这里总结一下二维几何基础知识！
常用定义：
//定义点的类型 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;}
点积：
//点积 double Dot(Vector A, Vector B) { return A.x*B.x + A.y*B.y; } //求点积 double Length(Vector A) { return sqrt(Dot(A, A)); } //求向量长度 double Angle(Vector A, Vector B) { return acos(Dot(A, B) / Length(A) / Length(B)); }//求向量之间的夹角
叉积：
//叉积 double Cross(Vector A, Vector B) { return A.x*B.y – A.y*B.x; }//求叉积 double Area2(Point A, Point B, Point C) { return Cross(B-A, C-A); }//根据叉积求三角形面积的两倍
旋转：
//旋转 Vector Rotate(Vector A, double rad) {//rad是弧度 return Vector(A.x*cos(rad) – A.y*sin(rad), A.x*sin(rad)+A.y*cos(rad) );}
向量的单位法线：
//向量单位法向量，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); }
二直线交点：
//二直线交点（参数式） 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;}
点到直线距离 ：
//点到直线距离 double DistanceToLine(Point P, Point A, Point B) {Vector v1 = B-A, v2 = P – A;return fabs(Cross(v1,v2) / Length(v1)); //如果不取绝对值，得到的是有向距离 }
点到线段距离 ：
//点到线段距离 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(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); }
点在直线上的投影：
//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：
//线段相交判定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;}
判断一个点是否在一条线段上：
//判断一个点是否在一条线段上bool OnSegment(Point p, Point a1, Point a2) {return dcmp(Cross(a1 – p, a2 – p)) == 0 && dcmp(Dot(a1 – p, a2 – p)) < 0;}
多边形面积：
//多边形面积 double ConvexPolygonArea(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; }
总结：
//定义点的类型 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;}//点积 double Dot(Vector A, Vector B) { return A.x*B.x + A.y*B.y; } //求点积 double Length(Vector A) { return sqrt(Dot(A, A)); } //求向量长度 double Angle(Vector A, Vector B) { return acos(Dot(A, B) / Length(A) / Length(B)); }//求向量之间的夹角 //叉积 double Cross(Vector A, Vector B) { return A.x*B.y – A.y*B.x; }//求叉积 double Area2(Point A, Point B, Point C) { return Cross(B-A, C-A); }//根据叉积求三角形面积的两倍 //旋转 Vector Rotate(Vector A, double rad) {//rad是弧度 return Vector(A.x*cos(rad) – A.y*sin(rad), A.x*sin(rad)+A.y*cos(rad) );} //向量单位法向量，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); }//二直线交点（参数式） 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;} //点到直线距离 double DistanceToLine(Point P, Point A, Point B) {Vector v1 = B-A, v2 = P – A;return fabs(Cross(v1,v2) / Length(v1)); //如果不取绝对值，，得到的是有向距离 }//点到线段距离 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(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); }//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;} //判断一个点是否在一条线段上bool OnSegment(Point p, Point a1, Point a2) {return dcmp(Cross(a1 – p, a2 – p)) == 0 && dcmp(Dot(a1 – p, a2 – p)) < 0;} //多边形面积 double ConvexPolygonArea(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; }
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