ) ); 63 } 64 65 double Angle( Vector A, Vector B ) //向量夹角 66 { 67 return acos( Dot(A, B) / Length(A) / Length(B) ); 68 } 69 70 double Cross( Vector A, Vector B ) //向量叉积 71 { 72 return A.x * B.y - A.y * B.x; 73 } 74 75 double Area2( Point A, Point B, Point C ) //向量有向面积 76 { 77 return Cross( B - A, C - A ); 78 } 79 80 Vector Rotate( Vector A, double rad ) //向量旋转 81 { 82 return Vector( A.x * cos(rad) - A.y * sin(rad), A.x * sin(rad) + A.y * cos(rad) ); 83 } 84 85 Vector Normal( Vector A ) //向量单位法向量 86 { 87 double L = Length(A); 88 return Vector( -A.y / L, A.x / L ); 89 } 90 91 Point GetLineIntersection( Point P, Vector v, Point Q, Vector w ) //两直线交点 92 { 93 Vector u = P - Q; 94 double t = Cross( w, u ) / Cross( v, w ); 95 return P + v * t; 96 } 97 98 double DistanceToLine( Point P, Point A, Point B ) //点到直线的距离 99 { 100 Vector v1 = B - A, v2 = P - A; 101 return fabs( Cross( v1, v2 ) ) / Length(v1); 102 } 103 104 double DistanceToSegment( Point P, Point A, Point B ) //点到线段的距离 105 { 106 if ( A == B ) return Length( P - A ); 107 Vector v1 = B - A, v2 = P - A, v3 = P - B; 108 if ( dcmp( Dot(v1, v2) ) < 0 ) return Length(v2); 109 else if ( dcmp( Dot(v1, v3) ) > 0 ) return Length(v3); 110 else return fabs( Cross( v1, v2 ) ) / Length(v1); 111 } 112 113 Point GetLineProjection( Point P, Point A, Point B ) // 点在直线上的投影 114 { 115 Vector v = B - A; 116 return A + v*( Dot(v, P - A) / Dot( v, v ) ); 117 } 118 119 bool SegmentProperIntersection( Point a1, Point a2, Point b1, Point b2 ) //线段相交,交点不在端点 120 { 121 double c1 = Cross( a2 - a1, b1 - a1 ), c2 = Cross( a2 - a1, b2 - a1 ), 122 c3 = Cross( b2 - b1, a1 - b1 ), c4 = Cross( b2 - b1, a2 - b1 ); 123 return dcmp(c1)*dcmp(c2) < 0 && dcmp(c3) * dcmp(c4) < 0; 124 } 125 126 bool OnSegment( Point p, Point a1, Point a2 ) //点在线段上,不包含端点 127 { 128 return dcmp( Cross(a1 - p, a2 - p) ) == 0 && dcmp( Dot( a1 - p, a2 - p ) ) < 0; 129 } 130 131 double toRad( double deg ) //角度转弧度 132 { 133 return deg / 180.0 * acos( -1.0 ); 134 } 135 136 int ConvexHull( Point *p, int n, Point *ch ) //求凸包 137 { 138 sort( p, p + n ); 139 n = unique( p, p + n ) - p; 140 int m = 0; 141 for ( int i = 0; i < n; ++i ) 142 { 143 while ( m > 1 && Cross( ch[m - 1] - ch[m - 2], p[i] - ch[m - 2] ) <= 0 ) --m; 144 ch[m++] = p[i]; 145 } 146 147 int k = m; 148 for ( int i = n - 2; i >= 0; --i ) 149 { 150 while ( m > k && Cross( ch[m - 1] - ch[m - 2], p[i] - ch[m - 2] ) <= 0 ) --m; 151 ch[m++] = p[i]; 152 } 153 154 if ( n > 1 ) --m; 155 return m; 156 } 157 158 int isPointInPolygon( Point p, Point *poly, int n ) //判断一点是否在凸包内 159 { 160 int wn = 0; 161 162 for ( int i = 0; i < n; ++i ) 163 { 164 Point& p1 = poly[i], p2 = poly[ (i + 1)%n ]; 165 if ( p == p1 || p == p2 || OnSegment( p, p1, p2 ) ) return -1; //在边界上 166 int k = dcmp( Cross( p2 - p1, p - p1 ) ); 167 int d1 = dcmp( p1.y - p.y ); 168 int d2 = dcmp( p2.y - p.y ); 169 if ( k > 0 && d1 <= 0 && d2 > 0 ) ++wn; 170 if ( k < 0 && d2 <= 0 && d1 > 0 ) --wn; 171 } 172 173 if ( wn ) return 1; //内部 174 return 0; //外部 175 } 176 177 double PolygonArea( Point *p, int n ) //多边形有向面积 178 { 179 double area = 0; 180 for ( int i = 1; i < n - 1; ++i ) 181 area += Cross( p[i] - p[0], p[i + 1] - p[0] ); 182 return area / 2.0; 183 } 184 185 bool checkConvexHullIntersection( Point *a, Point *b, int na, int nb ) 186 { 187 for ( int i = 0; i < na; ++i ) 188 if ( isPointInPolygon( a[i], b, nb ) ) return true; 189 190 for ( int i = 0; i < nb; ++i ) 191 if ( isPointInPolygon( b[i], a, na ) ) return true; 192 193 for ( int i = 0; i < na; ++i ) 194 for ( int j = 0; j < nb; ++j ) 195 if ( SegmentProperIntersection(a[i], a[ (i + 1) % na ], b[j], b[ (j + 1) % nb ] ) ) return true; 196 197 return false; 198 } 199 200 Point M[MAXN], chM[MAXN]; 201 Point C[MAXN], chC[MAXN]; 202 203 int main() 204 { 205 int Mn, Cn; 206 while ( scanf( "%d%d", &Mn, &Cn ), Mn || Cn ) 207 { 208 for ( int i = 0; i < Mn; ++i ) 209 scanf( "%lf%lf", &M[i].x, &M[i].y ); 210 211 for ( int i = 0; i < Cn; ++i ) 212 scanf( "%lf%lf", &C[i].x, &C[i].y ); 213 214 int Mcnt = ConvexHull( M, Mn, chM ); 215 int Ccnt = ConvexHull( C, Cn, chC ); 216 217 if ( checkConvexHullIntersection( chM, chC, Mcnt, Ccnt ) ) puts("No"); 218 else puts("Yes"); 219 } 220 return 0; 221 }