《训练指南》上的计算几何模板

不完整，待补充

#include <cstdio>

#include <cmath>

#include <algorithm>



using namespace std;



const double eps = 1e-10;



struct Point

{

    double x, y;

    Point( double x = 0, double y = 0 ):x(x), y(y) { }

};



typedef Point Vector;



Vector operator+( Vector A, Vector B )       //向量加

{

    return Vector( A.x + B.x, A.y + B.y );

}



Vector operator-( Vector A, Vector B )       //向量减

{

    return Vector( A.x - B.x, A.y - B.y );

}



Vector operator*( Vector A, double p )      //向量数乘

{

    return Vector( A.x * p, A.y * p );

}



Vector operator/( Vector A, double p )      //向量数除

{

    return Vector( A.x / p, A.y / p );

}



int dcmp( double x )    //控制精度

{

    if ( fabs(x) < eps ) return 0;

    else return x < 0 ? -1 : 1;

}



bool operator<( const Point& A, const Point& B )   //两点比较小于

{

    return dcmp( A.x - B.x) < 0 || ( dcmp(A.x - B.x ) == 0 && dcmp( A.y - B.y ) < 0 );

}



bool operator>( const Point& A, const Point& B )   //两点比较大于

{

    return dcmp( A.x - B.x) > 0 || ( dcmp(A.x - B.x ) == 0 && dcmp( A.y - B.y ) > 0 );

}



bool operator==( const Point& a, const Point& b )   //两点相等

{

    return dcmp( a.x - b.x ) == 0 && dcmp( a.y - b.y ) == 0;

}



double Dot( Vector A, Vector B )    //向量点乘

{

    return A.x * B.x + A.y * B.y;

}



double Length( Vector A )           //向量模

{

    return sqrt( Dot( A, A ) );

}



double Angle( Vector A, Vector B )    //向量夹角

{

    return acos( Dot(A, B) / Length(A) / Length(B) );

}



double Cross( Vector A, Vector B )   //向量叉积

{

    return A.x * B.y - A.y * B.x;

}



double Area2( Point A, Point B, Point C )    //向量有向面积

{

    return Cross( B - A, C - A );

}



Vector Rotate( Vector A, double rad )    //向量旋转

{

    return Vector( A.x * cos(rad) - A.y * sin(rad), A.x * sin(rad) + A.y * cos(rad) );

}



Vector Normal( Vector A )    //向量单位法向量

{

    double L = Length(A);

    return Vector( -A.y / L, A.x / L );

}



Point GetLineIntersection( Point P, Vector v, Point Q, Vector w )   //两直线交点

{

    Vector u = P - Q;

    double t = Cross( w, u ) / Cross( v, w );

    return P + v * t;

}



double DistanceToLine( Point P, Point A, Point B )    //点到直线的距离

{

    Vector v1 = B - A, v2 = P - A;

    return fabs( Cross( v1, v2 ) ) / Length(v1);

}



double DistanceToSegment( Point P, Point A, Point B )   //点到线段的距离

{

    if ( A == B ) return Length( P - A );

    Vector v1 = B - A, v2 = P - A, v3 = P - B;

    if ( dcmp( Dot(v1, v2) ) < 0 ) return Length(v2);

    else if ( dcmp( Dot(v1, v3) ) > 0 ) return Length(v3);

    else return fabs( Cross( v1, v2 ) ) / Length(v1);

}



Point GetLineProjection( Point P, Point A, Point B )    // 点在直线上的投影

{

    Vector v = B - A;

    return A + v*( Dot(v, P - A) / Dot( v, v ) );

}



bool SegmentProperIntersection( Point a1, Point a2, Point b1, Point b2 )  //线段相交，交点不在端点

{

    double c1 = Cross( a2 - a1, b1 - a1 ), c2 = Cross( a2 - a1, b2 - a1 ),

           c3 = Cross( b2 - b1, a1 - b1 ), c4 = Cross( b2 - b1, a2 - b1 );

    return dcmp(c1)*dcmp(c2) < 0 && dcmp(c3) * dcmp(c4) < 0;

}



bool OnSegment( Point p, Point a1, Point a2 )   //点在线段上，不包含端点

{

    return dcmp( Cross(a1 - p, a2 - p) ) == 0 && dcmp( Dot( a1 - p, a2 - p ) ) < 0;

}



double toRad( double deg )   //角度转弧度

{

    return deg / 180.0 * acos( -1.0 );

}



int ConvexHull( Point *p, int n, Point *ch )    //求凸包，卷包裹法，O(n2)

{

    sort( p, p + n );

    int m = 0;

    for ( int i = 0; i < n; ++i )

    {

        while ( m > 1 && Cross( ch[m - 1] - ch[m - 2], p[i] - ch[m - 2] ) <= 0 ) --m;

        ch[m++] = p[i];

    }



    int k = m;

    for ( int i = n - 2; i >= 0; --i )

    {

        while ( m > k && Cross( ch[m - 1] - ch[m - 2], p[i] - ch[m - 2] ) <= 0 ) --m;

        ch[m++] = p[i];

    }



    if ( n > 1 ) --m;

    return m;

}



double PolygonArea( Point *p, int n )   //多边形有向面积

{

    double area = 0;

    for ( int i = 1; i < n - 1; ++i )

        area += Cross( p[i] - p[0], p[i + 1] - p[0] );

    return area / 2.0;

}

 graham算法求凸包

//求凸包，graham算法，O(nlogn)，返回凸包点的个数

int graham( Point *p, int n, Point *ch )

{

    if ( n <= 2 ) return 0;

    int top = 0;

    sort( p, p + n );



    ch[ top ] = p[0];

    ch[ ++top ] = p[1];

    ch[ ++top ] = p[2];



    top = 1;



    for ( int i = 2; i < n; ++i )

    {

        while ( top && dcmp( Cross( ch[top] - ch[top - 1], p[i] - ch[top - 1] ) ) <= 0 ) --top;

        ch[++top] = p[i];

    }

    int len = top;

    ch[++top] = p[n - 2];

    for ( int i = n - 3; i >= 0; --i )

    {

        while ( top > len && dcmp( Cross( ch[top] - ch[top - 1], p[i] - ch[top - 1] ) ) <= 0 ) --top;

        ch[++top] = p[i];

    }

    return top;

}

关于圆的一些模板（不完整，待补充）

struct Circle

{

    Point c;   //圆心坐标

    double r;  //半径

    Circle() {}

    Circle( Point c, double r ): c(c), r(r) {}

    Point getPoint( double theta )   //根据极角返回圆上一点的坐标

    {

        return Point( c.x + cos(theta)*r, c.y + sin(theta)*r );

    }

    void readCircle()

    {

        scanf("%lf%lf%lf", &c.x, &c.y, &r );

        return;

    }

};



//过定点做圆的切线,得到切点，返回切点个数

//tps保存切点坐标

int getTangentPoints( Point p, Circle C, Point *tps )

{

    int cnt = 0;



    double dis = sqrt( PointDis( p, C.c ) );

    int aa = dcmp( dis - C.r );

    if ( aa < 0 ) return 0;  //点在圆内

    else if ( aa == 0 ) //点在圆上，该点就是切点

    {

        tps[cnt] = p;

        ++cnt;

        return cnt;

    }



    //点在圆外，有两个切点

    double base = atan2( p.y - C.c.y, p.x - C.c.x );

    double ang = acos( C.r / dis );

    //printf( "base = %f ang=%f\n", base, ang );

    //printf( "base-ang=%f  base+ang=%f \n", base - ang, base + ang );



    tps[cnt] = C.getPoint( base - ang ), ++cnt;

    tps[cnt] = C.getPoint( base + ang ), ++cnt;



    return cnt;

}



//求两圆外公切线切点，返回切线个数

//p是圆c2在圆c1上的切点

int makeCircle( Circle c1, Circle c2, Point *p )

{

    int cnt = 0;

    double d = sqrt( PointDis(c1.c, c2.c) ), dr = c1.r - c2.r;

    double b = acos(dr / d);

    double a = atan2( c2.c.y - c1.c.y, c2.c.x - c1.c.x );

    double a1 = a - b, a2 = a + b;

    p[cnt++] = Point(cos(a1) * c1.r, sin(a1) * c1.r) + c1.c;

    p[cnt++] = Point(cos(a2) * c1.r, sin(a2) * c1.r) + c1.c;

    return cnt;

}



//求三角形的外心

Point GetMid( Point *p )

{

    Point tmp1 = p[0] + ( p[1] - p[0] ) / 2.0;

    Point tmp2 = p[1] + ( p[2] - p[1] ) / 2.0;



    Vector v1 = Normal( p[1] - p[0] );

    Vector v2 = Normal( p[2] - p[1] );



    return GetLineIntersection( tmp1, v1, tmp2, v2 );

}