This documentation is automatically generated by online-judge-tools/verification-helper
#include "geometry/template.hpp"
using Real = double;
using Point = complex<Real>;
const Real EPS = 1e-8, PI = acos(-1);
inline bool eq(Real a, Real b) { return fabs(b - a) < EPS; }
Point operator*(const Point &p, const Real &d) {
return Point(real(p) * d, imag(p) * d);
}
istream &operator>>(istream &is, Point &p) {
Real a, b;
is >> a >> b;
p = Point(a, b);
return is;
}
ostream &operator<<(ostream &os, Point &p) {
return os << fixed << setprecision(10) << p.real() << " " << p.imag();
}
// rotate point p counterclockwise by theta rad
Point rotate(Real theta, const Point &p) {
return Point(cos(theta) * p.real() - sin(theta) * p.imag(),
sin(theta) * p.real() + cos(theta) * p.imag());
}
Real radian_to_degree(Real r) { return (r * 180.0 / PI); }
Real degree_to_radian(Real d) { return (d * PI / 180.0); }
// smaller angle of the a-b-c
Real get_angle(const Point &a, const Point &b, const Point &c) {
const Point v(b - a), w(c - b);
Real alpha = atan2(v.imag(), v.real()), beta = atan2(w.imag(), w.real());
if (alpha > beta) swap(alpha, beta);
Real theta = (beta - alpha);
return min(theta, 2 * acos(-1) - theta);
}
namespace std {
bool operator<(const Point &a, const Point &b) {
return a.real() != b.real() ? a.real() < b.real() : a.imag() < b.imag();
}
} // namespace std
struct Line {
Point a, b;
Line() = default;
Line(Point a, Point b) : a(a), b(b) {}
Line(Real A, Real B, Real C) // Ax + By = C
{
if (eq(A, 0))
a = Point(0, C / B), b = Point(1, C / B);
else if (eq(B, 0))
b = Point(C / A, 0), b = Point(C / A, 1);
else
a = Point(0, C / B), b = Point(C / A, 0);
}
friend ostream &operator<<(ostream &os, Line &p) {
return os << p.a << " to " << p.b;
}
friend istream &operator>>(istream &is, Line &a) { return is >> a.a >> a.b; }
};
struct Segment : Line {
Segment() = default;
Segment(Point a, Point b) : Line(a, b) {}
};
struct Circle {
Point p;
Real r;
Circle() = default;
Circle(Point p, Real r) : p(p), r(r) {}
};
using Points = vector<Point>;
using Polygon = vector<Point>;
using Segments = vector<Segment>;
using Lines = vector<Line>;
using Circles = vector<Circle>;
Real cross(const Point &a, const Point &b) {
return real(a) * imag(b) - imag(a) * real(b);
}
Real dot(const Point &a, const Point &b) {
return real(a) * real(b) + imag(a) * imag(b);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_C
int ccw(const Point &a, Point b, Point c) {
b = b - a, c = c - a;
if (cross(b, c) > EPS) return +1; // "COUNTER_CLOCKWISE"
if (cross(b, c) < -EPS) return -1; // "CLOCKWISE"
if (dot(b, c) < 0) return +2; // "ONLINE_BACK" c-a-b
if (norm(b) < norm(c)) return -2; // "ONLINE_FRONT" a-b-c
return 0; // "ON_SEGMENT" a-c-b
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_A
bool parallel(const Line &a, const Line &b) {
return eq(cross(a.b - a.a, b.b - b.a), 0.0);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_A
bool orthogonal(const Line &a, const Line &b) {
return eq(dot(a.a - a.b, b.a - b.b), 0.0);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_A
Point projection(const Line &l, const Point &p) {
double t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b);
return l.a + (l.a - l.b) * t;
}
Point projection(const Segment &l, const Point &p) {
double t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b);
return l.a + (l.a - l.b) * t;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_B
Point reflection(const Line &l, const Point &p) {
return p + (projection(l, p) - p) * 2.0;
}
bool intersect(const Line &l, const Point &p) {
return abs(ccw(l.a, l.b, p)) != 1;
}
bool intersect(const Line &l, const Line &m) {
return abs(cross(l.b - l.a, m.b - m.a)) > EPS ||
abs(cross(l.b - l.a, m.b - l.a)) < EPS;
}
bool intersect(const Segment &s, const Point &p) {
return ccw(s.a, s.b, p) == 0;
}
bool intersect(const Line &l, const Segment &s) {
return cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < EPS;
}
Real distance(const Line &l, const Point &p);
bool intersect(const Circle &c, const Line &l) {
return distance(l, c.p) <= c.r + EPS;
}
bool intersect(const Circle &c, const Point &p) {
return abs(abs(p - c.p) - c.r) < EPS;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_B
bool intersect(const Segment &s, const Segment &t) {
return ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 &&
ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0;
}
int intersect(const Circle &c, const Segment &l) {
if (norm(projection(l, c.p) - c.p) - c.r * c.r > EPS) return 0;
auto d1 = abs(c.p - l.a), d2 = abs(c.p - l.b);
if (d1 < c.r + EPS && d2 < c.r + EPS) return 0;
if (d1 < c.r - EPS && d2 > c.r + EPS || d1 > c.r + EPS && d2 < c.r - EPS)
return 1;
const Point h = projection(l, c.p);
if (dot(l.a - h, l.b - h) < 0) return 2;
return 0;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_A&lang=jp
int intersect(Circle c1, Circle c2) {
if (c1.r < c2.r) swap(c1, c2);
Real d = abs(c1.p - c2.p);
if (c1.r + c2.r < d) return 4;
if (eq(c1.r + c2.r, d)) return 3;
if (c1.r - c2.r < d) return 2;
if (eq(c1.r - c2.r, d)) return 1;
return 0;
}
Real distance(const Point &a, const Point &b) { return abs(a - b); }
Real distance(const Line &l, const Point &p) {
return abs(p - projection(l, p));
}
Real distance(const Line &l, const Line &m) {
return intersect(l, m) ? 0 : distance(l, m.a);
}
Real distance(const Segment &s, const Point &p) {
Point r = projection(s, p);
if (intersect(s, r)) return abs(r - p);
return min(abs(s.a - p), abs(s.b - p));
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_D
Real distance(const Segment &a, const Segment &b) {
if (intersect(a, b)) return 0;
return min(
{distance(a, b.a), distance(a, b.b), distance(b, a.a), distance(b, a.b)});
}
Real distance(const Line &l, const Segment &s) {
if (intersect(l, s)) return 0;
return min(distance(l, s.a), distance(l, s.b));
}
Point crosspoint(const Line &l, const Line &m) {
Real A = cross(l.b - l.a, m.b - m.a);
Real B = cross(l.b - l.a, l.b - m.a);
if (eq(abs(A), 0.0) && eq(abs(B), 0.0)) return m.a;
return m.a + (m.b - m.a) * B / A;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_C
Point crosspoint(const Segment &l, const Segment &m) {
return crosspoint(Line(l), Line(m));
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_D
pair<Point, Point> crosspoint(const Circle &c, const Line l) {
Point pr = projection(l, c.p);
Point e = (l.b - l.a) / abs(l.b - l.a);
if (eq(distance(l, c.p), c.r)) return {pr, pr};
double base = sqrt(c.r * c.r - norm(pr - c.p));
return {pr - e * base, pr + e * base};
}
pair<Point, Point> crosspoint(const Circle &c, const Segment &l) {
Line aa = Line(l.a, l.b);
if (intersect(c, l) == 2) return crosspoint(c, aa);
auto ret = crosspoint(c, aa);
if (dot(l.a - ret.first, l.b - ret.first) < 0)
ret.second = ret.first;
else
ret.first = ret.second;
return ret;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_E
pair<Point, Point> crosspoint(const Circle &c1, const Circle &c2) {
Real d = abs(c1.p - c2.p);
Real a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));
Real t = atan2(c2.p.imag() - c1.p.imag(), c2.p.real() - c1.p.real());
Point p1 = c1.p + Point(cos(t + a) * c1.r, sin(t + a) * c1.r);
Point p2 = c1.p + Point(cos(t - a) * c1.r, sin(t - a) * c1.r);
return {p1, p2};
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_F
// tangent of circle c through point p
pair<Point, Point> tangent(const Circle &c1, const Point &p2) {
return crosspoint(c1, Circle(p2, sqrt(norm(c1.p - p2) - c1.r * c1.r)));
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_G
// common tangent of circles c1 and c2
Lines tangent(Circle c1, Circle c2) {
Lines ret;
if (c1.r < c2.r) swap(c1, c2);
Real g = norm(c1.p - c2.p);
if (eq(g, 0)) return ret;
Point u = (c2.p - c1.p) / sqrt(g);
Point v = rotate(PI * 0.5, u);
for (int s : {-1, 1}) {
Real h = (c1.r + s * c2.r) / sqrt(g);
if (eq(1 - h * h, 0)) {
ret.emplace_back(c1.p + u * c1.r, c1.p + (u + v) * c1.r);
} else if (1 - h * h > 0) {
Point uu = u * h, vv = v * sqrt(1 - h * h);
ret.emplace_back(c1.p + (uu + vv) * c1.r, c2.p - (uu + vv) * c2.r * s);
ret.emplace_back(c1.p + (uu - vv) * c1.r, c2.p - (uu - vv) * c2.r * s);
}
}
return ret;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_B
bool is_convex(const Polygon &p) {
int n = (int)p.size();
for (int i = 0; i < n; i++) {
if (ccw(p[(i + n - 1) % n], p[i], p[(i + 1) % n]) == -1) return false;
}
return true;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_A
Polygon convex_hull(Polygon &p) {
int n = (int)p.size(), k = 0;
if (n <= 2) return p;
sort(p.begin(), p.end());
vector<Point> ch(2 * n);
for (int i = 0; i < n; ch[k++] = p[i++]) {
while (k >= 2 && cross(ch[k - 1] - ch[k - 2], p[i] - ch[k - 1]) < EPS) --k;
}
for (int i = n - 2, t = k + 1; i >= 0; ch[k++] = p[i--]) {
while (k >= t && cross(ch[k - 1] - ch[k - 2], p[i] - ch[k - 1]) < EPS) --k;
}
ch.resize(k - 1);
return ch;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_C
enum { OUT, ON, IN };
int contains(const Polygon &Q, const Point &p) {
bool in = false;
for (int i = 0; i < Q.size(); i++) {
Point a = Q[i] - p, b = Q[(i + 1) % Q.size()] - p;
if (a.imag() > b.imag()) swap(a, b);
if (a.imag() <= 0 && 0 < b.imag() && cross(a, b) < 0) in = !in;
if (cross(a, b) == 0 && dot(a, b) <= 0) return ON;
}
return in ? IN : OUT;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=0412
int convex_contains(const Polygon &Q, const Point &p) {
int N = (int)Q.size();
Point g = (Q[0] + Q[N / 3] + Q[N * 2 / 3]) / 3.0;
if (g == p) return IN;
Point gp = p - g;
int l = 0, r = N;
while (r - l > 1) {
int mid = (l + r) / 2;
Point gl = Q[l] - g;
Point gm = Q[mid] - g;
if (cross(gl, gm) > 0) {
if (cross(gl, gp) >= 0 && cross(gm, gp) <= 0)
r = mid;
else
l = mid;
} else {
if (cross(gl, gp) <= 0 && cross(gm, gp) >= 0)
l = mid;
else
r = mid;
}
}
r %= N;
Real v = cross(Q[l] - p, Q[r] - p);
return eq(v, 0.0) ? ON : v < 0.0 ? OUT : IN;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=1033
// deduplication of line segments
void merge_segments(vector<Segment> &segs) {
auto merge_if_able = [](Segment &s1, const Segment &s2) {
if (abs(cross(s1.b - s1.a, s2.b - s2.a)) > EPS) return false;
if (ccw(s1.a, s2.a, s1.b) == 1 || ccw(s1.a, s2.a, s1.b) == -1) return false;
if (ccw(s1.a, s1.b, s2.a) == -2 || ccw(s2.a, s2.b, s1.a) == -2)
return false;
s1 = Segment(min(s1.a, s2.a), max(s1.b, s2.b));
return true;
};
for (int i = 0; i < segs.size(); i++) {
if (segs[i].b < segs[i].a) swap(segs[i].a, segs[i].b);
}
for (int i = 0; i < segs.size(); i++) {
for (int j = i + 1; j < segs.size(); j++) {
if (merge_if_able(segs[i], segs[j])) {
segs[j--] = segs.back(), segs.pop_back();
}
}
}
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=1033
// construct a graph with the vertex of the intersection of any two line
// segments
vector<vector<int> > segment_arrangement(vector<Segment> &segs,
vector<Point> &ps) {
vector<vector<int> > g;
int N = (int)segs.size();
for (int i = 0; i < N; i++) {
ps.emplace_back(segs[i].a);
ps.emplace_back(segs[i].b);
for (int j = i + 1; j < N; j++) {
const Point p1 = segs[i].b - segs[i].a;
const Point p2 = segs[j].b - segs[j].a;
if (cross(p1, p2) == 0) continue;
if (intersect(segs[i], segs[j])) {
ps.emplace_back(crosspoint(segs[i], segs[j]));
}
}
}
sort(begin(ps), end(ps));
ps.erase(unique(begin(ps), end(ps)), end(ps));
int M = (int)ps.size();
g.resize(M);
for (int i = 0; i < N; i++) {
vector<int> vec;
for (int j = 0; j < M; j++) {
if (intersect(segs[i], ps[j])) {
vec.emplace_back(j);
}
}
for (int j = 1; j < vec.size(); j++) {
g[vec[j - 1]].push_back(vec[j]);
g[vec[j]].push_back(vec[j - 1]);
}
}
return (g);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_C
// cut with a straight line l and return a convex polygon on the left
Polygon convex_cut(const Polygon &U, Line l) {
Polygon ret;
for (int i = 0; i < U.size(); i++) {
Point now = U[i], nxt = U[(i + 1) % U.size()];
if (ccw(l.a, l.b, now) != -1) ret.push_back(now);
if (ccw(l.a, l.b, now) * ccw(l.a, l.b, nxt) < 0) {
ret.push_back(crosspoint(Line(now, nxt), l));
}
}
return (ret);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_A
Real area(const Polygon &p) {
Real A = 0;
for (int i = 0; i < p.size(); ++i) {
A += cross(p[i], p[(i + 1) % p.size()]);
}
return A * 0.5;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_H
Real area(const Polygon &p, const Circle &c) {
if (p.size() < 3) return 0.0;
function<Real(Circle, Point, Point)> cross_area =
[&](const Circle &c, const Point &a, const Point &b) {
Point va = c.p - a, vb = c.p - b;
Real f = cross(va, vb), ret = 0.0;
if (eq(f, 0.0)) return ret;
if (max(abs(va), abs(vb)) < c.r + EPS) return f;
if (distance(Segment(a, b), c.p) > c.r - EPS)
return c.r * c.r * arg(vb * conj(va));
auto u = crosspoint(c, Segment(a, b));
vector<Point> tot{a, u.first, u.second, b};
for (int i = 0; i + 1 < tot.size(); i++) {
ret += cross_area(c, tot[i], tot[i + 1]);
}
return ret;
};
Real A = 0;
for (int i = 0; i < p.size(); i++) {
A += cross_area(c, p[i], p[(i + 1) % p.size()]);
}
return A;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_B
Real convex_diameter(const Polygon &p) {
int N = (int)p.size();
int is = 0, js = 0;
for (int i = 1; i < N; i++) {
if (p[i].imag() > p[is].imag()) is = i;
if (p[i].imag() < p[js].imag()) js = i;
}
Real maxdis = norm(p[is] - p[js]);
int maxi, maxj, i, j;
i = maxi = is;
j = maxj = js;
do {
if (cross(p[(i + 1) % N] - p[i], p[(j + 1) % N] - p[j]) >= 0) {
j = (j + 1) % N;
} else {
i = (i + 1) % N;
}
if (norm(p[i] - p[j]) > maxdis) {
maxdis = norm(p[i] - p[j]);
maxi = i;
maxj = j;
}
} while (i != is || j != js);
return sqrt(maxdis);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_5_A
Real closest_pair(Points ps) {
if (ps.size() <= 1) throw(0);
sort(begin(ps), end(ps));
auto compare_y = [&](const Point &a, const Point &b) {
return imag(a) < imag(b);
};
vector<Point> beet(ps.size());
const Real INF = 1e18;
function<Real(int, int)> rec = [&](int left, int right) {
if (right - left <= 1) return INF;
int mid = (left + right) >> 1;
auto x = real(ps[mid]);
auto ret = min(rec(left, mid), rec(mid, right));
inplace_merge(begin(ps) + left, begin(ps) + mid, begin(ps) + right,
compare_y);
int ptr = 0;
for (int i = left; i < right; i++) {
if (abs(real(ps[i]) - x) >= ret) continue;
for (int j = 0; j < ptr; j++) {
auto luz = ps[i] - beet[ptr - j - 1];
if (imag(luz) >= ret) break;
ret = min(ret, abs(luz));
}
beet[ptr++] = ps[i];
}
return ret;
};
return rec(0, (int)ps.size());
}
#line 1 "geometry/template.hpp"
using Real = double;
using Point = complex<Real>;
const Real EPS = 1e-8, PI = acos(-1);
inline bool eq(Real a, Real b) { return fabs(b - a) < EPS; }
Point operator*(const Point &p, const Real &d) {
return Point(real(p) * d, imag(p) * d);
}
istream &operator>>(istream &is, Point &p) {
Real a, b;
is >> a >> b;
p = Point(a, b);
return is;
}
ostream &operator<<(ostream &os, Point &p) {
return os << fixed << setprecision(10) << p.real() << " " << p.imag();
}
// rotate point p counterclockwise by theta rad
Point rotate(Real theta, const Point &p) {
return Point(cos(theta) * p.real() - sin(theta) * p.imag(),
sin(theta) * p.real() + cos(theta) * p.imag());
}
Real radian_to_degree(Real r) { return (r * 180.0 / PI); }
Real degree_to_radian(Real d) { return (d * PI / 180.0); }
// smaller angle of the a-b-c
Real get_angle(const Point &a, const Point &b, const Point &c) {
const Point v(b - a), w(c - b);
Real alpha = atan2(v.imag(), v.real()), beta = atan2(w.imag(), w.real());
if (alpha > beta) swap(alpha, beta);
Real theta = (beta - alpha);
return min(theta, 2 * acos(-1) - theta);
}
namespace std {
bool operator<(const Point &a, const Point &b) {
return a.real() != b.real() ? a.real() < b.real() : a.imag() < b.imag();
}
} // namespace std
struct Line {
Point a, b;
Line() = default;
Line(Point a, Point b) : a(a), b(b) {}
Line(Real A, Real B, Real C) // Ax + By = C
{
if (eq(A, 0))
a = Point(0, C / B), b = Point(1, C / B);
else if (eq(B, 0))
b = Point(C / A, 0), b = Point(C / A, 1);
else
a = Point(0, C / B), b = Point(C / A, 0);
}
friend ostream &operator<<(ostream &os, Line &p) {
return os << p.a << " to " << p.b;
}
friend istream &operator>>(istream &is, Line &a) { return is >> a.a >> a.b; }
};
struct Segment : Line {
Segment() = default;
Segment(Point a, Point b) : Line(a, b) {}
};
struct Circle {
Point p;
Real r;
Circle() = default;
Circle(Point p, Real r) : p(p), r(r) {}
};
using Points = vector<Point>;
using Polygon = vector<Point>;
using Segments = vector<Segment>;
using Lines = vector<Line>;
using Circles = vector<Circle>;
Real cross(const Point &a, const Point &b) {
return real(a) * imag(b) - imag(a) * real(b);
}
Real dot(const Point &a, const Point &b) {
return real(a) * real(b) + imag(a) * imag(b);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_C
int ccw(const Point &a, Point b, Point c) {
b = b - a, c = c - a;
if (cross(b, c) > EPS) return +1; // "COUNTER_CLOCKWISE"
if (cross(b, c) < -EPS) return -1; // "CLOCKWISE"
if (dot(b, c) < 0) return +2; // "ONLINE_BACK" c-a-b
if (norm(b) < norm(c)) return -2; // "ONLINE_FRONT" a-b-c
return 0; // "ON_SEGMENT" a-c-b
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_A
bool parallel(const Line &a, const Line &b) {
return eq(cross(a.b - a.a, b.b - b.a), 0.0);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_A
bool orthogonal(const Line &a, const Line &b) {
return eq(dot(a.a - a.b, b.a - b.b), 0.0);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_A
Point projection(const Line &l, const Point &p) {
double t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b);
return l.a + (l.a - l.b) * t;
}
Point projection(const Segment &l, const Point &p) {
double t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b);
return l.a + (l.a - l.b) * t;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_B
Point reflection(const Line &l, const Point &p) {
return p + (projection(l, p) - p) * 2.0;
}
bool intersect(const Line &l, const Point &p) {
return abs(ccw(l.a, l.b, p)) != 1;
}
bool intersect(const Line &l, const Line &m) {
return abs(cross(l.b - l.a, m.b - m.a)) > EPS ||
abs(cross(l.b - l.a, m.b - l.a)) < EPS;
}
bool intersect(const Segment &s, const Point &p) {
return ccw(s.a, s.b, p) == 0;
}
bool intersect(const Line &l, const Segment &s) {
return cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < EPS;
}
Real distance(const Line &l, const Point &p);
bool intersect(const Circle &c, const Line &l) {
return distance(l, c.p) <= c.r + EPS;
}
bool intersect(const Circle &c, const Point &p) {
return abs(abs(p - c.p) - c.r) < EPS;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_B
bool intersect(const Segment &s, const Segment &t) {
return ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 &&
ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0;
}
int intersect(const Circle &c, const Segment &l) {
if (norm(projection(l, c.p) - c.p) - c.r * c.r > EPS) return 0;
auto d1 = abs(c.p - l.a), d2 = abs(c.p - l.b);
if (d1 < c.r + EPS && d2 < c.r + EPS) return 0;
if (d1 < c.r - EPS && d2 > c.r + EPS || d1 > c.r + EPS && d2 < c.r - EPS)
return 1;
const Point h = projection(l, c.p);
if (dot(l.a - h, l.b - h) < 0) return 2;
return 0;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_A&lang=jp
int intersect(Circle c1, Circle c2) {
if (c1.r < c2.r) swap(c1, c2);
Real d = abs(c1.p - c2.p);
if (c1.r + c2.r < d) return 4;
if (eq(c1.r + c2.r, d)) return 3;
if (c1.r - c2.r < d) return 2;
if (eq(c1.r - c2.r, d)) return 1;
return 0;
}
Real distance(const Point &a, const Point &b) { return abs(a - b); }
Real distance(const Line &l, const Point &p) {
return abs(p - projection(l, p));
}
Real distance(const Line &l, const Line &m) {
return intersect(l, m) ? 0 : distance(l, m.a);
}
Real distance(const Segment &s, const Point &p) {
Point r = projection(s, p);
if (intersect(s, r)) return abs(r - p);
return min(abs(s.a - p), abs(s.b - p));
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_D
Real distance(const Segment &a, const Segment &b) {
if (intersect(a, b)) return 0;
return min(
{distance(a, b.a), distance(a, b.b), distance(b, a.a), distance(b, a.b)});
}
Real distance(const Line &l, const Segment &s) {
if (intersect(l, s)) return 0;
return min(distance(l, s.a), distance(l, s.b));
}
Point crosspoint(const Line &l, const Line &m) {
Real A = cross(l.b - l.a, m.b - m.a);
Real B = cross(l.b - l.a, l.b - m.a);
if (eq(abs(A), 0.0) && eq(abs(B), 0.0)) return m.a;
return m.a + (m.b - m.a) * B / A;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_C
Point crosspoint(const Segment &l, const Segment &m) {
return crosspoint(Line(l), Line(m));
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_D
pair<Point, Point> crosspoint(const Circle &c, const Line l) {
Point pr = projection(l, c.p);
Point e = (l.b - l.a) / abs(l.b - l.a);
if (eq(distance(l, c.p), c.r)) return {pr, pr};
double base = sqrt(c.r * c.r - norm(pr - c.p));
return {pr - e * base, pr + e * base};
}
pair<Point, Point> crosspoint(const Circle &c, const Segment &l) {
Line aa = Line(l.a, l.b);
if (intersect(c, l) == 2) return crosspoint(c, aa);
auto ret = crosspoint(c, aa);
if (dot(l.a - ret.first, l.b - ret.first) < 0)
ret.second = ret.first;
else
ret.first = ret.second;
return ret;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_E
pair<Point, Point> crosspoint(const Circle &c1, const Circle &c2) {
Real d = abs(c1.p - c2.p);
Real a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));
Real t = atan2(c2.p.imag() - c1.p.imag(), c2.p.real() - c1.p.real());
Point p1 = c1.p + Point(cos(t + a) * c1.r, sin(t + a) * c1.r);
Point p2 = c1.p + Point(cos(t - a) * c1.r, sin(t - a) * c1.r);
return {p1, p2};
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_F
// tangent of circle c through point p
pair<Point, Point> tangent(const Circle &c1, const Point &p2) {
return crosspoint(c1, Circle(p2, sqrt(norm(c1.p - p2) - c1.r * c1.r)));
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_G
// common tangent of circles c1 and c2
Lines tangent(Circle c1, Circle c2) {
Lines ret;
if (c1.r < c2.r) swap(c1, c2);
Real g = norm(c1.p - c2.p);
if (eq(g, 0)) return ret;
Point u = (c2.p - c1.p) / sqrt(g);
Point v = rotate(PI * 0.5, u);
for (int s : {-1, 1}) {
Real h = (c1.r + s * c2.r) / sqrt(g);
if (eq(1 - h * h, 0)) {
ret.emplace_back(c1.p + u * c1.r, c1.p + (u + v) * c1.r);
} else if (1 - h * h > 0) {
Point uu = u * h, vv = v * sqrt(1 - h * h);
ret.emplace_back(c1.p + (uu + vv) * c1.r, c2.p - (uu + vv) * c2.r * s);
ret.emplace_back(c1.p + (uu - vv) * c1.r, c2.p - (uu - vv) * c2.r * s);
}
}
return ret;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_B
bool is_convex(const Polygon &p) {
int n = (int)p.size();
for (int i = 0; i < n; i++) {
if (ccw(p[(i + n - 1) % n], p[i], p[(i + 1) % n]) == -1) return false;
}
return true;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_A
Polygon convex_hull(Polygon &p) {
int n = (int)p.size(), k = 0;
if (n <= 2) return p;
sort(p.begin(), p.end());
vector<Point> ch(2 * n);
for (int i = 0; i < n; ch[k++] = p[i++]) {
while (k >= 2 && cross(ch[k - 1] - ch[k - 2], p[i] - ch[k - 1]) < EPS) --k;
}
for (int i = n - 2, t = k + 1; i >= 0; ch[k++] = p[i--]) {
while (k >= t && cross(ch[k - 1] - ch[k - 2], p[i] - ch[k - 1]) < EPS) --k;
}
ch.resize(k - 1);
return ch;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_C
enum { OUT, ON, IN };
int contains(const Polygon &Q, const Point &p) {
bool in = false;
for (int i = 0; i < Q.size(); i++) {
Point a = Q[i] - p, b = Q[(i + 1) % Q.size()] - p;
if (a.imag() > b.imag()) swap(a, b);
if (a.imag() <= 0 && 0 < b.imag() && cross(a, b) < 0) in = !in;
if (cross(a, b) == 0 && dot(a, b) <= 0) return ON;
}
return in ? IN : OUT;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=0412
int convex_contains(const Polygon &Q, const Point &p) {
int N = (int)Q.size();
Point g = (Q[0] + Q[N / 3] + Q[N * 2 / 3]) / 3.0;
if (g == p) return IN;
Point gp = p - g;
int l = 0, r = N;
while (r - l > 1) {
int mid = (l + r) / 2;
Point gl = Q[l] - g;
Point gm = Q[mid] - g;
if (cross(gl, gm) > 0) {
if (cross(gl, gp) >= 0 && cross(gm, gp) <= 0)
r = mid;
else
l = mid;
} else {
if (cross(gl, gp) <= 0 && cross(gm, gp) >= 0)
l = mid;
else
r = mid;
}
}
r %= N;
Real v = cross(Q[l] - p, Q[r] - p);
return eq(v, 0.0) ? ON : v < 0.0 ? OUT : IN;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=1033
// deduplication of line segments
void merge_segments(vector<Segment> &segs) {
auto merge_if_able = [](Segment &s1, const Segment &s2) {
if (abs(cross(s1.b - s1.a, s2.b - s2.a)) > EPS) return false;
if (ccw(s1.a, s2.a, s1.b) == 1 || ccw(s1.a, s2.a, s1.b) == -1) return false;
if (ccw(s1.a, s1.b, s2.a) == -2 || ccw(s2.a, s2.b, s1.a) == -2)
return false;
s1 = Segment(min(s1.a, s2.a), max(s1.b, s2.b));
return true;
};
for (int i = 0; i < segs.size(); i++) {
if (segs[i].b < segs[i].a) swap(segs[i].a, segs[i].b);
}
for (int i = 0; i < segs.size(); i++) {
for (int j = i + 1; j < segs.size(); j++) {
if (merge_if_able(segs[i], segs[j])) {
segs[j--] = segs.back(), segs.pop_back();
}
}
}
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=1033
// construct a graph with the vertex of the intersection of any two line
// segments
vector<vector<int> > segment_arrangement(vector<Segment> &segs,
vector<Point> &ps) {
vector<vector<int> > g;
int N = (int)segs.size();
for (int i = 0; i < N; i++) {
ps.emplace_back(segs[i].a);
ps.emplace_back(segs[i].b);
for (int j = i + 1; j < N; j++) {
const Point p1 = segs[i].b - segs[i].a;
const Point p2 = segs[j].b - segs[j].a;
if (cross(p1, p2) == 0) continue;
if (intersect(segs[i], segs[j])) {
ps.emplace_back(crosspoint(segs[i], segs[j]));
}
}
}
sort(begin(ps), end(ps));
ps.erase(unique(begin(ps), end(ps)), end(ps));
int M = (int)ps.size();
g.resize(M);
for (int i = 0; i < N; i++) {
vector<int> vec;
for (int j = 0; j < M; j++) {
if (intersect(segs[i], ps[j])) {
vec.emplace_back(j);
}
}
for (int j = 1; j < vec.size(); j++) {
g[vec[j - 1]].push_back(vec[j]);
g[vec[j]].push_back(vec[j - 1]);
}
}
return (g);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_C
// cut with a straight line l and return a convex polygon on the left
Polygon convex_cut(const Polygon &U, Line l) {
Polygon ret;
for (int i = 0; i < U.size(); i++) {
Point now = U[i], nxt = U[(i + 1) % U.size()];
if (ccw(l.a, l.b, now) != -1) ret.push_back(now);
if (ccw(l.a, l.b, now) * ccw(l.a, l.b, nxt) < 0) {
ret.push_back(crosspoint(Line(now, nxt), l));
}
}
return (ret);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_A
Real area(const Polygon &p) {
Real A = 0;
for (int i = 0; i < p.size(); ++i) {
A += cross(p[i], p[(i + 1) % p.size()]);
}
return A * 0.5;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_H
Real area(const Polygon &p, const Circle &c) {
if (p.size() < 3) return 0.0;
function<Real(Circle, Point, Point)> cross_area =
[&](const Circle &c, const Point &a, const Point &b) {
Point va = c.p - a, vb = c.p - b;
Real f = cross(va, vb), ret = 0.0;
if (eq(f, 0.0)) return ret;
if (max(abs(va), abs(vb)) < c.r + EPS) return f;
if (distance(Segment(a, b), c.p) > c.r - EPS)
return c.r * c.r * arg(vb * conj(va));
auto u = crosspoint(c, Segment(a, b));
vector<Point> tot{a, u.first, u.second, b};
for (int i = 0; i + 1 < tot.size(); i++) {
ret += cross_area(c, tot[i], tot[i + 1]);
}
return ret;
};
Real A = 0;
for (int i = 0; i < p.size(); i++) {
A += cross_area(c, p[i], p[(i + 1) % p.size()]);
}
return A;
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_B
Real convex_diameter(const Polygon &p) {
int N = (int)p.size();
int is = 0, js = 0;
for (int i = 1; i < N; i++) {
if (p[i].imag() > p[is].imag()) is = i;
if (p[i].imag() < p[js].imag()) js = i;
}
Real maxdis = norm(p[is] - p[js]);
int maxi, maxj, i, j;
i = maxi = is;
j = maxj = js;
do {
if (cross(p[(i + 1) % N] - p[i], p[(j + 1) % N] - p[j]) >= 0) {
j = (j + 1) % N;
} else {
i = (i + 1) % N;
}
if (norm(p[i] - p[j]) > maxdis) {
maxdis = norm(p[i] - p[j]);
maxi = i;
maxj = j;
}
} while (i != is || j != js);
return sqrt(maxdis);
}
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_5_A
Real closest_pair(Points ps) {
if (ps.size() <= 1) throw(0);
sort(begin(ps), end(ps));
auto compare_y = [&](const Point &a, const Point &b) {
return imag(a) < imag(b);
};
vector<Point> beet(ps.size());
const Real INF = 1e18;
function<Real(int, int)> rec = [&](int left, int right) {
if (right - left <= 1) return INF;
int mid = (left + right) >> 1;
auto x = real(ps[mid]);
auto ret = min(rec(left, mid), rec(mid, right));
inplace_merge(begin(ps) + left, begin(ps) + mid, begin(ps) + right,
compare_y);
int ptr = 0;
for (int i = left; i < right; i++) {
if (abs(real(ps[i]) - x) >= ret) continue;
for (int j = 0; j < ptr; j++) {
auto luz = ps[i] - beet[ptr - j - 1];
if (imag(luz) >= ret) break;
ret = min(ret, abs(luz));
}
beet[ptr++] = ps[i];
}
return ret;
};
return rec(0, (int)ps.size());
}