目前基本都是从蓝书上摘录的。
有一部分需要线性代数的知识，但是蓝书作者并没有解释，个人觉得用数学知识推出来更有助于记忆，死记硬背板子容易忘。以后有机会的话我在这里写点注解。
二维基础操作：
1 struct Point 2 { 3 double x, y; 4 Point(double x = 0, double y = 0):x(x), y(y){} 5 }; 6 7 typedef Point Vector; 8 9 int dcmp(double x)//比较 10 { 11 const double eps = 1e-10; 12 if (fabs(x) < eps) return 0; 13 else return x < 0 ? -1 : 1; 14 } 15 Vector operator + (Vector A, Vector B) { return Vector(A.x + B.x, A.y + B.y); } 16 Vector operator - (Vector A, Vector B) { return Vector(A.x - B.x, A.y - B.y); } 17 Vector operator * (Vector A, double p) { return Vector(A.x * p, A.y * p); } 18 Vector operator / (Vector A, double p) { return Vector(A.x / p, A.y / p); } 19 bool operator < (const Point& A, const Point& B) { return A.x < B.x || (A.x == B.x && A.y < B. y); } 20 bool operator == (const Point& A, const Point& B) { return dcmp(A.x - B.x) == 0 && dcmp(A.y - B.y) == 0; } 21 22 const double PI = acos(-1.0); 23 double torad(double deg) { return deg/180 * PI; }//角度转弧度 24 25 double Dot(Vector A, Vector B) { return A.x * B.x + A.y * B.y; }//点积 26 27 double Length(Vector A) { return sqrt(Dot(A, A)); }//长度 28 29 double Angle(Vector A, Vector B) { return acos(Dot(A, B)/Length(A)/Length(B)); }//角度 30 31 Vector Rotate(Vector A, double rad) { return Vector(A.x*cos(rad) - A.y*sin(rad), A.x*sin(rad) + A.y*cos(rad)); }//旋转 32 33 double Cross(Vector A, Vector B) { return A.x * B.y - A.y * B.x; }//叉积 34 35 double Area2(Point A, Point B, Point C) { return Cross(B-A, C-A);}//二倍的三角形面积 36 37 Vector Normal(Vector A) { double L = Length(A); return Vector(-A.y/L, A.x/L); }//向量的单位法线，旋转的特殊情况 38 39 Vector Get_Line_Projection(Point P, Point A, Point B)//P在AB方向上的映射 40 { 41 Vector v = B - A; 42 return A + v*(Dot(v, P-A) / Dot(v, v)); 43 } 44 45 Vector Get_Line_Intersect(Vector A, Vector v, Vector B, Vector w)//两直线交点 46 { 47 Vector u = A - B; 48 double t = Cross(w, u) / Cross(v, w); 49 return A + v*t; 50 } 51 52 double Distance_to_Line(Point P, Point A, Point B)//点P到直线AB的距离 53 { 54 Vector v1 = B - A, v2 = P - A; 55 return fabs(Cross(v1, v2)) / Length(v1); 56 } 57 58 double Distance_to_Segment(Point P, Point A, Point B)//点P到线段AB的距离 59 { 60 Vector v1 = B - A, v2 = P - A, v3 = P - B; 61 if (A == B || dcmp(Dot(v1, v2)) < 0) return Length(v2);//PA 62 else if (dcmp(Dot(v1, v3)) > 0) return Length(v3);//PB 63 else return fabs(Cross(v1, v2) / Length(v1));//PQ 64 } 65 66 bool On_Segment(Point P, Point A, Point B) { return dcmp(Cross(A - P, B - P)) == 0 && dcmp(Dot(A - P, B - P)) < 0; }//点在线段上 67 68 bool Segment_Proper_Intersection(Point a1, Point a2, Point b1, Point b2)//线段a1a2与b1b2相交（不包含端点） 69 { 70 double c1 = Cross(a2 - a1, b1 - a1), c2 = Cross(a2 - a1, b2 - a1); 71 double c3 = Cross(b2 - b1, a1 - b1), c2 = Cross(b2 - b1, a2 - b1); 72 return dcmp(c1)*dcmp(c2) < 0 && dcmp(c3)*dcmp(c4) < 0; 73 } 74 75 double Polygon_Area(Point *p, int n)//多边形面积，逆时针取点0~n-1 76 { 77 double area = 0; 78 for (int i = 1; i < n-1; i++) 79 area += Cross(p[i] - p[0], p[i+1] - p[0]); 80 return area/2; 81 } 82 83 //基于复数的几何运算，效率不是很高 84 #include <complex> 85 typedef complex<double> Point; 86 typedef Point Vector; 87 88 double Dot(Vector A, Vector B) { return real(conj(A) * B); }// A（x+yi）的共轭（x-yi）乘以B，取实部 89 double Cross(Vector A, Vector B) { return imag(conj(A) * B); }//取虚部 90 Vector Rotate(Vector A, double rad) { return A * exp(Point(0, rad)); }//使用了欧拉公式：e^(0 + rad*i) = cos(rad) + sin(rad)*i，求完确实是旋转
圆相关（他那个两圆公切线看着有点奇怪先不贴了）：
1 struct Circle 2 { 3 Point c; 4 double r; 5 // Circle(Point a = (0, 0), double x = 0):c(a), r(x){} 6 Point point(double seta) { return Point(c.x + cos(seta)*r, c.y + sin(seta)*r); }//已知角度求圆上某点 7 }; 8 9 struct Line 10 { 11 Point p; 12 Vector v; 13 Line(Point p, Vector v):p(p), v(v) { } 14 15 Point point(double t) { return p + v*t; }//已知参数t求直线上一点 16 Line move(double d) { return Line(p + Normal(v)*d, v); }//平移 17 }; 18 19 int get_Tangents(Point P, Circle C, vector<Vector> &v)//过定点做圆的切线 20 { 21 Vector u = C.c - P; 22 double d = Length(u); 23 24 if (dcmp(d - C.r) < 0) return 0;//点在圆内部 25 else if (dcmp(d - C.r) == 0)//点在圆上 26 { 27 v.push_back(Rotate(u, PI/2)); 28 return 1; 29 } 30 else//两条切线 31 { 32 double a = asin(C.r / d); 33 v.push_back(Rotate(u, a)); 34 v.push_back(Rotate(u, -a)); 35 return 2; 36 } 37 } 38 39 int get_Line_Circle_Intersection(Line L, Circle C, vector<Point> &ans)//方程求解直线与圆的交点 40 { 41 double t1, t2; 42 double a = L.v.x, b = L.p.x - C.c.x, c = L.v.y, d = L.p.y - C.c.y; 43 double e = a*a + c*c, f = 2*(a*b + c*d), g = b*b + d*d - C.r*C.r; 44 double delta = f*f - 4*e*g;//判别式 45 46 if (dcmp(delta) < 0) return 0;//相离 47 else if (dcmp(delta) == 0)//相切 48 { 49 t1 = t2 = -f/2/e; 50 ans.push_back(L.point(t1)); 51 return 1; 52 } 53 else//相交 54 { 55 t1 = (-f + sqrt(delta)) / 2 / e; 56 t2 = (-f - sqrt(delta)) / 2 / e; 57 ans.push_back(L.point(t1)), ans.push_back(L.point(t2)); 58 return 2; 59 } 60 } 61 62 inline double angle(Vector A) { return atan2(A.y, A.x); }//某向量极角 63 64 int get_Circle_Circle_Intersection(Circle C1, Circle C2, vector<Point> &v)//圆与圆的交点 65 { 66 double d = Length(C1.c - C2.c); 67 if (dcmp(d) == 0) 68 { 69 if (dcmp(C1.r - C2.r) == 0) return -1;//重合 70 return 0;//同心 71 } 72 if (dcmp(C1.r + C2.r - d) < 0) return 0;//外离 73 if (dcmp(fabs(C1.r - C2.r) - d) > 0) return 0;//内含 74 75 double a = angle(C2.c - C1.c); 76 double da = acos((C1.r*C1.r + d*d - C2.r*C2.r) / (2*C1.r*d));//余弦公式 77 Point p1 = C1.point(a - da), p2 = C1.point(a + da); 78 79 v.push_back(p1); 80 if (p1 == p2) return 1; 81 v.push_back(p2); 82 return 2; 83 }