struct Point {
double x, y;
Point() {}
Point(double a, double b) {
x = a, y = b;
}
bool operator == (Point a) {
return sgn(x-a.x) == 0 && sgn(y-a.y) == 0;
}
bool operator < (Point a) { //最左下角
return sgn(y-a.y) == 0 ? sgn(x-a.x)<0 : y<a.y;
}
Point operator - (Point a) {
return Point(x-a.x, y-a.y); //向量减
}
Point operator + (Point a) {
return Point(x+a.x, y+a.y); //向量加
}
double operator * (Point a) { //点乘(内积)
return x*a.x + y*a.y;
}
double operator ^ (Point a) { //叉乘(外积)
return x*a.y - y*a.x;
}
Point operator * (double k) {
return Point(x*k, y*k);
}
Point operator / (double k) {
return Point(x/k, y/k);
}
Point trunc(double r) {
//化为长度为 r 的向量
double l = len();
if(!sgn(l)) return *this;
r /= l;
return Point(x*r, y*r);
}
Point rotleft() {
//原向量逆时针旋转90度
return Point(-y, x);
}
Point rotright() {
//原向量顺时针旋转90度
return Point(y, -x);
}
Point rotate(Point p, double rad) {
//原向量绕着 p 点逆时针旋转 rad 度
Point v = (*this) - p;
double c = cos(rad), s = sin(rad);
return Point(p.x + v.x*c - v.y*s, p.y + v.x*s + v.y*c);
}
double len() { //向量长度
return hypot(x, y);
}
double len2() { //向量长度平方
return x*x + y*y;
}
double distance(Point p) { //两点之间距离
return hypot(x-p.x, y-p.y);
}
double rad(Point a, Point b) {
//三个点:p, a, b
//计算 pa 与 pb 所成的夹角
Point p = *this;
return fabs(atan2(fabs((a-p) ^ (b-p)), (a-p) * (b-p)));
}
};
struct Line {
Point s, e;
Line() {};
Line(Point _s, Point _e) {
// 两点确定直线
s = _s, e = _e;
}
Line(Point p, double angle) {
//根据点 p 和倾斜角 angle 确定直线, 0 <= angle <= pi
s = p;
if(sgn(angle - pi/2) == 0) {
e = (s + Point(0, 1));
} else {
e = (s + Point(1, tan(angle)));
}
}
Line(double a, double b, double c) {
//根据 ax+by+c = 0 确定直线
if(sgn(a) == 0) {
s = Point(0, -c/b);
e = Point(1, -c/b);
} else if(sgn(b) == 0) {
s = Point(-c/a, 0);
e = Point(-c/a, 1);
} else {
s = Point(0, -c/b);
e = Point(1, (-c-a)/b);
}
}
void adjust() {
if(e < s) swap(s, e);
}
bool operator == (Line v) {
return (s == v.s) && (e == v.e);
}
double length() {
return s.distance(e);
}
double angle() {
// 直线倾斜角
double k = atan2(e.y-s.y, e.x-s.x);
if(sgn(k) < 0) k += pi;
if(sgn(k-pi) == 0) k -= pi;
return k;
}
int relation(Point p) {
/*
判断直线和点关系
this line, p point
1 在左侧
2 在右侧
3 在直线上
*/
int c = sgn((p-s) ^ (e-s));
if(c < 0) return 1;
else if(c > 0) return 2;
else return 3;
}
bool pointonseg(Point p) {
/*
判断点和线段关系
this seg, p point
0 不在线段上
1 在线段上
*/
return sgn((p-s) ^ (e-s)) == 0 && sgn((p-s) * (p-e)) <= 0;
}
bool parallel(Line v) {
//判断两向量平行
return sgn((e-s) ^ (v.e-v.s)) == 0;
}
int segcrossseg(Line v) {
/*
两线段关系
2 规范相交
1 不规范相交
0 不相交
*/
int d1 = sgn((e-s) ^ (v.s-s));
int d2 = sgn((e-s) ^ (v.e-s));
int d3 = sgn((v.e-v.s) ^ (s-v.s));
int d4 = sgn((v.e-v.s) ^ (e-v.s));
if((d1 ^ d2) == -2 && (d3 ^ d4) == -2)
return 2;
return (d1 == 0 && sgn((v.s-s) * (v.s-e)) <= 0) ||
(d2 == 0 && sgn((v.e-s) * (v.e-e)) <= 0) ||
(d3 == 0 && sgn((s-v.s) * (s-v.e)) <= 0) ||
(d4 == 0 && sgn((e-v.s) * (e-v.e)) <= 0);
}
int linecrossseg(Line v) {
/*
判断直线和线段关系
this line, v seg
2 规范相交
1 非规范相交
0 不相交
*/
int d1 = sgn((e-s) ^ (v.s-s));
int d2 = sgn((e-s) ^ (v.e-s));
if((d1 ^ d2) == -2) return 2;
else return (d1 == 0 || d2 == 0);
}
int linecrossline(Line v) {
/*
两直线关系
0 平行
1 重合
2 相交
*/
if((*this).parallel(v))
return v.relation(s) == 3;
return 2;
}
Point crosspoint(Line v) {
//求两直线的交点
//要求两直线不平行，不重合
double a1 = (v.e-v.s) ^ (s-v.s);
double a2 = (v.e-v.s) ^ (e-v.s);
return Point((s.x*a2-e.x*a1), (s.y*a2-e.y*a1))/(a2-a1);
}
double dispointtoline(Point p) {
//求点到直线的距离
return fabs((p-s)^(e-s))/length();
}
double dispointtoseg(Point p) {
if(sgn((p-s)*(e-s))<0 || sgn((p-e)*(s-e)) < 0)
return min(p.distance(s), p.distance(e));
return dispointtoline(p);
}
double dissegtoseg(Line v) {
/*
返回线段到线段的距离
前提是两线段不相交，相交为0
*/
return min(min(dispointtoseg(v.s), dispointtoseg(v.e)),
min(v.dispointtoseg(s), v.dispointtoseg(e)));
}
Point lineprog(Point p) {
//返回点 p 在直线上的投影点
return s + (((e-s)*((e-s)*(p-s))) / ((e-s).len2()));
}
Point symmetrypoint(Point p) {
//返回点 p 关于直线p的对称点
Point q = lineprog(p);
return Point(2*q.x-p.x, 2*q.y-p.y);
}
};
struct Circle {
Point p; //圆心
double r; //半径
Circle() {}
Circle(Point _p, double _r) {
p = _p, r = _r;
}
Circle(double x, double y, double _r) {
p = Point(x, y), r = _r;
}
Circle(Point a, Point b, Point c) {
//求三角形的外接圆
Line u = Line((a+b)/2, ((a+b)/2) + ((b-a).rotleft()));
Line v = Line((b+c)/2, ((b+c)/2) + ((c-b).rotleft()));
p = u.crosspoint(v);
r = p.distance(a);
}
Circle(Point a, Point b, Point c, bool t) {
//求三角形的内接圆
//bool t用来区别三角形外接圆的函数
Line u, v;
double m = atan2(b.y-a.y, b.x-a.x);
double n = atan2(c.y-a.y, c.x-a.x);
u.s = a;
u.e = u.s + Point(cos((n+m)/2), sin((n+m)/2));
v.s = b;
m = atan2(a.y-b.y, a.x-b.x);
n = atan2(c.y-b.y, c.x-b.x);
v.e = v.s + Point(cos((n+m)/2), sin((n+m)/2));
p = u.crosspoint(v);
r = Line(a, b).dispointtoseg(p);
}
bool operator == (Circle v) {
return p==v.p && sgn(r-v.r)==0;
}
bool operator < (Circle v) {
return (p<v.p) || (p==v.p && sgn(r-v.r)<0);
}
double area() {
//面积
return pi*r*r;
}
double cirperimeter() {
//周长
return 2*pi*r;
}
int relation(Point b) {
/*
点和圆的关系
0 圆外
1 圆上
2 圆内
*/
double dst = b.distance(p);
if(sgn(dst-r) < 0) return 2;
else if(sgn(dst-r) == 0) return 1;
else return 0;
}
int relationseg(Line v) {
/*
线段和圆的关系
0 圆外
1 圆上
2 圆内
*/
double dst = v.dispointtoseg(p);
if(sgn(dst-r) < 0) return 2;
else if(sgn(dst-r) == 0) return 1;
else return 0;
}
int relationline(Line v) {
/*
直线和圆的关系
0 圆外
1 圆上
2 圆内
*/
double dst = v.dispointtoline(p);
if(sgn(dst-r) < 0) return 2;
else if(sgn(dst-r) == 0) return 1;
else return 0;
}
int relationcircle(Circle v) {
/*
两圆的关系
5 相离
4 外切
3 相交
2 内切
1 内含
*/
double d = p.distance(v.p);
if(sgn(d-r-v.r) > 0) return 5;
if(sgn(d-r-v.r) == 0) return 4;
double l = fabs(r-v.r);
if(sgn(d-r-v.r) < 0 && sgn(d-l) > 0) return 3;
if(sgn(d-l) == 0) return 2;
if(sgn(d-l) < 0) return 1;
}
int pointcrosscircle(Circle v, Point &p1, Point &p2) {
//求两个圆的交点，返回交点数
int rel = relationcircle(v);
if(rel == 1 || rel == 5) return 0;
double d = p.distance(v.p);
double l = (d*d + r*r - v.r*v.r) / (2*d);
double h = sqrt(r*r-l*l);
Point tmp = p + (v.p-p).trunc(l);
p1 = tmp + ((v.p-p).rotleft().trunc(h));
p2 = tmp + ((v.p-p).rotright().trunc(h));
if(rel == 2 || rel == 4)
return 1;
return 2;
}
int pointcrossline(Line v, Point &p1, Point &p2) {
//求直线和圆的交点，返回交点数
if(!(*this).relationline(v)) return 0;
Point a = v.lineprog(p);
double d = v.dispointtoline(p);
d = sqrt(r*r-d*d);
if(sgn(d) == 0) {
p1 = a;
p2 = a;
return 1;
}
p1 = a + (v.e-v.s).trunc(d);
p2 = a - (v.e-v.s).trunc(d);
return 2;
}
int getcircle(Point a, Point b, double r1, Circle &c1, Circle &c2) {
//过a, b两点, 半径为 r1 的圆, 返回圆数
Circle x(a, r1), y(b, r1);
int t = x.pointcrosscircle(y, c1.p, c2.p);
if(!t) return 0;
c1.r = c2.r = r1;
return t;
}
int getcircle(Line u, Point q, double r1, Circle &c1, Circle &c2) {
//与直线 u 相切, 过点 q, 半径为 r1 的圆,返回圆数
double dis = u.dispointtoline(q);
if(sgn(dis-2*r1) > 0) return 0;
if(sgn(dis) == 0) {
c1.p = q + ((u.e-u.s).rotleft().trunc(r1));
c2.p = q + ((u.e-u.s).rotright().trunc(r1));
c1.r = c2.r = r1;
return 2;
}
Line u1 = Line((u.s + (u.e-u.s).rotleft().trunc(r1)), (u.e + (u.e-u.s).rotleft().trunc(r1)));
Line u2 = Line((u.s + (u.e-u.s).rotright().trunc(r1)), (u.e + (u.e-u.s).rotright().trunc(r1)));
Circle cc = Circle(q, r1);
Point p1, p2;
if(!cc.pointcrossline(u1, p1, p2))
cc.pointcrossline(u2, p1, p2);
c1 = Circle(p1, r1);
if(p1 == p2) {
c2 = c1;
return 1;
}
c2 = Circle(p2, r1);
return 2;
}
int getcircle(Line u, Line v, double r1, Circle &c1, Circle &c2, Circle &c3, Circle &c4) {
//同时与直线u, v相切, 半径为 r1 的圆
if(u.parallel(v))return 0;//两直线平行
Line u1 = Line(u.s + (u.e-u.s).rotleft().trunc(r1),u.e + (u.e-u.s).rotleft().trunc(r1));
Line u2 = Line(u.s + (u.e-u.s).rotright().trunc(r1),u.e + (u.e-u.s).rotright().trunc(r1));
Line v1 = Line(v.s + (v.e-v.s).rotleft().trunc(r1),v.e + (v.e-v.s).rotleft().trunc(r1));
Line v2 = Line(v.s + (v.e-v.s).rotright().trunc(r1),v.e + (v.e-v.s).rotright().trunc(r1));
c1.r = c2.r = c3.r = c4.r = r1;
c1.p = u1.crosspoint(v1);
c2.p = u1.crosspoint(v2);
c3.p = u2.crosspoint(v1);
c4.p = u2.crosspoint(v2);
return 4;
}
int getcircle(Circle cx, Circle cy, double r1, Circle &c1, Circle &c2) {
//同时与不相交圆cx, cy相切，半径为 r1 的圆
Circle x(cx.p, r1+cx.r), y(cy.p, r1+cy.r);
int t = x.pointcrosscircle(y, c1.p, c2.p);
if(!t) return 0;
c1.r = c2.r = r1;
return t;
}
int tangentline(Point q, Line &u, Line &v) {
//过一点做圆的切线,求切线
int x = relation(q);
if(x == 2)return 0;
if(x == 1) {
u = Line(q, q + (q-p).rotleft());
v = u;
return 1;
}
double d = p.distance(q);
double l = r*r/d;
double h = sqrt(r*r-l*l);
u = Line(q, p + ((q-p).trunc(l) + (q-p).rotleft().trunc(h)));
v = Line(q, p + ((q-p).trunc(l) + (q-p).rotright().trunc(h)));
return 2;
}
double areacircle(Circle v) {
//两相交圆的面积
int rel = relationcircle(v);
if(rel >= 4) return 0.0;
if(rel <= 2) return min(area(), v.area());
double d = p.distance(v.p);
double hf = (r+v.r+d)/2.0;
double ss = 2*sqrt(hf*(hf-r)*(hf-v.r)*(hf-d));
double a1 = acos((r*r+d*d-v.r*v.r) / (2.0*r*d));
a1 = a1*r*r;
double a2 = acos((v.r*v.r+d*d-r*r) / (2.0*v.r*d));
a2 = a2*v.r*v.r;
return a1+a2-ss;
}
double areatriangle(Point a, Point b) {
//求圆和三角形 pab 的相交面积
if(sgn((p-a)^(p-b)) == 0) return 0.0;
Point q[5];
int len = 0;
q[len++] = a;
Line l(a,b);
Point p1,p2;
if(pointcrossline(l,q[1],q[2]) == 2) {
if(sgn((a-q[1])*(b-q[1])) < 0) q[len++] = q[1];
if(sgn((a-q[2])*(b-q[2])) < 0) q[len++] = q[2];
}
q[len++] = b;
if(len == 4 && sgn((q[0]-q[1])*(q[2]-q[1])) > 0)
swap(q[1], q[2]);
double res = 0;
for(int i = 0; i < len-1; i++) {
if(relation(q[i]) == 0 || relation(q[i+1]) == 0) {
double arg = p.rad(q[i], q[i+1]);
res += r*r*arg/2.0;
} else {
res += fabs((q[i]-p)^(q[i+1]-p)) / 2.0;
}
}
return res;
}
};
struct Polygon {
int n;
Point p[maxn];
Line l[maxn];
Polygon() {
n = 0;
}
void add(Point q) {
p[++n] = q;
}
void getline() {
for(int i=1; i<=n; i++) {
l[i] = Line(p[i], p[i%n+1]);
}
}
struct cmp {
//极角排序的比较函数
Point p;
cmp(Point _p) {
p = _p;
}
bool operator()(Point _a, Point _b) const {
Point a = _a, b = _b;
int d = sgn((a-p) ^ (b-p));
if(d == 0) {
return sgn(a.distance(p) - b.distance(p)) < 0;
} else {
return d > 0;
}
}
};
void norm() {
//极角排序
int id = 1;
for(int i=2; i<=n; i++) {
if(p[i] < p[id])
id = i;
}
swap(p[id], p[1]);
sort(p+1, p+1+n, cmp(p[1]));
}
void Graham(Polygon &convex) { //引用
//求解凸包
//convex凸包
norm();
mes(convex.p, 0);
int &top = convex.n = 0; //引用
if(n == 1) {
convex.p[++top] = p[1];
} else if(n == 2) {
convex.p[++top] = p[1];
convex.p[++top] = p[2];
if(convex.p[1] == convex.p[2]) top--;
} else {
convex.p[++top] = p[1];
convex.p[++top] = p[2];
for(int i=3; i<=n; i++) {
while(top>1 && sgn((convex.p[top]-convex.p[top-1]) ^ (p[i]-convex.p[top-1])) <= 0)
top--;
convex.p[++top] = p[i];
}
if(top == 2 && convex.p[1] == convex.p[2])
top--;
}
}
bool isconvex() {
//判断多边形是不是凸的
//1是凸的 0不是凸的
bool s[3];
mes(s, false);
for(int i=1; i<=n; i++) {
int j = i%n + 1;
int k = j%n + 1;
s[sgn((p[j]-p[i]) ^ (p[k]-p[i])) + 1] = true;
if(s[0] && s[2]) return false;
}
return true;
}
int relationpoint(Point q) {
/*
点和多边形关系
3 点上
2 边上
1 内部
0 外部
*/
for(int i=1; i<=n; i++) {
if(p[i] == q) return 3;
}
getline();
for(int i=1; i<=n; i++) {
if(l[i].pointonseg(q)) return 2;
}
int cnt = 0;
for(int i=1; i<=n; i++) {
int j = i%n + 1;
int k = sgn((q-p[j]) ^ (p[i]-p[j]));
int u = sgn(p[i].y - q.y);
int v = sgn(p[j].y - q.y);
if(k > 0 && u < 0 && v >= 0) cnt++;
if(k < 0 && v < 0 && u >= 0) cnt--;
}
return cnt != 0;
}
bool getdir() {
//得到方向
//1表示逆时针，0表示顺时针
double sum = 0;
for(int i=1; i<=n; i++) {
sum += p[i] ^ p[i%n+1];
}
if(sgn(sum) > 0) return 1;
return 0;
}
Point getbarycentre() {
//得到重心
Point ans(0, 0);
double area = 0;
for(int i=2; i<=n-1; i++) {
double tmp = (p[i]-p[1]) ^ (p[i+1]-p[0]);
if(sgn(tmp) == 0) continue;
area += tmp;
ans.x = (p[1].x + p[i].x + p[i+1].x) / 3 * tmp;
ans.y = (p[0].y + p[i].y + p[i+1].y) / 3 * tmp;
}
if(sgn(area)) ans = ans / area;
return ans;
}
void convexcut(Line u, Polygon &po) {
//直线切割凸多边形左侧,注意直线方向
int &top = po.n = 0; //引用
for(int i=1; i<=n; i++) {
int d1 = sgn((u.e-u.s) ^ (p[i]-u.s));
int d2 = sgn((u.e-u.s) ^ (p[i%n+1]-u.s));
if(d1 >= 0) po.p[++top] = p[i];
if(d1*d2<0) po.p[++top] = u.crosspoint(Line(p[i], p[i%n+1]));
}
}
double getperimeter() {
//多边形的周长
double ans = 0.0;
if(n == 1) {
return ans;
} else if(n == 2) {
ans += p[1].distance(p[2]);
} else {
for(int i=1; i<=n; i++) {
ans += p[i].distance(p[i%n+1]);
}
}
return ans;
}
double getarea() {
//多边形面积
double sum = 0;
for(int i=1; i<=n; i++) {
sum += (p[i] ^ p[i%n+1]);
}
return fabs(sum)/2.0;
}
double areacircle(Circle c) {
//多边形和圆相交的面积
double ans = 0;
for(int i=1; i<=n; i++) {
int j = i%n+1;
if(sgn((p[j]-c.p) ^ (p[i]-c.p)) >= 0)
ans += c.areatriangle(p[i], p[j]);
else
ans -= c.areatriangle(p[i], p[j]);
}
return fabs(ans);
}
int relationcircle(Circle c) {
/*
多边形和圆的关系
2 圆完全在多边形内
1 圆在多边形内，但是碰到了多边形边界
0 其他
*/
getline();
int x = 2;
if(relationpoint(c.p) != 1) return 0;
for(int i=1; i<=n; i++) {
if(c.relationseg(l[i]) == 2) return 0;
if(c.relationseg(l[i]) == 1) x = 1;
}
return x;
}
};
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