计算几何——向量的叉乘、点乘、夹角

 汇总篇：计算几何汇总

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一、向量的叉乘

向量p=(x1,y1), q=(x2,y2)

则 pxq=x1.y2-x2.y1

pxq= - qxp

叉乘的大小等于于2倍三角形面积.

右手法则：手掌表示p向量，手指表示q向量，方向均指向指尖

pxq > 0, 则p在q的顺时针方向（q,p），即大拇指朝上，手指与手掌弯曲成九十度，手指弯向左边，p逆时针方向旋转到q

pxq<0,  则p在q的逆时针方向   (p,q)   ，即大拇指朝下，手指与手掌弯曲成九十度，手指弯向右边，p顺时针方向旋转到q

pxq=0.  则pq 重合

代码计算叉乘

class point{
 	public:
	double x;
	double y;
	point(double x_=0,double y_=0):x(x_),y(y_){} 
	friend const point operator+(const point& p1,const point& p2){
		return point(p1.x+p2.x,p1.y+p2.y);
	};
	friend const point operator-(const point& p1,const point& p2){
		return point(p1.x-p2.x,p1.y-p2.y);
	};
	friend const point operator*(const point& p,const double& m){
		return point(p.x*m,p.y*m);
	};
	friend const point operator*(const double& m,const point& p){
		return point(p.x*m,p.y*m);
	};
	friend const point operator/(const point& p,const double& m){
		return point(p.x/m,p.y/m);
	};
	friend ostream& operator <<(ostream& out,point& a){
		printf("(%lf,%lf)",a.x,a.y);
		return out;
	};
};
typedef point vect2;//重命名，向量也是用坐标表示 

class line{
	public:
	point start;
	point end; 
	line(point s=point(0,0),point e=point(0,0)):start(s),end(e){}
};

double cross(point O,point A,point B){//叉乘 
	double oa_x=A.x-O.x;
	double oa_y=A.y-O.y;
	double ob_x=B.x-O.x;
	double ob_y=B.y-O.y;
	return oa_x*ob_y-oa_y*ob_x;
}

二、向量的点乘

向量p=(x1,y1), q=(x2,y2)

pq=x1*x2+y1*y2

double dot(point O,point A,point B){//点乘 
	double oa_x=A.x-O.x;
	double oa_y=A.y-O.y;
	double ob_x=B.x-O.x;
	double ob_y=B.y-O.y;
	return oa_x*ob_x+oa_y*ob_y;
}

三、向量的夹角

cos(alpha)=(pq)/(|p|*|q|)

#include

double dot(point O,point A,point B){//点乘 
	double oa_x=A.x-O.x;
	double oa_y=A.y-O.y;
	double ob_x=B.x-O.x;
	double ob_y=B.y-O.y;
	return oa_x*ob_x+oa_y*ob_y;
}
double dis(const point &p1,const point &p2){//求两点之间距离
	double ans=(p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y);
	return sqrt(ans);
}
double angle(point O,point A,point B){//两向量OA,OB的夹角
	return acos(dot(O,A,B)/(dis(O,A)*dis(O,B)));
}