二维几何基础模板(一)

点和向量都用两个数x,y表示：

一、常定义用 ：

struct Point{  
    double x,y;  
    Point(double x=0,double y=0):x(x),y(y){} // 构造函数，方便代码编写  
};  
typedef Point Vector;  //从程序实现上，Vector只是Point的别名

//向量+向量=向量，点+向量=点
Vector operator + (Vector A,Vector B){
    return Vector(A.x+B.x,A.y+B.y);
}

//点-点=向量
Vector operator - (Vector A,Vector B){
    return Vector(A.x-B.x,A.y-B.y);
}

//向量*数=向量
Vector operator * (Vector A,double p){
    return Vector(A.x*p,A.y*p);
}

//向量/数=向量
Vector operator / (Vector A,double p){
    return Vector(A.x/p,A.y/p);
}

bool operator < (const Point& a,const Point& b){
return a.x<b.x||(a.x==b.x&&a.y<b.y);
}

const double eps=1e-10;
int dcmp(double x){
    if(fabs(x)<eps)
        return 0;
    else
        return x<0?-1:1;
}

bool operator == (const Point& a,const Point& b){
    return dcmp(a.x-b.x)==0&&dcmp(a.y-b.y)==0;
}

二、基本运算

1.利用点积计算向量长度和夹角的函数

double Dot(Vector A,Vector B){
    return A.x*B.x+A.y*B.y;
}
double Length(Vector A){
    return sqrt((Dot(A,A)));
}

double Angle(Vector A,Vector B){
    return acos(Dot(A,B)/Length(A)/length(B));
}

2.叉积

double Cross(Vector A,Vector B){
    return A.x*B.y-A.y*B.x;
}

double Area2(Point A,Point B,Point C){
    return Corss(B-A,C-A);
}

3.向量旋转

//rad是弧度
Vector Rotate(Vector A,double rad){
    return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad));
}

4.计算向量的单位法线，即转90°

//调用前请确保A不是零向量
Vector Normal(Vector A){
    double L=Length(A);
    return Vector(-A.y/L,A.x/L);
}

三、基于复数的几何计算

//调用前请确保A不是零向量
Vector Normal(Vector A){
    double L=Length(A);
    return Vector(-A.y/L,A.x/L);
}

//用real(p)和imag(p)访问实部和虚部，conj(p)返回共轭复数，即conj(a+bj)=a-bj;
double Dot(Vector A,Vector B){
    return real(conj(A)*B);
}

double Corss(Vector A,Vector B){
    return imag(conj(A)*B);
}

Vector Rotate(Vector A,double rad){
    return A*exp(Point (0,rad));
}

//const double pi=4*atan(1.0);

//const double pi=acos(-1.0);