#define _USE_MATH_DEFINES #include using namespace std; #define FOR(i,m,n) for(int i=(m);i<(n);++i) #define REP(i,n) FOR(i,0,n) #define ALL(v) (v).begin(),(v).end() using ll = long long; constexpr int INF = 0x3f3f3f3f; constexpr long long LINF = 0x3f3f3f3f3f3f3f3fLL; constexpr double EPS = 1e-8; constexpr int MOD = 1000000007; // constexpr int MOD = 998244353; constexpr int dy[] = {1, 0, -1, 0}, dx[] = {0, -1, 0, 1}; constexpr int dy8[] = {1, 1, 0, -1, -1, -1, 0, 1}, dx8[] = {0, -1, -1, -1, 0, 1, 1, 1}; template inline bool chmax(T &a, U b) { return a < b ? (a = b, true) : false; } template inline bool chmin(T &a, U b) { return a > b ? (a = b, true) : false; } struct IOSetup { IOSetup() { std::cin.tie(nullptr); std::ios_base::sync_with_stdio(false); std::cout << fixed << setprecision(20); } } iosetup; namespace geometry { using Real = double; constexpr long double PI = 3.14159265358979323846; int sgn(Real x) { static constexpr Real EPS = 1e-8; return x > EPS ? 1 : x < -EPS ? -1 : 0; } Real degree_to_radian(Real d) { return d * PI / 180; } Real radian_to_degree(Real r) { return r * 180 / PI; } struct Point { Real x, y; Point(Real x = 0, Real y = 0) : x(x), y(y) {} Real abs() const { return std::sqrt(norm()); } Real arg() const { Real res = std::atan2(y, x); return res < 0 ? res + PI * 2 : res; } Real norm() const { return x * x + y * y; } Point rotate(Real angle) const { Real cs = std::cos(angle), sn = std::sin(angle); return Point(x * cs - y * sn, x * sn + y * cs); } Point unit_vector() const { Real a = abs(); return Point(x / a, y / a); } std::pair normal_unit_vector() const { Point p = unit_vector(); return {Point(-p.y, p.x), Point(p.y, -p.x)}; } Point &operator+=(const Point &p) { x += p.x; y += p.y; return *this; } Point &operator-=(const Point &p) { x -= p.x; y -= p.y; return *this; } Point &operator*=(Real k) { x *= k; y *= k; return *this; } Point &operator/=(Real k) { x /= k; y /= k; return *this; } bool operator<(const Point &p) const { int x_sgn = sgn(p.x - x); return x_sgn != 0 ? x_sgn == 1 : sgn(p.y - y) == 1; } bool operator<=(const Point &p) const { return !(p < *this); } bool operator>(const Point &p) const { return p < *this; } bool operator>=(const Point &p) const { return !(*this < p); } Point operator+() const { return *this; } Point operator-() const { return Point(-x, -y); } Point operator+(const Point &p) const { return Point(*this) += p; } Point operator-(const Point &p) const { return Point(*this) -= p; } Point operator*(Real k) const { return Point(*this) *= k; } Point operator/(Real k) const { return Point(*this) /= k; } friend std::ostream &operator<<(std::ostream &os, const Point &p) { return os << '(' << p.x << ", " << p.y << ')'; } friend std::istream &operator>>(std::istream &is, Point &p) { Real x, y; is >> x >> y; p = Point(x, y); return is; } }; struct Segment { Point s, t; Segment(const Point &s = {0, 0}, const Point &t = {0, 0}) : s(s), t(t) {} }; struct Line : Segment { using Segment::Segment; Line(Real a, Real b, Real c) { if (sgn(a) == 0) { s = Point(0, -c / b); t = Point(1, s.y); } else if (sgn(b) == 0) { s = Point(-c / a, 0); t = Point(s.x, 1); } else if (sgn(c) == 0) { s = Point(0, 0); t = Point(1, -a / b); } else { s = Point(0, -c / b); t = Point(-c / a, 0); } } }; struct Circle { Point p; Real r; Circle(const Point &p = {0, 0}, Real r = 0) : p(p), r(r) {} }; Real cross(const Point &a, const Point &b) { return a.x * b.y - a.y * b.x; } Real dot(const Point &a, const Point &b) { return a.x * b.x + a.y * b.y; } int ccw(const Point &a, const Point &b, const Point &c) { Point ab = b - a, ac = c - a; int sign = sgn(cross(ab, ac)); if (sign == 0) { if (sgn(dot(ab, ac)) == -1) return 2; if (sgn(ac.norm() - ab.norm()) == 1) return -2; } return sign; } Real get_angle(const Point &a, const Point &b, const Point &c) { Real ba_arg = (a - b).arg(), bc_arg = (c - b).arg(); if (ba_arg > bc_arg) std::swap(ba_arg, bc_arg); return std::min(bc_arg - ba_arg, static_cast(PI * 2 - (bc_arg - ba_arg))); } Real closest_pair(std::vector ps) { int n = ps.size(); assert(n > 1); std::sort(ps.begin(), ps.end()); std::function rec = [&ps, &rec](int left, int right) -> Real { int mid = (left + right) >> 1; Real x_mid = ps[mid].x, d = std::numeric_limits::max(); if (left + 1 < mid) { Real tmp = rec(left, mid); if (tmp < d) d = tmp; } if (mid + 1 < right) { Real tmp = rec(mid, right); if (tmp < d) d = tmp; } std::inplace_merge(ps.begin() + left, ps.begin() + mid, ps.begin() + right, [&](const Point &a, const Point &b) -> bool { return sgn(b.y - a.y) == 1; }); std::vector tmp; for (int i = left; i < right; ++i) { if (sgn(std::abs(ps[i].x - x_mid) - d) == 1) continue; for (int j = static_cast(tmp.size()) - 1; j >= 0; --j) { Point now = ps[i] - tmp[j]; if (sgn(now.y - d) == 1) break; Real tmp = now.abs(); if (tmp < d) d = tmp; } tmp.emplace_back(ps[i]); } return d; }; return rec(0, n); } Point projection(const Segment &a, const Point &b) { return a.s + (a.t - a.s) * dot(a.t - a.s, b - a.s) / (a.t - a.s).norm(); } Point reflection(const Segment &a, const Point &b) { return projection(a, b) * 2 - b; } bool is_parallel(const Segment &a, const Segment &b) { return sgn(cross(a.t - a.s, b.t - b.s)) == 0; } bool is_orthogonal(const Segment &a, const Segment &b) { return sgn(dot(a.t - a.s, b.t - b.s)) == 0; } Real distance(const Point&, const Point&); Real distance(const Segment&, const Point&); Real distance(const Line&, const Point&); int sizeof_common_tangent(const Circle&, const Circle&); bool has_intersected(const Segment &a, const Point &b) { return ccw(a.s, a.t, b) == 0; } bool has_intersected(const Segment &a, const Segment &b) { return ccw(a.s, a.t, b.s) * ccw(a.s, a.t, b.t) <= 0 && ccw(b.s, b.t, a.s) * ccw(b.s, b.t, a.t) <= 0; } bool has_intersected(const Line &a, const Point &b) { int c = ccw(a.s, a.t, b); return c != 1 && c != -1; } bool has_intersected(const Line &a, const Segment &b) { return ccw(a.s, a.t, b.s) * ccw(a.s, a.t, b.t) != 1; } bool has_intersected(const Line &a, const Line &b) { return sgn(cross(a.t - a.s, b.t - b.s)) != 0 || sgn(cross(a.t - a.s, b.s - a.s)) == 0; } bool has_intersected(const Circle &a, const Point &b) { return sgn(distance(a.p, b) - a.r) == 0; } bool has_intersected(const Circle &a, const Segment &b) { return sgn(a.r - distance(b, a.p)) != -1 && sgn(std::max(distance(a.p, b.s), distance(a.p, b.t)) - a.r) != -1; } bool has_intersected(const Circle &a, const Line &b) { return sgn(a.r - distance(b, a.p)) != -1; } bool has_intersected(const Circle &a, const Circle &b) { return sizeof_common_tangent(a, b) > 0; } Point intersection(const Line &a, const Line &b) { assert(has_intersected(a, b) && !is_parallel(a, b)); return a.s + (a.t - a.s) * cross(b.t - b.s, b.s - a.s) / cross(b.t - b.s, a.t - a.s); } Point intersection(const Segment &a, const Segment &b) { assert(has_intersected(a, b)); if (is_parallel(a, b)) { if (sgn(distance(a.s, b.s)) == 0) { assert(sgn(dot(a.t - a.s, b.t - a.s)) == -1); return a.s; } else if (sgn(distance(a.s, b.t)) == 0) { assert(sgn(dot(a.t - a.s, b.s - a.s)) == -1); return a.s; } else if (sgn(distance(a.t, b.s)) == 0) { assert(sgn(dot(a.s - a.t, b.t - a.t)) == -1); return a.t; } else if (sgn(distance(a.t, b.t)) == 0) { assert(sgn(dot(a.s - a.t, b.s - a.t)) == -1); return a.t; } else { assert(false); } } else { return intersection(Line(a.s, a.t), Line(b.s, b.t)); } } Point intersection(const Line &a, const Segment &b) { assert(has_intersected(a, b)); return intersection(a, Line(b.s, b.t)); } std::vector intersection(const Circle &a, const Line &b) { Point pro = projection(b, a.p); Real nor = (a.p - pro).norm(); int sign = sgn(a.r - std::sqrt(nor)); if (sign == -1) return {}; if (sign == 0) return {pro}; Point v = (b.t - b.s).unit_vector() * std::sqrt(a.r * a.r - nor); return {pro + v, pro - v}; } std::vector intersection(const Circle &a, const Segment &b) { if (!has_intersected(a, b)) return {}; std::vector res = intersection(a, Line(b.s, b.t)); if (sgn(distance(a.p, b.s) - a.r) != -1 && sgn(distance(a.p, b.t) - a.r) != -1) return res; return {sgn(dot(res[0] - b.s, res[0] - b.t)) == 1 ? res[1] : res[0]}; } std::vector intersection(const Circle &a, const Circle &b) { int sz = sizeof_common_tangent(a, b); if (sz == 0 || sz == 4) return {}; Real alpha = (b.p - a.p).arg(); if (sz == 1 || sz == 3) return {Point(a.p.x + a.r * std::cos(alpha), a.p.y + a.r * std::sin(alpha))}; Real dist = (b.p - a.p).norm(), beta = std::acos((dist + a.r * a.r - b.r * b.r) / (2 * std::sqrt(dist) * a.r)); return {a.p + Point(a.r * std::cos(alpha + beta), a.r * std::sin(alpha + beta)), a.p + Point(a.r * std::cos(alpha - beta), a.r * std::sin(alpha - beta))}; } Real distance(const Point &a, const Point &b) { return (b - a).abs(); } Real distance(const Segment &a, const Point &b) { Point foot = projection(a, b); return has_intersected(a, foot) ? distance(foot, b) : std::min(distance(a.s, b), distance(a.t, b)); } Real distance(const Segment &a, const Segment &b) { return has_intersected(a, b) ? 0 : std::min({distance(a, b.s), distance(a, b.t), distance(b, a.s), distance(b, a.t)}); } Real distance(const Line &a, const Point &b) { return distance(projection(a, b), b); } Real distance(const Line &a, const Segment &b) { return has_intersected(a, b) ? 0 : std::min(distance(a, b.s), distance(a, b.t)); } Real distance(const Line &a, const Line &b) { return has_intersected(a, b) ? 0 : distance(a, b.s); } std::vector tangency(const Circle &a, const Point &b) { Real dist = distance(a.p, b); int sign = sgn(dist - a.r); if (sign == -1) return {}; if (sign == 0) return {b}; Real alpha = (b - a.p).arg(), beta = std::acos(a.r / dist); return {a.p + Point(a.r * std::cos(alpha + beta), a.r * std::sin(alpha + beta)), a.p + Point(a.r * std::cos(alpha - beta), a.r * std::sin(alpha - beta))}; } int sizeof_common_tangent(const Circle &a, const Circle &b) { Real dist = distance(a.p, b.p); int sign = sgn(a.r + b.r - dist); if (sign == -1) return 4; if (sign == 0) return 3; sign = sgn((sgn(a.r - b.r) == -1 ? b.r - a.r : a.r - b.r) - dist); if (sign == -1) return 2; if (sign == 0) return 1; return 0; } std::vector common_tangent(const Circle &a, const Circle &b) { std::vector tangents; Real dist = distance(a.p, b.p), argument = (b.p - a.p).arg(); int sign = sgn(a.r + b.r - dist); if (sign == -1) { Real ac = std::acos((a.r + b.r) / dist), alpha = argument + ac, cs = std::cos(alpha), sn = std::sin(alpha); tangents.emplace_back(a.p + Point(a.r * cs, a.r * sn), b.p + Point(-b.r * cs, -b.r * sn)); alpha = argument - ac; cs = std::cos(alpha); sn = std::sin(alpha); tangents.emplace_back(a.p + Point(a.r * cs, a.r * sn), b.p + Point(-b.r * cs, -b.r * sn)); } else if (sign == 0) { Point s = a.p + Point(a.r * std::cos(argument), a.r * std::sin(argument)); tangents.emplace_back(s, s + (b.p - a.p).normal_unit_vector().first); } if (sgn(b.r - a.r) == -1) { sign = sgn(a.r - b.r - dist); if (sign == -1) { Real at = std::acos((a.r - b.r) / dist), alpha = argument + at, cs = std::cos(alpha), sn = std::sin(alpha); tangents.emplace_back(a.p + Point(a.r * cs, a.r * sn), b.p + Point(b.r * cs, b.r * sn)); alpha = argument - at; cs = std::cos(alpha); sn = std::sin(alpha); tangents.emplace_back(a.p + Point(a.r * cs, a.r * sn), b.p + Point(b.r * cs, b.r * sn)); } else if (sign == 0) { Point s = a.p + Point(a.r * std::cos(argument), a.r * std::sin(argument)); tangents.emplace_back(s, s + (b.p - a.p).normal_unit_vector().first); } } else { sign = sgn(b.r - a.r - dist); if (sign == -1) { Real at = std::acos((b.r - a.r) / dist), alpha = argument - at, cs = std::cos(alpha), sn = std::sin(alpha); tangents.emplace_back(a.p + Point(-a.r * cs, -a.r * sn), b.p + Point(-b.r * cs, -b.r * sn)); alpha = argument + at; cs = std::cos(alpha); sn = std::sin(alpha); tangents.emplace_back(a.p + Point(-a.r * cs, -a.r * sn), b.p + Point(-b.r * cs, -b.r * sn)); } else if (sign == 0) { Point s = b.p + Point(-b.r * std::cos(argument), -b.r * std::sin(argument)); tangents.emplace_back(s, s + (a.p - b.p).normal_unit_vector().first); } } return tangents; } Real intersection_area(const Circle &a, const Circle &b) { Real nor = (b.p - a.p).norm(), dist = std::sqrt(nor); if (sgn(a.r + b.r - dist) != 1) return 0; if (sgn(std::abs(a.r - b.r) - dist) != -1) return std::min(a.r, b.r) * std::min(a.r, b.r) * PI; Real alpha = std::acos((nor + a.r * a.r - b.r * b.r) / (2 * dist * a.r)), beta = std::acos((nor + b.r * b.r - a.r * a.r) / (2 * dist * b.r)); return (alpha - std::sin(alpha + alpha) * 0.5) * a.r * a.r + (beta - std::sin(beta + beta) * 0.5) * b.r * b.r; } using Polygon = std::vector; Real area(const Polygon &a) { int n = a.size(); Real res = 0; for (int i = 0; i < n; ++i) res += cross(a[i], a[(i + 1) % n]); return res * 0.5; } Point centroid(const Polygon &a) { Point res(0, 0); int n = a.size(); Real den = 0; for (int i = 0; i < n; ++i) { Real cro = cross(a[i], a[(i + 1) % n]); res += (a[i] + a[(i + 1) % n]) / 3 * cro; den += cro; } return res / den; } int is_contained(const Polygon &a, const Point &b) { int n = a.size(); bool is_in = false; for (int i = 0; i < n; ++i) { Point p = a[i] - b, q = a[(i + 1) % n] - b; if (sgn(q.y - p.y) == -1) std::swap(p, q); int sign = sgn(cross(p, q)); if (sign == 1 && sgn(p.y) != 1 && sgn(q.y) == 1) is_in = !is_in; if (sign == 0 && sgn(dot(p, q)) != 1) return 1; } return is_in ? 2 : 0; } bool is_convex(const Polygon &a) { int n = a.size(); for (int i = 0; i < n; ++i) { if (ccw(a[(i - 1 + n) % n], a[i], a[(i + 1) % n]) == -1) return false; } return true; } Polygon monotone_chain(std::vector ps, bool tight = true) { std::sort(ps.begin(), ps.end()); int n = ps.size(), idx = 0; Polygon convex_hull(n << 1); for (int i = 0; i < n; convex_hull[idx++] = ps[i++]) { while (idx >= 2 && sgn(cross(convex_hull[idx - 1] - convex_hull[idx - 2], ps[i] - convex_hull[idx - 1])) < tight) --idx; } for (int i = n - 2, border = idx + 1; i >= 0; convex_hull[idx++] = ps[i--]) { while (idx >= border && sgn(cross(convex_hull[idx - 1] - convex_hull[idx - 2], ps[i] - convex_hull[idx - 1])) < tight) --idx; } convex_hull.resize(idx - 1); return convex_hull; } Polygon cut_convex(const Polygon &a, const Line &b) { int n = a.size(); Polygon res; for (int i = 0; i < n; ++i) { int c = ccw(b.s, b.t, a[i]); if (c != -1) res.emplace_back(a[i]); if (c * ccw(b.s, b.t, a[(i + 1) % n]) == -1) res.emplace_back(intersection(Line(a[i], a[(i + 1) % n]), b)); } if (res.size() < 3) res.clear(); return res; } std::pair rotating_calipers(const Polygon &a) { int n = a.size(), high = 0, low = 0; if (n <= 2) { assert(n == 2); return {a[0], a[1]}; } for (int i = 1; i < n; ++i) { if (a[i].y > a[high].y) high = i; if (a[i].y < a[low].y) low = i; } Real max_norm = (a[high] - a[low]).norm(); int i = high, j = low, max_i = i, max_j = j; do { ((sgn(cross(a[(i + 1) % n] - a[i], a[(j + 1) % n] - a[j])) != -1 ? j : i) += 1) %= n; Real tmp = (a[j] - a[i]).norm(); if (sgn(tmp - max_norm) == 1) { max_norm = tmp; max_i = i; max_j = j; } } while (i != high || j != low); return {a[max_i], a[max_j]}; } } // geometry int main() { int n; cin >> n; geometry::Polygon poly(n); REP(i, n) cin >> poly[i].x >> poly[i].y; cout << static_cast(round(geometry::area(poly) * 2)) << '\n'; return 0; }