计算几何的简单模板
const double eps = 1e-8;
struct Point
{
double x,y;
Point(double tx = 0,double ty = 0) : x(tx),y(ty){}
};
typedef Point Vtor;
//向量的加减乘除
Vtor operator + (Vtor A,Vtor B) { return Vtor(A.x + B.x,A.y + B.y); }
Vtor operator - (Point A,Point B) { return Vtor(A.x - B.x,A.y - B.y); }
Vtor operator * (Vtor A,double p) { return Vtor(A.x*p,A.y*p); }
Vtor operator / (Vtor A,double p) { return Vtor(A.x/p,A.y/p); }
bool operator < (Point A,Point B) { return A.x < B.x || (A.x == B.x && A.y < B.y);}
int dcmp(double x){ if (fabs(x) < eps) return 0; else return x < 0 ? -1 : 1; }
bool operator == (Point A,Point B) {return dcmp(A.x - B.x) == 0 && dcmp(A.y - B.y) == 0; }
//向量的点积，长度，夹角
double Dot(Vtor A,Vtor B) { return A.x*B.x + A.y*B.y; }
double Length(Vtor A) { return sqrt(Dot(A,A)); }
double Angle(Vtor A,Vtor B) { return acos(Dot(A,B)/Length(A)/Length(B)); }
//叉积，三角形面积
double Cross(Vtor A,Vtor B) { return A.x*B.y - A.y*B.x; }
double Area2(Point A,Point B,Point C) { return Cross(B - A,C - A); }
//向量的旋转,求向量的单位法线（即左转90度，然后长度归一）
Vtor Rotate(Vtor A,double rad){ return Vtor(A.x*cos(rad) - A.y*sin(rad),A.x*sin(rad) + A.y*cos(rad)); }
Vtor Normal(Vtor A)
{
double L = Length(A);
return Vtor(-A.y/L, A.x/L);
}
//直线的交点
Point GetLineIntersection(Point P,Vtor v,Point Q,Vtor w)
{
Vtor 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)
{
Vtor v1 = B - A;
return fabs(Cross(P - A,v1))/Length(v1);
}
//点到线段的距离
double DistanceToSegment(Point P,Point A,Point B)
{
if (A == B) return Length(P - A);
Vtor 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);
}
//点到直线的映射
Point GetLineProjection(Point P,Point A,Point B)
{
Vtor v = B - A;
return A + v*Dot(v,P - A)/Dot(v,v);
}
//判断线段是否规范相交
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 PolgonArea(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;
}
和圆有关的计算
struct Line
{
Point p;
Vtor v;
Line(Point p,Vtor v) : p(p),v(v){}
Point point(double t) { return p + v*t; }
};
struct Circle
{
Point c;
double r;
Circle(Point tc,double tr) : c(tc),r(tr){}
Point point(double a)
{
return Point(c.x + cos(a)*r + c.y + sin(a)*r);
}
};
//判断圆与直线是否相交以及求出交点
int getLineCircleIntersection(Line L,Circle C,double t1,double t2,vector<Point> &sol)
{
//注意sol没有清空哦
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;
double delta = f*f - 4*e*g;
if (dcmp(delta) < 0) return 0;
else if (dcmp(delta) == 0)
{
t1 = t2 = -f/(2*e);
sol.push_back(L.point(t1));
return 1;
}
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;
}
//判断并求出两圆的交点
double angle(Vtor v) { return atan2(v.y, v.x); }
int getCircleIntersection(Circle C1,Circle C2,vector<Point> &sol)
{
double d = Length(C2.c - C1.c);
// 圆心重合
if (dcmp(d) == 0)
{
if (dcmp(C1.r - C2.r) == 0) return -1; // 两圆重合
return 0; // 包含
}
// 圆心不重合
if (dcmp(C1.r + C2.r - d) < 0) return 0; // 相离
if (dcmp(fabs(C1.r - C2.r) - d) > 0) return 0; // 包含
double a = angle(C2.c - C1.c);
double da = acos(C1.r*C1.r + d*d - C2.r*C2.r) / (2*C1.r*d);
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;
}
//求点到圆的切线
int getTangents(Point p,Circle C,Vtor *v)
{
Vtor 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);
return 1;
}
else
{
double ang = asin(C.r / dis);
v[0] = Rotate(u, -ang);
v[1] = Rotate(u, ang);
return 2;
}
}
//求两圆的切线
int getCircleTangents(Circle A,Circle B,Point *a,Point *b)
{
int cnt = 0;
if (A.r < B.r) { swap(A,B); swap(a, b) ; }
//圆心距的平方
double 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);
double rdiff = A.r - B.r;
double rsum = A.r + B.r;
double base = angle(B.c - A.c);
//重合有无限多条
if (d2 == 0 && dcmp(A.r - B.r) == 0) return -1;
//内切
if (dcmp(d2 - rdiff*rdiff) == 0)
{
a[cnt] = A.point(base);
b[cnt] = B.point(base); cnt++;
return 1;
}
//有外公切线
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++;
//一条内切线
if (dcmp(d2 - rsum*rsum) == 0)
{
a[cnt] = A.point(base); b[cnt] = B.point(PI + base); cnt++;
}//两条内切线
else if (dcmp(d2 - rsum*rsum) > 0)
{
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++;
}
return cnt;
}