此模板包含了一些基本简单的二维几何问题,
1三角形外接圆 2三角形内切圆
3过圆外某点切线的角度 4过某条直线外一点半径为r的圆
5和两条相交直线相切的半径为r的圆 6和两个相离的圆相切的圆
1.计算向量点积, 叉积, 长度, 夹角, 向量的旋转(逆时针), 向量的单位法线
2.计算两点距离, 点到直线距离,两直线交点, 点到线段距离, 点在直线的投影,将直线AB沿法线方向平移d得到的直线EF
3. 圆与直线的交点(相离,没有交点, 相切一个交点, 相交两个交点), 计算两圆相交(返回交点和个数), 过某点圆的切线(一条或两条), 两圆的切线(相离,内切,内含,外切)
//Autor LJH; // #include <cstdio> #include <iostream> #include <cmath> #include <cstdlib> #include <cstring> #include <vector> #include <algorithm> using namespace std; #define PI acos(-1) const double eps = 1e-6; struct Point { double x, y; Point (double x = 0, double y = 0) : x(x), y(y) { } //构造函数, 方便代码书写 }; typedef Point myvector; // 向量 + 向量 = 向量 myvector operator + (myvector A, myvector B) { return myvector(A.x + B.x, A.y + B.y); } // 点 - 点 = 向量 myvector operator - (Point A, Point B) { return myvector(A.x - B.x, A.y - B.y); } //向量 * 数 = 向量 myvector operator * (myvector A, double p) { return myvector(A.x * p, A.y * p); } //向量/数 = 向量 myvector operator / (myvector A, double p) { return myvector(A.x / p, A.y / p); } // 小于号 bool operator < (const Point & a, const Point & b) { if (a.x == b.x) return a.y < b.y; return a.x < b.x; } //比较 int dcmp(double x) { if(fabs (x) < eps) return 0; else return x < 0 ? -1 : 1; } // 恒等于号 bool operator == (const Point & a, const Point & b) { return dcmp(a.x - b.x) == 0 && dcmp(a.y - b.y) == 0; } //计算向量 A B 的点积, A*B = |A| * |B| * cosß double Dot (myvector A, myvector B) { return A.x*B.x + A.y*B.y; } //计算向量 A 的长度 double Length (myvector A) { return sqrt (Dot(A, A)); } //返回弧度, 计算向量 A,B 的夹角,是cos, 有公式 double Angle (myvector A, myvector B) { return acos(Dot(A, B) / Length(A) / Length(B)); } // 计算叉积,AxB = |A| * |B| * sinß, 得到的是与这两个向量垂直的向量 double Cross(myvector A, myvector B) { return A.x * B.y - A.y * B.x; } //计算三角形面积 double Area2(Point A, Point B, Point C) { return fabs(Cross(B - A, C - A)); } //计算两点距离 double DistancePoint(Point A, Point B) { return sqrt((A.x-B.x)*(A.x-B.x) + (A.y-B.y)*(A.y-B.y)); } // 计算向量旋转后变成的另一个向量, rad 是弧度 //公式 x1 = x * cosß - y * sinß, y1 = x * sinß + y * cosß; myvector Rotate(myvector A, double rad) { return myvector(A.x * cos(rad) - A.y * sin(rad), A.x * sin(rad) + A.y * cos(rad)); } //计算向量的单位法线, 在调用前确保 A 不是零向量 myvector Normal(myvector A) { double L = Length(A); return myvector(-A.y / L, A.x / L); } //直线可以用直线上一点p1, 和方向向量V表示, 即 向量P = 点p1 + V; //计算两直线的 交点 , 调用前确保两直线有交点 Point GetLineInstersection(Point P, myvector v, Point Q, myvector w) { myvector 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) { myvector v1 = B - A, v2 = P - A; return fabs(Cross(v1, v2) / Length(v1)); } // 点到线段的距离, 有两种可能, 一种点在线段上方, 这时候算垂直, 不在线段上方; double DistanceToSegment(Point P, Point A, Point B) { if( A == B) return Length(P-A); //如果线段是一个点 myvector 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); } //计算点在直线上投影的点 Point GetLineProjectoin(Point P, Point A, Point B) { myvector v = B - A; return A + v * (Dot(v, P-A) / Dot(v, v)); } struct Line //线,由 { Point v, p, e; Point point(double t) { return (p + v * t); } }; struct Circle { Point c; double r; Circle(Point _c=0,double _r=0):c(_c),r(_r){} Point point(double a)///根据圆心角算圆上的点 { return Point(c.x+cos(a)*r,c.y+sin(a)*r); } }; //返回向量的极角,极坐标系, 有时候需要按照极角排序 double PolarAngle(myvector V) {return atan2(V.y, V.x);} //将直线AB沿法线方向平移d得到的直线EF, myvector move_d(Point A, Point B, double d, Line& L) { myvector C = B - A; C = C/Length(C); C = Rotate(C, PI/2); L.p = A + C * d; L.e = B + C * d; L.v = L.e - L.p; return (L.v); } //圆与直线的交点, 相离,没有交点, 相切一个交点, 相交两个交点 int getLineCircleInteresection(Line L, Circle C, double& t1, double& t2, vector<Point>& sol) { //printf("##%f %f %f %f\n", L.p.x, L.p.y, L.e.x, L.e.y); double a = L.v.x, b = L.p.x - C.c.x, c = L.v.y, d = L.p.y - C.c.y; double e = a*a + c*c, f = 2 * (a*b + c*d), g = b*b + d*d - C.r*C.r; double delta = f*f - 4*e*g; // printf("delta = %.16f, esp = %.16f\n", delta, eps); if(dcmp(delta) < 0) { return 0; } if(dcmp(delta) == 0) { t1 = t2 = -f / (2 * e); sol.push_back(L.point(t1)); return 1; } //相交 t1 = (-f - sqrt(delta)) / (2 * e); sol.push_back(L.point(t1)); t2 = (-f + sqrt(delta)) / (2 * e); sol.push_back(L.point(t2)); return 2; } //计算两圆相交 int getCircleCircleIntersection(Circle C1, Circle C2, vector<Point> &sol) { double d = Length(C1.c - C2.c); if(dcmp(d) == 0) { if(dcmp(C1.r - C2.r) == 0) return -1; //两圆重合 return -2;//圆心一样,小圆在大圆内 } if(dcmp(C1.r + C2.r - d) < 0) return 0; //相离 if(dcmp(fabs(C1.r - C2.r) - d) > 0) return -3;//圆心不同,大圆在小圆内 double a = PolarAngle(C2.c - C1.c);//计算向量C1C2的极角 double da = acos((C1.r * C1.r + d * d - C2.r * C2.r) / (2 * C1.r * d)); //C1C2到C1P1的角 Point p1 = C1.point(a - da), p2 = C1.point(a + da); sol.push_back(p1); if(p1 == p2) return 1; sol.push_back(p2); return 2; } //过点p到圆 C 的切线,v[i]是第i条切线的向量,返回切线条数 int getTangents(Point p, Circle C, myvector* v) { myvector u = C.c - p; double dis = Length(u); if (dis < C.r) return 0; else if (dcmp(dis - C.r) == 0) { v[0] = Rotate(u, PI / 2.0); return 1; } else { double ang = asin(C.r / dis); v[0] = Rotate(u, -ang); v[1] = Rotate(u, +ang); return 2; } } //两圆的切线条数, (1)重合,无数条,(2)两圆内含没有公共点没有切线,(3)两圆内切,有1条, //(4)两圆相交有2条, (5)两圆外切,3条, (6)两圆相离,4条公切线 //返回切线条数, a[i],b[i]分别是第i条切线在圆A和B的切点 int getTangentsCircle(Circle A, Circle B, Point* a, Point* b) { int cnt = 0; if(A.r < B.r) //swap { Circle temp; Point *temp1 = NULL; A = temp; A = B; B = temp; a = temp1; a = b; b = temp1; } int d2 = (A.c.x - B.c.x) * (A.c.x - B.c.x) + (A.c.y - B.c.y) * (A.c.y - B.c.y); int rdiff = A.r - B.r; int rsum = A.r + B.r; if(d2 < rdiff * rdiff) return 0; // 内含 double base = atan2(B.c.y - A.c.y, B.c.x - A.c.x); if(d2 == 0 && A.r == B.r) return -1; //无限多条 if(d2 == rdiff * rdiff) { a[cnt] = A.point(base); b[cnt] = B.point(base); cnt++; } //有外共切线 double ang = acos((A.r - B.r) / sqrt(d2)); a[cnt] = A.point(base + ang); b[cnt] = B.point(base + ang); cnt++; a[cnt] = A.point(base - ang); b[cnt] = B.point(base - ang); cnt++; if(d2 == rsum * rsum) //一条内公切线 { a[cnt] = A.point(base); b[cnt] = B.point(PI + base); cnt++; } else if(d2 > rsum * rsum) { double ang = acos(A.r + B.r) / sqrt(d2); a[cnt] = A.point(base + ang); b[cnt] = B.point(PI + base + ang); cnt++; a[cnt] = A.point(base - ang); b[cnt] = B.point(PI + base - ang); cnt++; } return cnt; } //两圆相交的面积,返回面积 double CircleInterSectionArea(Circle A, Circle B, vector<Point>&sol) { sol.clear(); int f = getCircleCircleIntersection(A, B, sol); if(f == 0 || f == 1) return 0;//相离, 相切 if(f < 0) return (min(A.r,B.r)*min(A.r,B.r)*PI);//内含 Point p3 = sol[0]; double areatri = Area2(A.c, B.c, p3); double ang1 = Angle(sol[0]-A.c, B.c-A.c); double ang2 = Angle(sol[0]-B.c, A.c-B.c); double areacir1 = A.r * A.r * ang1; double areacir2 = B.r * B.r * ang2; return (areacir1+areacir2-areatri); } bool cmp(Point A, Point B) { if(A.x == B.x) return A.y < B.y; return A.x < B.x; } //三角形外接圆 void FUN1(double x1, double y1, double x2, double y2, double x3, double y3); //三角形内切圆 void FUN2(double x1, double y1, double x2, double y2, double x3, double y3); //过圆外某点切线的角度 void FUN3(double xc, double yc, double r, double xp, double yp); //过某条直线外一点半径为r的圆 void FUN4(double xp, double yp, double x1, double y1, double x2, double y2, double r); //和两条相交直线相切的半径为r的圆 void FUN5(double x1, double y1, double x2, double y2, double x3, double y3, double x4, double y4, double r); //和两个相离的圆相切的圆 void FUN6(double x1, double y1, double r1, double x2, double y2, double r2, double r); int main() { //freopen("1.txt", "r", stdin); /*double a1, b1, r1, a2, b2, r2; vector<Point> sol; while(~scanf("%lf %lf %lf %lf %lf %lf", &a1, &b1, &r1, &a2, &b2, &r2)) { Circle a, b; Point p1, p2; p1.x = a1; p1.y = b1; a.c = p1; a.r = r1; p2.x = a2; p2.y = b2; b.c = p2; b.r = r2; printf("%.3f\n", CircleInterSectionArea(a, b, sol)); }*/ char s[50], s1[50] = "CircumscribedCircle", s2[50] = "InscribedCircle", s3[50] = "TangentLineThroughPoint", s4[50] = "CircleThroughAPointAndTangentToALineWithRadius", s5[50] = "CircleTangentToTwoLinesWithRadius", s6[50] = "CircleTangentToTwoDisjointCirclesWithRadius"; while(~scanf("%s", s)) { if(!strcmp(s,s1)) { double x1, y1, x2, y2, x3, y3; cin >> x1 >> y1 >> x2 >> y2 >> x3 >> y3; FUN1(x1, y1, x2, y2, x3, y3); } else if(!strcmp(s, s2)) { double x1, y1, x2, y2, x3, y3; cin >> x1 >> y1 >> x2 >> y2 >> x3 >> y3; FUN2(x1, y1, x2, y2, x3, y3); } else if(!strcmp(s,s3)) { double xc, yc, r, xp, yp; cin >> xc >> yc >> r >> xp >> yp; FUN3(xc, yc, r, xp, yp); } else if(!strcmp(s,s4)) { double xp, yp, x1, y1, x2, y2, r; cin >> xp >> yp >> x1 >> y1 >> x2 >> y2 >> r; FUN4( xp, yp, x1, y1, x2, y2, r); } else if(!strcmp(s,s5)) { double x1, y1, x2, y2, x3, y3, x4, y4, r; cin >> x1 >> y1 >> x2 >> y2 >> x3 >> y3 >> x4 >> y4 >> r; FUN5(x1, y1, x2, y2, x3, y3, x4, y4, r); } else if(!strcmp(s,s6)) { double x1, y1, r1, x2, y2, r2, r; cin >> x1 >> y1 >> r1 >> x2 >> y2 >> r2 >> r; FUN6(x1, y1, r1, x2, y2, r2, r); } getchar(); } return 0; } void FUN1(double x1, double y1, double x2, double y2, double x3, double y3) {//三角形外接圆 Point A, B, C, D, E, F; myvector AB, BC, DE, DF; A.x = x1; A.y = y1; B.x = x2; B.y = y2; C.x = x3; C.y = y3; E.x = (A.x + B.x)/2.0; E.y = (A.y + B.y)/2.0; F.x = (B.x + C.x)/2.0; F.y = (B.y + C.y)/2.0; AB = B - A; BC = C - B; DE = Normal(AB); DF = Normal(BC); D = GetLineInstersection(E, DE, F, DF); double r = DistancePoint(B, D); printf("(%f,%f,%f)\n", D.x, D.y, r); return; } void FUN2(double x1, double y1, double x2, double y2, double x3, double y3) {//三角形内切圆 Point A, B, C; A.x = x1; A.y = y1; B.x = x2; B.y = y2; C.x = x3; C.y = y3; myvector v11 = B - A; myvector v12 = C - A; myvector v21 = A - B; myvector v22 = C - B; double ang1 = (PolarAngle(v11) + PolarAngle(v12)) / 2.0; double ang2 = (PolarAngle(v21) + PolarAngle(v22)) / 2.0; myvector vec1 = myvector(cos(ang1), sin(ang1)); myvector vec2 = myvector(cos(ang2), sin(ang2)); Point O = GetLineInstersection(A, vec1, B, vec2); double r = DistanceToLine(O, A, B); printf("(%f,%f,%f)\n", O.x, O.y, r); } void FUN3(double xc, double yc, double r, double xp, double yp) {//过圆外某点切线的角度 myvector vc[5]; int len = getTangents(Point(xp, yp), Circle(Point(xc, yc), r), vc); double tmp[5]; for (int i = 0; i < len; ++i) { double ang = PolarAngle(vc[i]); if (ang < 0) ang += PI; ang = fmod(ang, PI); tmp[i] = ang * 180 / PI; } sort(tmp, tmp + len); printf("["); for (int i = 0; i < len; ++i) { printf("%.6lf", tmp[i]); if (i != len - 1) printf(","); } printf("]\n"); return; } void FUN4(double xp, double yp, double x1, double y1, double x2, double y2, double r) {//过某条直线外一点半径为r的圆 Line L1, L2; Point X, Y, P, Q, pp[10]; double t1, t2; int k = 0; vector<Point>sol, sol2; X.x = x1; X.y = y1; Y.x = x2; Y.y = y2; P.x = xp; P.y = yp; Circle C(P, r); move_d(X, Y, -r, L1); move_d(X, Y, r, L2); int f = getLineCircleInteresection(L1, C, t1, t2, sol), f1 = getLineCircleInteresection(L2, C, t1, t2, sol2); printf("["); if(f == 1) { pp[k++] = sol[0]; // printf("(%f,%f)", sol[0].x, sol[0].y); } if(f == 2) { pp[k++] = sol[0]; pp[k++] = sol[1]; // printf("(%f,%f),(%f,%f)", sol[0].x, sol[0].y, sol[1].x, sol[1].y); } if(f1 == 1) { pp[k++] = sol2[0]; // if(f != 0) printf(","); // printf("(%f,%f)", sol2[0].x, sol2[0].y); } if(f1 == 2) { pp[k++] = sol2[0]; pp[k++] = sol2[1]; //if(f != 0) printf(","); //printf("(%f,%f),(%f,%f)", sol2[0].x, sol2[0].y, sol2[1].x, sol2[1].y); } sort(pp,pp+k); for(int i=0;i<k;i++) { printf("(%f,%f)", pp[i].x, pp[i].y); if(i != k-1) printf(","); } printf("]\n"); return; } void FUN5(double x1, double y1, double x2, double y2, double x3, double y3, double x4, double y4, double r) {//和两条相交直线相切的半径为r的圆 Line L1, L2; Point A, B, C, D, P, pp[10]; int k = 0; A.x = x1; A.y = y1; B.x = x2; B.y = y2; C.x = x3; C.y = y3; D.x = x4; D.y = y4; move_d(A, B, r, L1); move_d(C, D, r, L2); P = GetLineInstersection(L1.p, L1.v, L2.p, L2.v); pp[k++] = P; // printf("[(%f,%f),", P.x, P.y); move_d(A, B, r, L1); move_d(C, D, -r, L2); P = GetLineInstersection(L1.p, L1.v, L2.p, L2.v); pp[k++] = P; // printf("(%f,%f),", P.x, P.y); move_d(A, B, -r, L1); move_d(C, D, r, L2); P = GetLineInstersection(L1.p, L1.v, L2.p, L2.v); pp[k++] = P; // printf("(%f,%f),", P.x, P.y); move_d(A, B, -r, L1); move_d(C, D, -r, L2); P = GetLineInstersection(L1.p, L1.v, L2.p, L2.v); pp[k++] = P; // printf("(%f,%f)]\n", P.x, P.y); sort(pp, pp+k); printf("["); for(int i=0;i<k;i++) { printf("(%f,%f)", pp[i].x, pp[i].y); if(i != k-1) printf(","); } printf("]\n"); return; } void FUN6(double x1, double y1, double r1, double x2, double y2, double r2, double r) {//和两个相离的圆相切的圆 Point a, b, c, pp[10]; int k = 0; a.x = x1; a.y = y1;b.x = x2; b.y = y2; Circle A(a,r1), B(b,r2); Circle C(a,r1+r), D(b,r2+r); vector<Point>sol; int t = getCircleCircleIntersection(C, D, sol); if(t == 1) pp[k++] = sol[0]; //printf("[(%f,%f)]\n", sol[0].x, sol[0].y); else if(t == 2) { pp[k++] = sol[0]; pp[k++] = sol[1]; //printf("[(%f,%f),(%f,%f)]\n", sol[0].x, sol[0].y,sol[1].x, sol[1].y); } sort(pp, pp+k); printf("["); for(int i=0;i<k;i++) { printf("(%f,%f)", pp[i].x, pp[i].y); if(i != k-1) printf(","); } printf("]\n"); return; } //Power by LJH