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#include <bits/stdc++.h>
#define MP make_pair
#define PB push_back
#define int long long
#define st first
#define nd second
#define rd third
#define FOR(i, a, b) for(int i =(a); i <=(b); ++i)
#define RE(i, n) FOR(i, 1, n)
#define FORD(i, a, b) for(int i = (a); i >= (b); --i)
#define REP(i, n) for(int i = 0;i <(n); ++i)
#define VAR(v, i) __typeof(i) v=(i)
#define FORE(i, c) for(VAR(i, (c).begin()); i != (c).end(); ++i)
#define ALL(x) (x).begin(), (x).end()
#define SZ(x) ((int)(x).size())
#define __builtin_ctz __builtin_ctzll
#define __builtin_clz __builtin_clzll
#define __builtin_popcount __builtin_popcountll
using namespace std;
template<typename TH> void _dbg(const char* sdbg, TH h) { cerr<<sdbg<<"="<<h<<"\n"; }
template<typename TH, typename... TA> void _dbg(const char* sdbg, TH h, TA... t) {
while(*sdbg != ',') { cerr<<*sdbg++; } cerr<<"="<<h<<","; _dbg(sdbg+1, t...);
}
#ifdef LOCAL
#define debug(...) _dbg(#__VA_ARGS__, __VA_ARGS__)
#define debugv(x) {{cerr <<#x <<" = "; FORE(itt, (x)) cerr <<*itt <<", "; cerr <<"\n"; }}
#else
#define debug(...) (__VA_ARGS__)
#define debugv(x)
#define cerr if(0)cout
#endif
#define next ____next
#define prev ____prev
#define left ____left
#define hash ____hash
typedef long long ll;
typedef long double LD;
typedef pair<int, int> PII;
typedef pair<ll, ll> PLL;
typedef vector<int> VI;
typedef vector<VI> VVI;
typedef vector<ll> VLL;
typedef vector<pair<int, int> > VPII;
typedef vector<pair<ll, ll> > VPLL;
template<class C> void mini(C&a4, C b4){a4=min(a4, b4); }
template<class C> void maxi(C&a4, C b4){a4=max(a4, b4); }
template<class T1, class T2>
ostream& operator<< (ostream &out, pair<T1, T2> pair) { return out << "(" << pair.first << ", " << pair.second << ")";}
template<class A, class B, class C> struct Triple { A first; B second; C third;
bool operator<(const Triple& t) const { if (st != t.st) return st < t.st; if (nd != t.nd) return nd < t.nd; return rd < t.rd; } };
template<class T> void ResizeVec(T&, vector<int>) {}
template<class T> void ResizeVec(vector<T>& vec, vector<int> sz) {
vec.resize(sz[0]); sz.erase(sz.begin()); if (sz.empty()) { return; }
for (T& v : vec) { ResizeVec(v, sz); }
}
typedef Triple<int, int, int> TIII;
template<class A, class B, class C>
ostream& operator<< (ostream &out, Triple<A, B, C> t) { return out << "(" << t.st << ", " << t.nd << ", " << t.rd << ")"; }
template<class T> ostream& operator<<(ostream& out, vector<T> vec) { out<<"("; for (auto& v: vec) out<<v<<", "; return out<<")"; }
template<class T> ostream& operator<<(ostream& out, set<T> vec) { out<<"("; for (auto& v: vec) out<<v<<", "; return out<<")"; }
template<class L, class R> ostream& operator<<(ostream& out, map<L, R> vec) { out<<"("; for (auto& v: vec) out<<v<<", "; return out<<")"; }
typedef long double LD;
const LD kEps = 1e-9;
const LD kPi = 2 * acos(0);
LD Sq(LD x) {
return x * x;
}
struct Point {
LD x, y;
Point() {}
Point(LD a, LD b) : x(a), y(b) {}
Point(const Point& a) : x(a.x), y(a.y) {}
void operator=(const Point& a) { x = a.x; y = a.y; }
Point operator+(const Point& a) const { Point p(x + a.x, y + a.y); return p; }
Point operator-(const Point& a) const { Point p(x - a.x, y - a.y); return p; }
Point operator*(LD a) const { Point p(x * a, y * a); return p; }
Point operator/(LD a) const { assert(abs(a) > kEps); Point p(x / a, y / a); return p; }
Point& operator+=(const Point& a) { x += a.x; y += a.y; return *this; }
Point& operator-=(const Point& a) { x -= a.x; y -= a.y; return *this; }
Point& operator*=(LD a) { x *= a; y *= a; return *this;}
Point& operator/=(LD a) { assert(abs(a) > kEps); x /= a; y /= a; return *this; }
bool IsZero() const {
return abs(x) < kEps && abs(y) < kEps;
}
bool operator==(const Point& a) const {
return (*this - a).IsZero();
}
LD CrossProd(const Point& a) const {
return x * a.y - y * a.x;
}
LD CrossProd(Point a, Point b) const {
a -= *this;
b -= *this;
return a.CrossProd(b);
}
LD DotProd(const Point& a) const {
return x * a.x + y * a.y;
}
LD Norm() const {
return sqrt(Sq(x) + Sq(y));
}
void NormalizeSelf() {
*this /= Norm();
}
Point Normalize() {
Point res(*this);
res.NormalizeSelf();
return res;
}
LD Dist(const Point& a) const {
return (*this - a).Norm();
}
LD Angle() const {
return atan2(y, x);
}
void RotateSelf(LD angle) {
LD c = cos(angle);
LD s = sin(angle);
LD nx = x * c - y * s;
LD ny = y * c + x * s;
y = ny;
x = nx;
}
Point Rotate(LD angle) const {
Point res(*this);
res.RotateSelf(angle);
return res;
}
static bool LexCmp(const Point& a, const Point& b) {
if (abs(a.x - b.x) > kEps) {
return a.x < b.x;
}
return a.y < b.y;
}
LD SqNorm() {
return x * x + y * y;
}
friend ostream& operator<<(ostream& out, Point m);
};
ostream& operator<<(ostream& out, Point p) {
out << "(" << p.x << ", " << p.y << ")";
return out;
}
struct Circle {
Point center;
LD r;
Circle(LD x, LD y, LD rad) {
center = Point(x, y);
r = rad;
}
Circle(const Point& a, LD rad) : center(a), r(rad) {}
LD Area() const {
return kPi * Sq(r);
}
LD Perimeter() const {
return 2 * kPi * r;
}
LD Diameter() const {
return 2 * r;
}
Point RotateRightMost(LD ang) const {
return center + Point{r * cos(ang), r * sin(ang)};
}
bool operator==(const Circle& c) const {
return center == c.center && abs(r - c.r) < kEps;
}
};
struct Line {
Point p[2];
bool is_seg;
Line(Point a, Point b, bool is_seg_ = false) {
p[0] = a;
p[1] = b;
is_seg = is_seg_;
}
Line() {
}
Point& operator[](int a) {
return p[a];
}
Point NormalVector() {
Point perp = p[1] - p[0];
perp.RotateSelf(kPi / 2);
perp.NormalizeSelf();
return perp;
}
// (A, B, C) such that A^2 + B^2 = 1, (A, B) > (0, 0)
vector<LD> LineEqNormLD() { // seems ok
LD A = p[1].y - p[0].y;
LD B = p[0].x - p[1].x;
LD C = -(A * p[0].x + B * p[0].y);
assert(abs(A * p[1].x + B * p[1].y + C) < kEps);
LD norm = sqrt(Sq(A) + Sq(B));
vector<LD> res{A, B, C};
for (auto& x : res) { x /= norm; }
if (A < -kEps || (abs(A) < kEps && B < -kEps)) {
for (auto& x : res) { x *= -1; }
}
return res;
}
// assumes that coordinates are integers!
vector<int> LineEqNormInt() { // seems ok
int A = round(p[1].y - p[0].y);
int B = round(p[0].x - p[1].x);
int C = -(A * p[0].x + B * p[0].y);
int gcd = abs(__gcd(A, __gcd(B, C)));
vector<int> res{A, B, C};
for (auto& x : res) { x /= gcd; }
if (A < 0 || (A == 0 && B < 0)) {
for (auto& x : res) { x *= -1; }
}
return res;
}
};
struct Utils {
// 0, 1, 2 or 3 pts. In case of 3 pts it means they are equal
static vector<Point> InterCircleCircle(Circle a, Circle b) {
if (a.r + kEps < b.r) {
swap(a, b);
}
if (a == b) {
return vector<Point>{a.RotateRightMost(0), a.RotateRightMost(2 * kPi / 3),
a.RotateRightMost(4 * kPi / 3)};
}
Point diff = b.center - a.center;
LD dis = diff.Norm();
LD ang = diff.Angle();
LD longest = max(max(a.r, b.r), dis);
LD per = a.r + b.r + dis;
if (2 * longest > per + kEps) {
return vector<Point>();
}
if (abs(2 * longest - per) < 2 * kEps) {
return vector<Point>{a.RotateRightMost(ang)};
}
LD ang_dev = acos((Sq(a.r) + Sq(dis) - Sq(b.r)) / (2 * a.r * dis));
return vector<Point>{a.RotateRightMost(ang - ang_dev), a.RotateRightMost(ang + ang_dev)};
}
static vector<Point> InterLineLine(Line& a, Line& b) { // working fine
Point vec_a = a[1] - a[0];
Point vec_b1 = b[1] - a[0];
Point vec_b0 = b[0] - a[0];
LD tr_area = vec_b1.CrossProd(vec_b0);
LD quad_area = vec_b1.CrossProd(vec_a) + vec_a.CrossProd(vec_b0);
if (abs(quad_area) < kEps) { // parallel or coinciding
if (PtBelongToLine(b, a[0])) {
return {a[0], a[1]};
} else {
return {};
}
}
return {a[0] + vec_a * (tr_area / quad_area)};
}
static Point ProjPointToLine(Point p, Line l) { ///Tested
Point diff = l[1] - l[0];
return l[0] + diff * (diff.DotProd(p - l[0]) / diff.DotProd(diff));
}
static Point ReflectPtWRTLine(Point p, Line l) {
Point proj = ProjPointToLine(p, l);
return proj * 2 - p;
}
static vector<Point> InterCircleLine(Circle c, Line l) { /// Tested here: http://codeforces.com/gym/100554/submission/10197624
Point proj = ProjPointToLine(c.center, l);
LD dis_proj = c.center.Dist(proj);
if (dis_proj > c.r + kEps) { return vector<Point>(); }
LD a = sqrt(max((LD)0, Sq(c.r) - Sq(dis_proj)));
Point dir = l[1] - l[0];
LD dir_norm = dir.Norm();
vector<Point> cands{proj + dir * (a / dir_norm), proj - dir * (a / dir_norm)};
if (cands[0].Dist(cands[1]) < kEps) { return vector<Point>{proj}; }
return cands;
}
static bool PtBelongToLine(Line l, Point p) {
return abs(l[0].CrossProd(l[1], p)) < kEps;
}
static bool PtBelongToSeg(Line l, Point p) { // seems ok
return abs(p.Dist(l[0]) + p.Dist(l[1]) - l[0].Dist(l[1])) < kEps;
}
static vector<Point> InterCircleSeg(Circle c, Line l) { //seems ok
vector<Point> from_line = InterCircleLine(c, l);
vector<Point> res;
for (auto p : from_line) {
if (PtBelongToSeg(l, p)) { res.PB(p); }
}
return res;
}
static vector<Point> TangencyPtsToCircle(Circle c, Point p) { // seems ok
LD d = c.center.Dist(p);
if (d < c.r - kEps) { return {}; }
if (d < c.r + kEps) { return {p}; }
LD from_cent = (p - c.center).Angle();
LD ang_dev = acos(c.r / d);
return {c.RotateRightMost(from_cent - ang_dev), c.RotateRightMost(from_cent + ang_dev)};
}
// outer and inner tangents tested only locally (however I believe that rigorously)
static vector<Line> OuterTangents(Circle c1, Circle c2) {
if (c1 == c2) { return {}; } // is it surely best choice?
if (c1.r < c2.r) { swap(c1, c2); }
if (c2.r + c1.center.Dist(c2.center) < c1.r - kEps) { return {}; }
if (abs(c1.r - c2.r) < kEps) {
Point diff = c2.center - c1.center;
Point R = diff.Rotate(kPi / 2) * (c1.r / diff.Norm());
return {{c1.center + R, c2.center + R}, {c1.center - R, c2.center - R}};
}
Point I = c1.center + (c2.center - c1.center) * (c1.r / (c1.r - c2.r));
if (c2.r + c1.center.Dist(c2.center) < c1.r + kEps) {
return {{I, I + (c2.center - c1.center).Rotate(kPi / 2)}};
}
vector<Point> to1 = TangencyPtsToCircle(c1, I);
vector<Point> to2 = TangencyPtsToCircle(c2, I);
vector<Line> res{{to1[0], to2[0]}, {to1[1], to2[1]}};
assert(Utils::PtBelongToLine(res[0], I));
assert(Utils::PtBelongToLine(res[1], I));
return res;
}
// unfortunately big part of code is same as in previous function
// can be joined when putting appropriate signs in few places
// however those ifs differ a bit hence it may not be good idea
// to necessarily join them
static vector<Line> InnerTangents(Circle c1, Circle c2) {
if (c1 == c2) { return {}; } // this time surely best choice
if (c1.r < c2.r) { swap(c1, c2); }
LD d = c1.center.Dist(c2.center);
if (d < c1.r + c2.r - kEps) { return {}; }
Point I = c1.center + (c2.center - c1.center) * (c1.r / (c1.r + c2.r));
if (d < c1.r + c2.r + kEps) {
return {{I, I + (c2.center - c1.center).Rotate(kPi / 2)}};
}
vector<Point> to1 = TangencyPtsToCircle(c1, I);
vector<Point> to2 = TangencyPtsToCircle(c2, I);
vector<Line> res{{to1[0], to2[0]}, {to1[1], to2[1]}};
assert(Utils::PtBelongToLine(res[0], I));
assert(Utils::PtBelongToLine(res[1], I));
return res;
}
static bool AreParallel(Line l1, Line l2) { // seems ok
return abs(l1[0].CrossProd(l2[0], l1[1]) - l1[0].CrossProd(l2[1], l1[1])) < kEps;
}
// returns a vector of points such that their convex hull is intersection of those segments
// SZ(res) == 0 => empty intersection, SZ(res) == 1 => intersection is a point, SZ(res) == 2 => intersection is a segment
static vector<Point> InterSegs(Line l1, Line l2) { // seems ok
if (!Point::LexCmp(l1[0], l1[1])) { swap(l1[0], l1[1]); }
if (!Point::LexCmp(l2[0], l2[1])) { swap(l2[0], l2[1]); }
if (AreParallel(l1, l2)) {
if (!PtBelongToLine(l1, l2[0])) { return vector<Point>(); }
vector<Point> ends(2);
for (int tr = 0; tr < 2; tr++) {
if (Point::LexCmp(l1[tr], l2[tr]) ^ tr) {
ends[tr] = l2[tr];
} else {
ends[tr] = l1[tr];
}
}
if ((ends[1] - ends[0]).IsZero()) {
ends.pop_back();
}
if (SZ(ends) == 2 && Point::LexCmp(ends[1], ends[0])) { return vector<Point>(); }
return ends;
} else {
vector<Point> p = InterLineLine(l1, l2);
if (PtBelongToSeg(l1, p[0]) && PtBelongToSeg(l2, p[0])) { return p; }
return vector<Point>();
}
}
static LD Angle(Point P, Point Q, Point R) { // angle PQR
LD ang2 = (P - Q).Angle();
LD ang1 = (R - Q).Angle();
LD ans = ang1 - ang2;
if (ans < kEps) {
ans += 2 * kPi;
}
return ans;
}
// tested here: http://codeforces.com/contest/600/submission/14961583
// DON'T change anything as this will lead to precision errors
// don't know why, but this is the only version which works precisely even for very mean cases
static LD DiskInterArea(Circle c1, Circle c2) { // tested here: http://opentrains.snarknews.info/~ejudge/team.cgi?contest_id=006254 problem I
if (c1.r < c2.r) {
swap(c1, c2);
}
LD d = c1.center.Dist(c2.center);
if (c1.r + c2.r < d + kEps) {
return 0;
}
if (c1.r - c2.r > d - kEps) {
return kPi * Sq(c2.r);
}
LD alfa = acos((Sq(d) + Sq(c1.r) - Sq(c2.r)) / (2 * d * c1.r));
LD beta = acos((Sq(d) + Sq(c2.r) - Sq(c1.r)) / (2 * d * c2.r));
return alfa * Sq(c1.r) + beta * Sq(c2.r) - sin(2 * alfa) * Sq(c1.r) / 2 - sin(2 * beta) * Sq(c2.r) / 2;
}
static Line RadAxis(Circle c1, Circle c2) {
LD d = c1.center.Dist(c2.center);
LD a = (Sq(c1.r) - Sq(c2.r) + Sq(d)) / (2 * d);
Point Q = c1.center + (c2.center - c1.center) * (a / d);
Point R = Q + (c2.center - c1.center).Rotate(kPi / 2);
return Line(Q, R);
}
};
struct Polygon {
vector<Point> pts;
Polygon(vector<Point> pts_) : pts(pts_) {}
Polygon() : Polygon(vector<Point>()) {}
void Add(Point p) {
pts.push_back(p);
}
// positive for counterclockwise
double Area() {
double area = 0;
for (int i = 0; i < SZ(pts); i++) {
area += pts[i].CrossProd(pts[(i + 1) % SZ(pts)]);
}
area /= 2;
return area;
}
void OrientCounterclockwise() {
if (Area() < 0) {
reverse(pts.begin(), pts.end());
}
}
int next(int a) {
if (a + 1 < SZ(pts)) {
return a + 1;
}
return 0;
}
pair<int, int> FurthestPair() { // tested here: http://codeforces.com/contest/333/submission/11058065
MakeConvexHull();
OrientCounterclockwise();
int furth = 1;
pair<int, int> best_pair = make_pair(0, 0);
double best_dis = 0;
for (int i = 0; i < SZ(pts); i++) {
Point side = pts[next(i)] - pts[i];
while (side.CrossProd(pts[furth] - pts[i]) < side.CrossProd(pts[next(furth)] - pts[i])) {
furth = next(furth);
}
vector<int> vec{i, next(i)};
for (auto ind : vec) {
if (pts[ind].Dist(pts[furth]) > best_dis) {
best_pair = make_pair(ind, furth);
best_dis = pts[ind].Dist(pts[furth]);
}
}
cerr<<"Furthest from: "<<pts[i]<<"-"<<pts[next(i)]<<" is "<<pts[furth]<<endl;
}
return best_pair;
}
// for square 34
// 12 holds one_way_hull = {{1,3,4},{1,2,4}}
// resulting polygon is counterclockwise {1, 2, 4, 3}
vector<vector<Point>> MakeConvexHull() { // tested everywhere http://codeforces.com/contest/333/submission/11058065
vector<vector<Point>> one_way_hull(2);
sort(pts.begin(), pts.end(), Point::LexCmp);
for (int dir = -1; dir <= 1; dir += 2) {
int hull_num = (dir + 1) / 2;
auto& H = one_way_hull[hull_num];
one_way_hull[hull_num].push_back(pts[0]);
if (SZ(pts) > 1) {
H.push_back(pts[1]);
}
for (int i = 2; i < SZ(pts); i++) {
while (SZ(H) >= 2 &&
dir * (pts[i] - H[SZ(H) - 2]).CrossProd(H.back() - H[SZ(H) - 2]) > -kEps) {
H.pop_back();
}
H.push_back(pts[i]);
}
if (SZ(H) > 1 && (H[0] - H.back()).IsZero()) { H.pop_back(); }
}
pts.clear();
for (auto p : one_way_hull[1]) {
pts.push_back(p);
}
for (int i = SZ(one_way_hull[0]) - 2; i >= 1; i--) {
pts.push_back(one_way_hull[0][i]);
}
return one_way_hull;
}
// without sides
vector<vector<bool>> InsideDiagonalsMatrix() { // tested here: http://codeforces.com/contest/438/submission/11063385
int n = pts.size();
vector<vector<bool>> res(n, vector<bool>(n));
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
Line diag(pts[i], pts[j]);
if (i == j || abs(i - j) == 1 || abs(i - j) == n - 1) { continue; }
res[i][j] = 1;
for (int k = 0; k < n; k++) {
int kk = next(k);
Line side(pts[k], pts[kk]);
if (k == i || k == j || kk == i || kk == j) { continue; }
vector<Point> inter = Utils::InterSegs(diag, side);
if (SZ(inter)) { res[i][j] = 0; }
}
int act = next(i);
LD areas[2] = {0, 0};
int passed_j = 0;
while (act != i) {
passed_j |= (act == j);
areas[passed_j] += pts[i].CrossProd(pts[act], pts[next(act)]);
act = next(act);
}
if (areas[0] * areas[1] < kEps) {
res[i][j] = 0;
}
}
}
return res;
}
// P needs to be strictly outside polygon
// polygon needs to be STRICTLY convex and counterclockwise oriented (as MakeConvexHull does)
// returns {L, R} so that PL, PR are tangents and PL is on left
vector<Point> Tangents(Point p) { // tested here: https://icpc.kattis.com/problems/spin (1169964)
vector<Point> res;
REP (tr, 2) {
auto GrThan = [&](int fir, int sec) { // fir on sec's left
return p.CrossProd(pts[sec], pts[fir]) > kEps;
};
bool up = false;
int cr = 0;
if (SZ(pts) >= 2) { cr = p.CrossProd(pts[0], pts[1]); }
if (abs(cr) < kEps && SZ(pts) >= 3) { cr = p.CrossProd(pts[0], pts[2]); }
up = (cr > 0);
VI bd{1, SZ(pts) - 1};
int faj = 0;
while (bd[0] + 6 <= bd[1]) { // better don't replace with smaller constants
VI h(2);
REP (hh, 2) { h[hh] = (bd[0] + bd[1] + bd[hh]) / 3; }
if (!GrThan(h[up ^ tr], 0) ^ tr) { bd[up ^ tr] = h[up ^ tr]; }
else {
int gr = GrThan(h[0], h[1]);
bd[gr ^ tr] = h[gr ^ tr];
}
}
FOR (i, bd[0], bd[1]) {
if (GrThan(i, faj) ^ tr) {
faj = i;
}
}
res.PB(pts[faj]);
}
return res;
}
};
struct ConvexPolHalves { // tested here: https://icpc.kattis.com/problems/spin (1169964)
vector<vector<Point>> chains; // initialized by MakeConvexHull
bool BelongTo(Point p) { // including borders
if (SZ(chains[0]) == 1) {
return (chains[0][0] - p).IsZero();
}
if (p.x + kEps < chains[0][0].x || p.x - kEps > chains[0].back().x) { return false; }
REP (tr, 2) {
int kl = 0, kp = SZ(chains[tr]) - 2, faj = 0;
while (kl <= kp) {
int aktc = (kl + kp) / 2;
if (chains[tr][aktc].x < p.x + kEps) {
kl = aktc + 1;
faj = aktc;
} else {
kp = aktc - 1;
}
}
Point fir = chains[tr][faj], sec = chains[tr][faj + 1];
if (abs(fir.x - sec.x) < kEps) {
if (tr == 0) { if (sec.y + kEps < p.y) { return false; } }
else { if (fir.y - kEps > p.y) { return false; } }
} else {
LD cr = fir.CrossProd(sec, p);
if (abs(cr) < kEps) { return true; }
if ((cr > 0) ^ tr) { return false; }
}
}
return true;
}
};
// CLIP START
bool InUpper(Point a) {
if (abs(a.y) > kEps) {
return a.y > 0;
}
return a.x > 0;
}
bool angle_cmp(const Point a, const Point b) {
bool u = InUpper(a);
bool v = InUpper(b);
return u!=v ? u : a.CrossProd(b)>0;
}
/**
* @brief a+(b-a)*f \in c+lin(d-c)
* @returns f
*/
LD cross(Point a, Point b, Point c, Point d) {
return (d - c).CrossProd(a - c) / (d - c).CrossProd(a - b);
}
struct ClipLine { // valid side is on left
ClipLine(Point A, Point B) : al(A), bl(B), a(A), b(B) {};
Point al,bl; // original line points
mutable Point a,b; // actual intersection points
Point dir() const { return bl - al; }
bool operator<(const ClipLine& l) const { return angle_cmp(dir(),l.dir()); }
Point cross(const ClipLine& l) {
return al + (bl - al) * ::cross(al, bl, l.al, l.bl);
}
bool left(Point p) {
return (bl - al).CrossProd(p - al) > 0;
}
};
set<VI> put;
struct Clip {
Clip(LD r) : area(4*r*r) {
Point a{-r,-r}, b{r,-r}, c{r,r}, d{-r,r};
lines = {ClipLine(a,b), ClipLine(b,c), ClipLine(c,d), ClipLine(d,a)};
}
void insert(Line l) {
VI abc = l.LineEqNormInt();
if (put.count(abc)) { return; }
put.insert(abc);
insert(ClipLine(l[0], l[1]));
}
void insert(ClipLine l) {
assert(abs(l.dir().SqNorm()) > kEps);
find(l);
while (size() && !l.left(it->a) && !l.left(it->b)) { erase(); }
if (size()) {
while (prev(), size() && !l.left(it->a) && !l.left(it->b)) { erase(); }
}
if (size() && (!l.left(it->a) || !l.left(it->b))) {
l.a = l.cross(*it);
area -= l.a.CrossProd(it->b)*.5; it->b = l.a; next();
l.b = l.cross(*it);
if ((l.a-l.b).SqNorm() < kEps) {
l.b = l.a;
}
area -= it->a.CrossProd(l.b) * .5;
it->a = l.b;
if (!(l.a - l.b).IsZero()) {
area += l.a.CrossProd(l.b)*.5;
lines.insert(l);
}
}
//assert(l.dir().SqNorm()>1e-13);
}
void find(const ClipLine &l) {
it = lines.lower_bound(l);
if (it == lines.end()) { it = lines.begin(); }
}
void recalculate() {
area = 0; for (const ClipLine &l : lines) area += l.a.CrossProd(l.b);
area *= .5;
}
int size() { return lines.size(); }
void next() { if(++it==lines.end()) it = lines.begin(); }
void prev() { if(it==lines.begin()) it = lines.end(); --it; }
void erase() {
assert(it!=lines.end());
area -= it->a.CrossProd(it->b)*.5;
it = lines.erase(it);
if(it==lines.end()) it = lines.begin();
}
typename set<ClipLine>::iterator it;
set<ClipLine> lines;
LD area;
};
// CLIP ENDS
// CENTERS BEGIN
Point Bary(Point A, Point B, Point C, LD a, LD b, LD c) {
return (A * a + B * b + C * c) / (a + b + c);
}
Point Centroid(Point A, Point B, Point C) {
return Bary(A, B, C, 1, 1, 1);
}
Point Circumcenter(Point A, Point B, Point C) {
LD a = (B - C).SqNorm(), b = (C - A).SqNorm(), c = (A - B).SqNorm();
return Bary(A, B, C, a * (b + c - a), b * (c + a - b), c * (a + b - c));
}
Point Incenter(Point A, Point B, Point C) {
return Bary(A, B, C, (B - C).Norm(), (A - C).Norm(), (A - B).Norm());
}
Point Orthocenter(Point A, Point B, Point C) {
LD a = (B - C).SqNorm(), b = (C - A).SqNorm(), c = (A - B).SqNorm();
return Bary(A, B, C, (a+b-c)*(c+a-b), (b+c-a)*(a+b-c), (c+a-b)*(b+c-a));
}
Point Excenter(Point A, Point B, Point C) { // opposite to A
LD a = (B - C).Norm(), b = (A - C).Norm(), c = (A - B).Norm();
return Bary(A, B, C, -a, b, c);
}
// const LD kEps = 1e-9;
// const LD kPi = 2 * acos(0);
struct PointInt {
int x, y;
PointInt operator-(PointInt p) { return {x - p.x, y - p.y}; }
PointInt operator+(PointInt p) { return {x + p.x, y + p.y}; }
// Point operator*(LD f) { return {x * f, y * f}; }
// bool operator<(const Point& oth) const {
// if (abs(x - oth.x) > kEps) { return x < oth.x; }
// if (abs(y - oth.y) > kEps) { return y < oth.y; }
// return false;
// }
// LD Angle() {
// return atan2(y, x);
// }
};
// LD Normalize(LD ang) {
// if (ang < -kPi) {
// ang += 2 * kPi;
// }
// if (ang > kPi) {
// ang -= 2 * kPi;
// }
// return ang;
// }
int CrossProd(PointInt A, PointInt B, PointInt C) {
B = B - A;
C = C - A;
return B.x * C.y - B.y * C.x;
}
// vector<Point> InterLineLine(Point A, Point B, Point P, Point Q) {
// LD tr_area = CrossProd(A, P, Q);
// LD quad_area = CrossProd(A, P, B) + CrossProd(A, B, Q);
// if (abs(quad_area) < 1e-9) {
// return {};
// }
// return {A + (B - A) * (tr_area / quad_area)};
// }
const int N = 222;
int n;
vector<PointInt> segs[N];
// bool CheckPoint(Point p) {
// int bil = 0;
// vector<pair<LD, int>> evs;
// RE (i, n) {
// Point D1 = segs[i][0] - p;
// Point D2 = segs[i][1] - p;
// LD ang1 = D1.Angle();
// LD ang2 = D2.Angle();
// LD diff = Normalize(ang2 - ang1);
// if (diff < 0) {
// swap(D1, D2);
// swap(ang1, ang2);
// }
// LD from = Normalize(ang2 - kPi / 2);
// LD to = Normalize(ang1 + kPi / 2);
// evs.PB({from, 1});
// evs.PB({to, -1});
// if (from > to) {
// bil++;
// }
// }
// sort(ALL(evs));
// if (bil == 0) { return false; }
// for (auto ev : evs) {
// bil += ev.nd;
// if (bil == 0) { return false; }
// }
// return true;
// }
Point PFromPI(PointInt P) {
debug(P.x, P.y);
return {(LD)P.x, (LD)P.y};
}
int32_t main() {
ios_base::sync_with_stdio(0);
cout << fixed << setprecision(15);
cerr << fixed << setprecision(15);
cin.tie(0);
//double beg_clock = 1.0 * clock() / CLOCKS_PER_SEC;
cin>>n;
vector<Point> pts;
RE (i, n) {
int x1, y1, x2, y2;
cin>>x1>>y1>>x2>>y2;
segs[i].PB(PointInt{x1, y1});
segs[i].PB(PointInt{x2, y2});
//pts.PB(Point{LD(x1), LD(y1)});
//pts.PB(Point{LD(x2), LD(y2)});
}
//map<Point, int> ids;
Clip clip(1e4);
set<VI> put_lines;
RE (i, n) {
RE (j, i - 1) {
REP (ti, 2) {
REP (tj, 2) {
bool can_all_atmost0 = 1, can_all_atleast0 = 1;
PointInt P = segs[i][ti], Q = segs[j][tj];
RE (k, n) {
if (k == i || k == j) { continue; }
int cr0 = CrossProd(P, Q, segs[k][0]);
int cr1 = CrossProd(P, Q, segs[k][1]);
can_all_atleast0 &= (cr0 >= 0 || cr1 >= 0);
can_all_atmost0 &= (cr0 <= 0 || cr1 <= 0);
}
if (can_all_atleast0 && can_all_atmost0) {
cout<<(LD)0<<endl;
return 0;
}
if (can_all_atleast0) {
debug(">=0");
Line l = Line{PFromPI(P), PFromPI(Q)};
clip.insert(l);
}
if (can_all_atmost0) {
debug("<=0");
clip.insert(Line{PFromPI(Q), PFromPI(P)});
}
}
}
}
}
clip.recalculate();
cout<<clip.area<<endl;
return 0;
}
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TA> void _dbg(const char* sdbg, TH h, TA... t) { while(*sdbg != ',') { cerr<<*sdbg++; } cerr<<"="<<h<<","; _dbg(sdbg+1, t...); } #ifdef LOCAL #define debug(...) _dbg(#__VA_ARGS__, __VA_ARGS__) #define debugv(x) {{cerr <<#x <<" = "; FORE(itt, (x)) cerr <<*itt <<", "; cerr <<"\n"; }} #else #define debug(...) (__VA_ARGS__) #define debugv(x) #define cerr if(0)cout #endif #define next ____next #define prev ____prev #define left ____left #define hash ____hash typedef long long ll; typedef long double LD; typedef pair<int, int> PII; typedef pair<ll, ll> PLL; typedef vector<int> VI; typedef vector<VI> VVI; typedef vector<ll> VLL; typedef vector<pair<int, int> > VPII; typedef vector<pair<ll, ll> > VPLL; template<class C> void mini(C&a4, C b4){a4=min(a4, b4); } template<class C> void maxi(C&a4, C b4){a4=max(a4, b4); } template<class T1, class T2> ostream& operator<< (ostream &out, pair<T1, T2> pair) { return out << "(" << pair.first << ", " << pair.second << ")";} template<class A, class B, class C> struct Triple { A first; B second; C third; bool operator<(const Triple& t) const { if (st != t.st) return st < t.st; if (nd != t.nd) return nd < t.nd; return rd < t.rd; } }; template<class T> void ResizeVec(T&, vector<int>) {} template<class T> void ResizeVec(vector<T>& vec, vector<int> sz) { vec.resize(sz[0]); sz.erase(sz.begin()); if (sz.empty()) { return; } for (T& v : vec) { ResizeVec(v, sz); } } typedef Triple<int, int, int> TIII; template<class A, class B, class C> ostream& operator<< (ostream &out, Triple<A, B, C> t) { return out << "(" << t.st << ", " << t.nd << ", " << t.rd << ")"; } template<class T> ostream& operator<<(ostream& out, vector<T> vec) { out<<"("; for (auto& v: vec) out<<v<<", "; return out<<")"; } template<class T> ostream& operator<<(ostream& out, set<T> vec) { out<<"("; for (auto& v: vec) out<<v<<", "; return out<<")"; } template<class L, class R> ostream& operator<<(ostream& out, map<L, R> vec) { out<<"("; for (auto& v: vec) out<<v<<", "; return out<<")"; } typedef long double LD; const LD kEps = 1e-9; const LD kPi = 2 * acos(0); LD Sq(LD x) { return x * x; } struct Point { LD x, y; Point() {} Point(LD a, LD b) : x(a), y(b) {} Point(const Point& a) : x(a.x), y(a.y) {} void operator=(const Point& a) { x = a.x; y = a.y; } Point operator+(const Point& a) const { Point p(x + a.x, y + a.y); return p; } Point operator-(const Point& a) const { Point p(x - a.x, y - a.y); return p; } Point operator*(LD a) const { Point p(x * a, y * a); return p; } Point operator/(LD a) const { assert(abs(a) > kEps); Point p(x / a, y / a); return p; } Point& operator+=(const Point& a) { x += a.x; y += a.y; return *this; } Point& operator-=(const Point& a) { x -= a.x; y -= a.y; return *this; } Point& operator*=(LD a) { x *= a; y *= a; return *this;} Point& operator/=(LD a) { assert(abs(a) > kEps); x /= a; y /= a; return *this; } bool IsZero() const { return abs(x) < kEps && abs(y) < kEps; } bool operator==(const Point& a) const { return (*this - a).IsZero(); } LD CrossProd(const Point& a) const { return x * a.y - y * a.x; } LD CrossProd(Point a, Point b) const { a -= *this; b -= *this; return a.CrossProd(b); } LD DotProd(const Point& a) const { return x * a.x + y * a.y; } LD Norm() const { return sqrt(Sq(x) + Sq(y)); } void NormalizeSelf() { *this /= Norm(); } Point Normalize() { Point res(*this); res.NormalizeSelf(); return res; } LD Dist(const Point& a) const { return (*this - a).Norm(); } LD Angle() const { return atan2(y, x); } void RotateSelf(LD angle) { LD c = cos(angle); LD s = sin(angle); LD nx = x * c - y * s; LD ny = y * c + x * s; y = ny; x = nx; } Point Rotate(LD angle) const { Point res(*this); res.RotateSelf(angle); return res; } static bool LexCmp(const Point& a, const Point& b) { if (abs(a.x - b.x) > kEps) { return a.x < b.x; } return a.y < b.y; } LD SqNorm() { return x * x + y * y; } friend ostream& operator<<(ostream& out, Point m); }; ostream& operator<<(ostream& out, Point p) { out << "(" << p.x << ", " << p.y << ")"; return out; } struct Circle { Point center; LD r; Circle(LD x, LD y, LD rad) { center = Point(x, y); r = rad; } Circle(const Point& a, LD rad) : center(a), r(rad) {} LD Area() const { return kPi * Sq(r); } LD Perimeter() const { return 2 * kPi * r; } LD Diameter() const { return 2 * r; } Point RotateRightMost(LD ang) const { return center + Point{r * cos(ang), r * sin(ang)}; } bool operator==(const Circle& c) const { return center == c.center && abs(r - c.r) < kEps; } }; struct Line { Point p[2]; bool is_seg; Line(Point a, Point b, bool is_seg_ = false) { p[0] = a; p[1] = b; is_seg = is_seg_; } Line() { } Point& operator[](int a) { return p[a]; } Point NormalVector() { Point perp = p[1] - p[0]; perp.RotateSelf(kPi / 2); perp.NormalizeSelf(); return perp; } // (A, B, C) such that A^2 + B^2 = 1, (A, B) > (0, 0) vector<LD> LineEqNormLD() { // seems ok LD A = p[1].y - p[0].y; LD B = p[0].x - p[1].x; LD C = -(A * p[0].x + B * p[0].y); assert(abs(A * p[1].x + B * p[1].y + C) < kEps); LD norm = sqrt(Sq(A) + Sq(B)); vector<LD> res{A, B, C}; for (auto& x : res) { x /= norm; } if (A < -kEps || (abs(A) < kEps && B < -kEps)) { for (auto& x : res) { x *= -1; } } return res; } // assumes that coordinates are integers! vector<int> LineEqNormInt() { // seems ok int A = round(p[1].y - p[0].y); int B = round(p[0].x - p[1].x); int C = -(A * p[0].x + B * p[0].y); int gcd = abs(__gcd(A, __gcd(B, C))); vector<int> res{A, B, C}; for (auto& x : res) { x /= gcd; } if (A < 0 || (A == 0 && B < 0)) { for (auto& x : res) { x *= -1; } } return res; } }; struct Utils { // 0, 1, 2 or 3 pts. In case of 3 pts it means they are equal static vector<Point> InterCircleCircle(Circle a, Circle b) { if (a.r + kEps < b.r) { swap(a, b); } if (a == b) { return vector<Point>{a.RotateRightMost(0), a.RotateRightMost(2 * kPi / 3), a.RotateRightMost(4 * kPi / 3)}; } Point diff = b.center - a.center; LD dis = diff.Norm(); LD ang = diff.Angle(); LD longest = max(max(a.r, b.r), dis); LD per = a.r + b.r + dis; if (2 * longest > per + kEps) { return vector<Point>(); } if (abs(2 * longest - per) < 2 * kEps) { return vector<Point>{a.RotateRightMost(ang)}; } LD ang_dev = acos((Sq(a.r) + Sq(dis) - Sq(b.r)) / (2 * a.r * dis)); return vector<Point>{a.RotateRightMost(ang - ang_dev), a.RotateRightMost(ang + ang_dev)}; } static vector<Point> InterLineLine(Line& a, Line& b) { // working fine Point vec_a = a[1] - a[0]; Point vec_b1 = b[1] - a[0]; Point vec_b0 = b[0] - a[0]; LD tr_area = vec_b1.CrossProd(vec_b0); LD quad_area = vec_b1.CrossProd(vec_a) + vec_a.CrossProd(vec_b0); if (abs(quad_area) < kEps) { // parallel or coinciding if (PtBelongToLine(b, a[0])) { return {a[0], a[1]}; } else { return {}; } } return {a[0] + vec_a * (tr_area / quad_area)}; } static Point ProjPointToLine(Point p, Line l) { ///Tested Point diff = l[1] - l[0]; return l[0] + diff * (diff.DotProd(p - l[0]) / diff.DotProd(diff)); } static Point ReflectPtWRTLine(Point p, Line l) { Point proj = ProjPointToLine(p, l); return proj * 2 - p; } static vector<Point> InterCircleLine(Circle c, Line l) { /// Tested here: http://codeforces.com/gym/100554/submission/10197624 Point proj = ProjPointToLine(c.center, l); LD dis_proj = c.center.Dist(proj); if (dis_proj > c.r + kEps) { return vector<Point>(); } LD a = sqrt(max((LD)0, Sq(c.r) - Sq(dis_proj))); Point dir = l[1] - l[0]; LD dir_norm = dir.Norm(); vector<Point> cands{proj + dir * (a / dir_norm), proj - dir * (a / dir_norm)}; if (cands[0].Dist(cands[1]) < kEps) { return vector<Point>{proj}; } return cands; } static bool PtBelongToLine(Line l, Point p) { return abs(l[0].CrossProd(l[1], p)) < kEps; } static bool PtBelongToSeg(Line l, Point p) { // seems ok return abs(p.Dist(l[0]) + p.Dist(l[1]) - l[0].Dist(l[1])) < kEps; } static vector<Point> InterCircleSeg(Circle c, Line l) { //seems ok vector<Point> from_line = InterCircleLine(c, l); vector<Point> res; for (auto p : from_line) { if (PtBelongToSeg(l, p)) { res.PB(p); } } return res; } static vector<Point> TangencyPtsToCircle(Circle c, Point p) { // seems ok LD d = c.center.Dist(p); if (d < c.r - kEps) { return {}; } if (d < c.r + kEps) { return {p}; } LD from_cent = (p - c.center).Angle(); LD ang_dev = acos(c.r / d); return {c.RotateRightMost(from_cent - ang_dev), c.RotateRightMost(from_cent + ang_dev)}; } // outer and inner tangents tested only locally (however I believe that rigorously) static vector<Line> OuterTangents(Circle c1, Circle c2) { if (c1 == c2) { return {}; } // is it surely best choice? if (c1.r < c2.r) { swap(c1, c2); } if (c2.r + c1.center.Dist(c2.center) < c1.r - kEps) { return {}; } if (abs(c1.r - c2.r) < kEps) { Point diff = c2.center - c1.center; Point R = diff.Rotate(kPi / 2) * (c1.r / diff.Norm()); return {{c1.center + R, c2.center + R}, {c1.center - R, c2.center - R}}; } Point I = c1.center + (c2.center - c1.center) * (c1.r / (c1.r - c2.r)); if (c2.r + c1.center.Dist(c2.center) < c1.r + kEps) { return {{I, I + (c2.center - c1.center).Rotate(kPi / 2)}}; } vector<Point> to1 = TangencyPtsToCircle(c1, I); vector<Point> to2 = TangencyPtsToCircle(c2, I); vector<Line> res{{to1[0], to2[0]}, {to1[1], to2[1]}}; assert(Utils::PtBelongToLine(res[0], I)); assert(Utils::PtBelongToLine(res[1], I)); return res; } // unfortunately big part of code is same as in previous function // can be joined when putting appropriate signs in few places // however those ifs differ a bit hence it may not be good idea // to necessarily join them static vector<Line> InnerTangents(Circle c1, Circle c2) { if (c1 == c2) { return {}; } // this time surely best choice if (c1.r < c2.r) { swap(c1, c2); } LD d = c1.center.Dist(c2.center); if (d < c1.r + c2.r - kEps) { return {}; } Point I = c1.center + (c2.center - c1.center) * (c1.r / (c1.r + c2.r)); if (d < c1.r + c2.r + kEps) { return {{I, I + (c2.center - c1.center).Rotate(kPi / 2)}}; } vector<Point> to1 = TangencyPtsToCircle(c1, I); vector<Point> to2 = TangencyPtsToCircle(c2, I); vector<Line> res{{to1[0], to2[0]}, {to1[1], to2[1]}}; assert(Utils::PtBelongToLine(res[0], I)); assert(Utils::PtBelongToLine(res[1], I)); return res; } static bool AreParallel(Line l1, Line l2) { // seems ok return abs(l1[0].CrossProd(l2[0], l1[1]) - l1[0].CrossProd(l2[1], l1[1])) < kEps; } // returns a vector of points such that their convex hull is intersection of those segments // SZ(res) == 0 => empty intersection, SZ(res) == 1 => intersection is a point, SZ(res) == 2 => intersection is a segment static vector<Point> InterSegs(Line l1, Line l2) { // seems ok if (!Point::LexCmp(l1[0], l1[1])) { swap(l1[0], l1[1]); } if (!Point::LexCmp(l2[0], l2[1])) { swap(l2[0], l2[1]); } if (AreParallel(l1, l2)) { if (!PtBelongToLine(l1, l2[0])) { return vector<Point>(); } vector<Point> ends(2); for (int tr = 0; tr < 2; tr++) { if (Point::LexCmp(l1[tr], l2[tr]) ^ tr) { ends[tr] = l2[tr]; } else { ends[tr] = l1[tr]; } } if ((ends[1] - ends[0]).IsZero()) { ends.pop_back(); } if (SZ(ends) == 2 && Point::LexCmp(ends[1], ends[0])) { return vector<Point>(); } return ends; } else { vector<Point> p = InterLineLine(l1, l2); if (PtBelongToSeg(l1, p[0]) && PtBelongToSeg(l2, p[0])) { return p; } return vector<Point>(); } } static LD Angle(Point P, Point Q, Point R) { // angle PQR LD ang2 = (P - Q).Angle(); LD ang1 = (R - Q).Angle(); LD ans = ang1 - ang2; if (ans < kEps) { ans += 2 * kPi; } return ans; } // tested here: http://codeforces.com/contest/600/submission/14961583 // DON'T change anything as this will lead to precision errors // don't know why, but this is the only version which works precisely even for very mean cases static LD DiskInterArea(Circle c1, Circle c2) { // tested here: http://opentrains.snarknews.info/~ejudge/team.cgi?contest_id=006254 problem I if (c1.r < c2.r) { swap(c1, c2); } LD d = c1.center.Dist(c2.center); if (c1.r + c2.r < d + kEps) { return 0; } if (c1.r - c2.r > d - kEps) { return kPi * Sq(c2.r); } LD alfa = acos((Sq(d) + Sq(c1.r) - Sq(c2.r)) / (2 * d * c1.r)); LD beta = acos((Sq(d) + Sq(c2.r) - Sq(c1.r)) / (2 * d * c2.r)); return alfa * Sq(c1.r) + beta * Sq(c2.r) - sin(2 * alfa) * Sq(c1.r) / 2 - sin(2 * beta) * Sq(c2.r) / 2; } static Line RadAxis(Circle c1, Circle c2) { LD d = c1.center.Dist(c2.center); LD a = (Sq(c1.r) - Sq(c2.r) + Sq(d)) / (2 * d); Point Q = c1.center + (c2.center - c1.center) * (a / d); Point R = Q + (c2.center - c1.center).Rotate(kPi / 2); return Line(Q, R); } }; struct Polygon { vector<Point> pts; Polygon(vector<Point> pts_) : pts(pts_) {} Polygon() : Polygon(vector<Point>()) {} void Add(Point p) { pts.push_back(p); } // positive for counterclockwise double Area() { double area = 0; for (int i = 0; i < SZ(pts); i++) { area += pts[i].CrossProd(pts[(i + 1) % SZ(pts)]); } area /= 2; return area; } void OrientCounterclockwise() { if (Area() < 0) { reverse(pts.begin(), pts.end()); } } int next(int a) { if (a + 1 < SZ(pts)) { return a + 1; } return 0; } pair<int, int> FurthestPair() { // tested here: http://codeforces.com/contest/333/submission/11058065 MakeConvexHull(); OrientCounterclockwise(); int furth = 1; pair<int, int> best_pair = make_pair(0, 0); double best_dis = 0; for (int i = 0; i < SZ(pts); i++) { Point side = pts[next(i)] - pts[i]; while (side.CrossProd(pts[furth] - pts[i]) < side.CrossProd(pts[next(furth)] - pts[i])) { furth = next(furth); } vector<int> vec{i, next(i)}; for (auto ind : vec) { if (pts[ind].Dist(pts[furth]) > best_dis) { best_pair = make_pair(ind, furth); best_dis = pts[ind].Dist(pts[furth]); } } cerr<<"Furthest from: "<<pts[i]<<"-"<<pts[next(i)]<<" is "<<pts[furth]<<endl; } return best_pair; } // for square 34 // 12 holds one_way_hull = {{1,3,4},{1,2,4}} // resulting polygon is counterclockwise {1, 2, 4, 3} vector<vector<Point>> MakeConvexHull() { // tested everywhere http://codeforces.com/contest/333/submission/11058065 vector<vector<Point>> one_way_hull(2); sort(pts.begin(), pts.end(), Point::LexCmp); for (int dir = -1; dir <= 1; dir += 2) { int hull_num = (dir + 1) / 2; auto& H = one_way_hull[hull_num]; one_way_hull[hull_num].push_back(pts[0]); if (SZ(pts) > 1) { H.push_back(pts[1]); } for (int i = 2; i < SZ(pts); i++) { while (SZ(H) >= 2 && dir * (pts[i] - H[SZ(H) - 2]).CrossProd(H.back() - H[SZ(H) - 2]) > -kEps) { H.pop_back(); } H.push_back(pts[i]); } if (SZ(H) > 1 && (H[0] - H.back()).IsZero()) { H.pop_back(); } } pts.clear(); for (auto p : one_way_hull[1]) { pts.push_back(p); } for (int i = SZ(one_way_hull[0]) - 2; i >= 1; i--) { pts.push_back(one_way_hull[0][i]); } return one_way_hull; } // without sides vector<vector<bool>> InsideDiagonalsMatrix() { // tested here: http://codeforces.com/contest/438/submission/11063385 int n = pts.size(); vector<vector<bool>> res(n, vector<bool>(n)); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { Line diag(pts[i], pts[j]); if (i == j || abs(i - j) == 1 || abs(i - j) == n - 1) { continue; } res[i][j] = 1; for (int k = 0; k < n; k++) { int kk = next(k); Line side(pts[k], pts[kk]); if (k == i || k == j || kk == i || kk == j) { continue; } vector<Point> inter = Utils::InterSegs(diag, side); if (SZ(inter)) { res[i][j] = 0; } } int act = next(i); LD areas[2] = {0, 0}; int passed_j = 0; while (act != i) { passed_j |= (act == j); areas[passed_j] += pts[i].CrossProd(pts[act], pts[next(act)]); act = next(act); } if (areas[0] * areas[1] < kEps) { res[i][j] = 0; } } } return res; } // P needs to be strictly outside polygon // polygon needs to be STRICTLY convex and counterclockwise oriented (as MakeConvexHull does) // returns {L, R} so that PL, PR are tangents and PL is on left vector<Point> Tangents(Point p) { // tested here: https://icpc.kattis.com/problems/spin (1169964) vector<Point> res; REP (tr, 2) { auto GrThan = [&](int fir, int sec) { // fir on sec's left return p.CrossProd(pts[sec], pts[fir]) > kEps; }; bool up = false; int cr = 0; if (SZ(pts) >= 2) { cr = p.CrossProd(pts[0], pts[1]); } if (abs(cr) < kEps && SZ(pts) >= 3) { cr = p.CrossProd(pts[0], pts[2]); } up = (cr > 0); VI bd{1, SZ(pts) - 1}; int faj = 0; while (bd[0] + 6 <= bd[1]) { // better don't replace with smaller constants VI h(2); REP (hh, 2) { h[hh] = (bd[0] + bd[1] + bd[hh]) / 3; } if (!GrThan(h[up ^ tr], 0) ^ tr) { bd[up ^ tr] = h[up ^ tr]; } else { int gr = GrThan(h[0], h[1]); bd[gr ^ tr] = h[gr ^ tr]; } } FOR (i, bd[0], bd[1]) { if (GrThan(i, faj) ^ tr) { faj = i; } } res.PB(pts[faj]); } return res; } }; struct ConvexPolHalves { // tested here: https://icpc.kattis.com/problems/spin (1169964) vector<vector<Point>> chains; // initialized by MakeConvexHull bool BelongTo(Point p) { // including borders if (SZ(chains[0]) == 1) { return (chains[0][0] - p).IsZero(); } if (p.x + kEps < chains[0][0].x || p.x - kEps > chains[0].back().x) { return false; } REP (tr, 2) { int kl = 0, kp = SZ(chains[tr]) - 2, faj = 0; while (kl <= kp) { int aktc = (kl + kp) / 2; if (chains[tr][aktc].x < p.x + kEps) { kl = aktc + 1; faj = aktc; } else { kp = aktc - 1; } } Point fir = chains[tr][faj], sec = chains[tr][faj + 1]; if (abs(fir.x - sec.x) < kEps) { if (tr == 0) { if (sec.y + kEps < p.y) { return false; } } else { if (fir.y - kEps > p.y) { return false; } } } else { LD cr = fir.CrossProd(sec, p); if (abs(cr) < kEps) { return true; } if ((cr > 0) ^ tr) { return false; } } } return true; } }; // CLIP START bool InUpper(Point a) { if (abs(a.y) > kEps) { return a.y > 0; } return a.x > 0; } bool angle_cmp(const Point a, const Point b) { bool u = InUpper(a); bool v = InUpper(b); return u!=v ? u : a.CrossProd(b)>0; } /** * @brief a+(b-a)*f \in c+lin(d-c) * @returns f */ LD cross(Point a, Point b, Point c, Point d) { return (d - c).CrossProd(a - c) / (d - c).CrossProd(a - b); } struct ClipLine { // valid side is on left ClipLine(Point A, Point B) : al(A), bl(B), a(A), b(B) {}; Point al,bl; // original line points mutable Point a,b; // actual intersection points Point dir() const { return bl - al; } bool operator<(const ClipLine& l) const { return angle_cmp(dir(),l.dir()); } Point cross(const ClipLine& l) { return al + (bl - al) * ::cross(al, bl, l.al, l.bl); } bool left(Point p) { return (bl - al).CrossProd(p - al) > 0; } }; set<VI> put; struct Clip { Clip(LD r) : area(4*r*r) { Point a{-r,-r}, b{r,-r}, c{r,r}, d{-r,r}; lines = {ClipLine(a,b), ClipLine(b,c), ClipLine(c,d), ClipLine(d,a)}; } void insert(Line l) { VI abc = l.LineEqNormInt(); if (put.count(abc)) { return; } put.insert(abc); insert(ClipLine(l[0], l[1])); } void insert(ClipLine l) { assert(abs(l.dir().SqNorm()) > kEps); find(l); while (size() && !l.left(it->a) && !l.left(it->b)) { erase(); } if (size()) { while (prev(), size() && !l.left(it->a) && !l.left(it->b)) { erase(); } } if (size() && (!l.left(it->a) || !l.left(it->b))) { l.a = l.cross(*it); area -= l.a.CrossProd(it->b)*.5; it->b = l.a; next(); l.b = l.cross(*it); if ((l.a-l.b).SqNorm() < kEps) { l.b = l.a; } area -= it->a.CrossProd(l.b) * .5; it->a = l.b; if (!(l.a - l.b).IsZero()) { area += l.a.CrossProd(l.b)*.5; lines.insert(l); } } //assert(l.dir().SqNorm()>1e-13); } void find(const ClipLine &l) { it = lines.lower_bound(l); if (it == lines.end()) { it = lines.begin(); } } void recalculate() { area = 0; for (const ClipLine &l : lines) area += l.a.CrossProd(l.b); area *= .5; } int size() { return lines.size(); } void next() { if(++it==lines.end()) it = lines.begin(); } void prev() { if(it==lines.begin()) it = lines.end(); --it; } void erase() { assert(it!=lines.end()); area -= it->a.CrossProd(it->b)*.5; it = lines.erase(it); if(it==lines.end()) it = lines.begin(); } typename set<ClipLine>::iterator it; set<ClipLine> lines; LD area; }; // CLIP ENDS // CENTERS BEGIN Point Bary(Point A, Point B, Point C, LD a, LD b, LD c) { return (A * a + B * b + C * c) / (a + b + c); } Point Centroid(Point A, Point B, Point C) { return Bary(A, B, C, 1, 1, 1); } Point Circumcenter(Point A, Point B, Point C) { LD a = (B - C).SqNorm(), b = (C - A).SqNorm(), c = (A - B).SqNorm(); return Bary(A, B, C, a * (b + c - a), b * (c + a - b), c * (a + b - c)); } Point Incenter(Point A, Point B, Point C) { return Bary(A, B, C, (B - C).Norm(), (A - C).Norm(), (A - B).Norm()); } Point Orthocenter(Point A, Point B, Point C) { LD a = (B - C).SqNorm(), b = (C - A).SqNorm(), c = (A - B).SqNorm(); return Bary(A, B, C, (a+b-c)*(c+a-b), (b+c-a)*(a+b-c), (c+a-b)*(b+c-a)); } Point Excenter(Point A, Point B, Point C) { // opposite to A LD a = (B - C).Norm(), b = (A - C).Norm(), c = (A - B).Norm(); return Bary(A, B, C, -a, b, c); } // const LD kEps = 1e-9; // const LD kPi = 2 * acos(0); struct PointInt { int x, y; PointInt operator-(PointInt p) { return {x - p.x, y - p.y}; } PointInt operator+(PointInt p) { return {x + p.x, y + p.y}; } // Point operator*(LD f) { return {x * f, y * f}; } // bool operator<(const Point& oth) const { // if (abs(x - oth.x) > kEps) { return x < oth.x; } // if (abs(y - oth.y) > kEps) { return y < oth.y; } // return false; // } // LD Angle() { // return atan2(y, x); // } }; // LD Normalize(LD ang) { // if (ang < -kPi) { // ang += 2 * kPi; // } // if (ang > kPi) { // ang -= 2 * kPi; // } // return ang; // } int CrossProd(PointInt A, PointInt B, PointInt C) { B = B - A; C = C - A; return B.x * C.y - B.y * C.x; } // vector<Point> InterLineLine(Point A, Point B, Point P, Point Q) { // LD tr_area = CrossProd(A, P, Q); // LD quad_area = CrossProd(A, P, B) + CrossProd(A, B, Q); // if (abs(quad_area) < 1e-9) { // return {}; // } // return {A + (B - A) * (tr_area / quad_area)}; // } const int N = 222; int n; vector<PointInt> segs[N]; // bool CheckPoint(Point p) { // int bil = 0; // vector<pair<LD, int>> evs; // RE (i, n) { // Point D1 = segs[i][0] - p; // Point D2 = segs[i][1] - p; // LD ang1 = D1.Angle(); // LD ang2 = D2.Angle(); // LD diff = Normalize(ang2 - ang1); // if (diff < 0) { // swap(D1, D2); // swap(ang1, ang2); // } // LD from = Normalize(ang2 - kPi / 2); // LD to = Normalize(ang1 + kPi / 2); // evs.PB({from, 1}); // evs.PB({to, -1}); // if (from > to) { // bil++; // } // } // sort(ALL(evs)); // if (bil == 0) { return false; } // for (auto ev : evs) { // bil += ev.nd; // if (bil == 0) { return false; } // } // return true; // } Point PFromPI(PointInt P) { debug(P.x, P.y); return {(LD)P.x, (LD)P.y}; } int32_t main() { ios_base::sync_with_stdio(0); cout << fixed << setprecision(15); cerr << fixed << setprecision(15); cin.tie(0); //double beg_clock = 1.0 * clock() / CLOCKS_PER_SEC; cin>>n; vector<Point> pts; RE (i, n) { int x1, y1, x2, y2; cin>>x1>>y1>>x2>>y2; segs[i].PB(PointInt{x1, y1}); segs[i].PB(PointInt{x2, y2}); //pts.PB(Point{LD(x1), LD(y1)}); //pts.PB(Point{LD(x2), LD(y2)}); } //map<Point, int> ids; Clip clip(1e4); set<VI> put_lines; RE (i, n) { RE (j, i - 1) { REP (ti, 2) { REP (tj, 2) { bool can_all_atmost0 = 1, can_all_atleast0 = 1; PointInt P = segs[i][ti], Q = segs[j][tj]; RE (k, n) { if (k == i || k == j) { continue; } int cr0 = CrossProd(P, Q, segs[k][0]); int cr1 = CrossProd(P, Q, segs[k][1]); can_all_atleast0 &= (cr0 >= 0 || cr1 >= 0); can_all_atmost0 &= (cr0 <= 0 || cr1 <= 0); } if (can_all_atleast0 && can_all_atmost0) { cout<<(LD)0<<endl; return 0; } if (can_all_atleast0) { debug(">=0"); Line l = Line{PFromPI(P), PFromPI(Q)}; clip.insert(l); } if (can_all_atmost0) { debug("<=0"); clip.insert(Line{PFromPI(Q), PFromPI(P)}); } } } } } clip.recalculate(); cout<<clip.area<<endl; return 0; } |