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)
#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)); }
// 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; }
double Area2(Point A, Point B, Point C) { return 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
{
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);
}
};
double angle(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 0;
}
if(dcmp(C1.r + C2.r - d) < 0) return 0;
if(dcmp(fabs(C1.r - C2.r) - d) > 0) return 0;
double a = angle(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;
}
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;
}
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);
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 = (angle(v11) + angle(v12)) / 2.0;
double ang2 = (angle(v21) + angle(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 = angle(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
时间： 2024-10-29 02:48:45