计算几何入门——二维几何基础

//Part 1
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) { return Vector(A.x+B.x,A.y+B.y); }
Vector operator - (Vector A,Vector B) { return Vector(A.x-B.x,A.y-B.y); }
Vector operator * (Vector A,double p) { return Vector(A.x*p.A.y*p); }
Vector operator / (Vector A,double p) { return Vector(A.x/p.B.y/p); }

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

//Part 2 基本运算 
//点积 
double Dot(Vector A,Vector B) { return A.x*B.x+A.y*B.y; }
//模长 
double Length(Vector A) { return sqrt(Dot(A,A)); }
//夹角 
double Angle(Vector A,Vector B) { return acos(Dot(A,B)/Length(A)/Length(B)); }

//叉积 
double Cross(Vector A,Vector B) { return A.x*B.y-A.y*B.x; }
//三角形面积 
double Area2(Point A,Point B,Point C) { return Cross(B-A,C-A); }

//旋转 
Vector Rotate(Vector A,double rad) { return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad)); }
//单位法线 
Vector Normal(Vector A) { double L=Length(A); return Vector(-A.y/L,A.x/L); }

//Part 3 点和直线
//直线交点
Point GetLineIntersection(Point P,Vector v,Point Q,Vector w)
{
    Vector u=P-Q;
    double t=Cross(w,u)/Cross(v,w);
    return P+v*t;
} 
//点到直线的距离
double DistanceToLine(Point P,Point A,Point B)
{
    Vector v1=B-A,v2=P-A;
    return fabs(Cross(v1,v2)/Length(v1));
} 
//点到线段的距离
double DistanceToSegment(Point P,Point A,Point B)
{
    if(A==B) return Length(P-A);
    Vector v1=B-A,v2=P-A,v3=P-B;
    if(dcmp(Doc(v1,v2))<0) return Length(v2);
    if(dcmp(Doc(v1,v3))>0) return Length(v3);
    return fans(Cross(v1,v2)/Length(v1)); 
}
//点在直线上的投影
Point GetLineProjection(Point P,Point A,Point B)
{
     Vector v=B-A;
     return A+v*(Dot(v,P-A)/Dot(v,v));
} 
//线段相交判定（规范相交） 
bool SegmentProperIntersection(Point a1,Point a2,Point b1,Point b2)
{
    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;
} 
//点在线段上
bool OnSegment(Point p,Point a1,Point a2)
{
    return dcmp(Cross(a1-p,a2-p))==0 && dcmp(Dot(a1-p,a2-p)<0);
} 

//Part 4 多边形
double PolygonArea(Point* p,int n)
{
    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; 
}