《訓練指南》大白 二維幾何基礎 基本公式（模板）

《訓練指南》大白 二維幾何基礎 基本公式（模板）

#include<bits/stdc++.h>
using namespace std;
struct Point{
    double x,y;
    Point(double x=0,double y=0):x(x),y(y){}///構造函數
};
typedef Point Vector ;
///向量+向量=向量，點+向量=點
Vector operator + (Vector A,Vector B){return Vector(A.x+B.x,A.y+B.y);}
///點-點等於向量
Vector operator - (Point A,Point B){return Vector(A.x-B.x,A.y-B.y);}
///向量*數 = 向量
Vector operator * (Vector A,double p){return Vector(A.x*p,A.y*p);}
///向量/數 = 向量
Vector operator / (Vector A,double p){return Vector(A.x/p,A.y/p);}

///判斷小於關係，先判斷x後判斷y
bool operator < (const Point& a,const Point& b){
    return a.x<b.x||(a.x==b.x&&a.y<b.y) ;
}

///判斷一個數與0的關係
const double eps = 1e-10 ;
int dcmp(double x){
    if(fabs(x)<eps)return 0;///精度範圍內認爲等於0
    else return x<0?-1:1 ;///小於0返回-1，大於0返回1
}

///判斷兩個向量/點是否相等
bool operator == (const Point& a,const Point& b){
    return dcmp(a.x-b.x)==0&&dcmp(a.y-b.y)==0 ;
}

///點積
double Dot(Vector A,Vector B){return A.x*B.x + A.y*B.y ;}
///向量長度
double Length(Vector A){return sqrt(Dot(A,A));}
///兩向量夾角
double Angle(Vector A,Vector B){return acos(Dot(A,B)/Length(A)/Length(B)) ;}
///叉積
double Cross(Vector A,Vector B){return A.x*B.y - A.y*B.x ;}
///what
double Area2(Point A,Point B,Point C){return Cross(B-A,C-A) ;}
///向量旋轉，rad爲逆時針旋轉的角度
Vector Rotate(Vector A,double rad){return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad));}
///單位法向量，左轉90度，長度單位化
Vector Normal(Vector A){
    double L = Length(A) ;
    return Vector(-A.y/L,A.x/L) ;
}
///獲得兩條直線的交點,注意v，w不共線，即有唯一交點
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 ;
}
///獲得點到直線的距離，A,B是直線上的兩個點
double DistanceToLine(Point P,Point A,Point B){
    Vector v1 = B-A ,v2 = P-A ;
    return fabs(Cross(v1,v2))/Length(v1) ;
}
///點到線段的距離，要求三點不共線
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(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) ;
}
///點在直線上的投影,三點不共線
Point GetLineProjection(Point P,Point A,Point B){
    Vector v = B-A ;
    return A+v*(Dot(v,P-A)/Dot(v,v)) ;
}
///線段相交判定,規範相交，即兩個線段恰好有一個公共點，且不再任意一條線段的端點上
bool SegmentProperIntersection(Point a1,Point a2,Point b1,Point b2){
    double c1 = Cross(a2-a1,b1-a1) ;
    double c2 = Cross(a2-a1,b2-a1) ;
    double c3 = Cross(b2-b1,a1-b1) ;
    double c4 = Cross(b2-b1,a2-b1) ;
    return dcmp(c1)*dcmp(c2)<0&&dcmp(c3)*dcmp(c4)<0 ;
}
///判斷一個點是否在線段上
bool OnSegment(Point p,Point a1,Point a2){return dcmp(Cross(a1-p,a2-p))==0&&dcmp(Dot(a1-p,a2-p))<0 ;}

///多邊形的有向面積
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 ;
}

///以上爲可以在主函數調用的函數