二维几何基础

浮点数的比较

//符号函数
int sgn(double x)
{
    if(fabs(x) < eps) return 0;
    else return x<0?-1:1;
}

//浮点数的比较
int cmp(double x,double y)
{
    if(fabs(x-y) < eps) return 0;
    else return x<y?-1:1;
}

点和向量

struct Point
{
    double x,y;
    Point(double x,double y):x(x),y(y){}
};
typedef Point Vector;

向量的运算

struct Point
{
    double x,y;
    Point(double x,double y):x(x),y(y){}
    
    Point operator+(Point a)
    {
        return Point(x+a.x,y+a.y);
    }
    Point operator-(Point a)
    {
        return Point(x-a.x,y-a.y);
    }
    Point operator*(double k)
    {
        return Point(x*k,y*k);
    }
    Point operator/(double k)
    {
        return Point(x/k,y/k);
    }
    bool operator==(Point a)
    {
        return sgn(x-a.x)==0 && sgn(y-a.y)==0;
    }
};
typedef Point Vector;

点积和叉积

//点积
double dot(Vector a,Vector b)
{
    return a.x*b.x+a.y*b.y;
}

//叉积
double cross(Vector a,Vector b)
{
    return a.x*b.y-a.y*b.x;
}

点积的应用

判断\(A\)与\(B\)的夹角是钝角还是锐角。

点积有正负，利用正负号可以判断向量的夹角。

• 若\(dot(A,B)>0\)，\(A\)与\(B\)的夹角为锐角
• 若\(dot(A,B)<0\)，\(A\)与\(B\)的夹角为钝角
• 若\(dot(A,B)=0\)，\(A\)与\(B\)的夹角为直角

//求向量的长度
double len(Vector a)
{
    return sqrt(dot(a,a));
}

//求两向量的夹角
double angle(Vector a,Vector b)
{
    return acos(dot(a,b)/len(a)/len(b));
}

//两向量构成的平行四边形的有向面积
double area(Point a,Point b,Point c)
{
    return cross(b-a,c-a);
}

//向量逆时针旋转角度rad
Vector rotate(Vector a,double rad)
{
    return Vector(a.x*cos(rad)-a.y*sin(rad),a.x*sin(rad)+a.y*cos(rad));
}