《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 );
}