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#include "geo/minkowski_sum.hpp"
#include "geo/convex_polygon.hpp" #include "geo/angle_sort.hpp" #include "geo/convex_hull.hpp" // https://codeforces.com/contest/87/problem/E template <typename T> ConvexPolygon<T> minkowski_sum(ConvexPolygon<T> A, ConvexPolygon<T> B) { using P = Point<T>; vc<P> F; P p(0, 0); FOR(2) { swap(A, B); vc<P> point = A.point; int n = len(point); FOR(i, n) { int j = (i + 1) % n; F.eb(point[j] - point[i]); } p = p + MIN(point); } auto I = angle_sort(F); int n = len(I); F = rearrange(F, I); vc<P> point(n); FOR(i, n - 1) point[i + 1] = point[i] + F[i]; P add = p - MIN(point); for (auto& x: point) x = x + add; I = ConvexHull(point); point = rearrange(point, I); return ConvexPolygon<T>(point, true); }
#line 2 "geo/base.hpp" template <typename T> struct Point { T x, y; Point() : x(0), y(0) {} template <typename A, typename B> Point(A x, B y) : x(x), y(y) {} template <typename A, typename B> Point(pair<A, B> p) : x(p.fi), y(p.se) {} Point operator+(Point p) const { return {x + p.x, y + p.y}; } Point operator-(Point p) const { return {x - p.x, y - p.y}; } bool operator==(Point p) const { return x == p.x && y == p.y; } bool operator!=(Point p) const { return x != p.x || y != p.y; } Point operator-() const { return {-x, -y}; } Point operator*(T t) const { return {x * t, y * t}; } Point operator/(T t) const { return {x / t, y / t}; } bool operator<(Point p) const { if (x != p.x) return x < p.x; return y < p.y; } T dot(Point other) { return x * other.x + y * other.y; } T det(Point other) { return x * other.y - y * other.x; } double norm() { return sqrtl(x * x + y * y); } double angle() { return atan2(y, x); } Point rotate(double theta) { static_assert(!is_integral<T>::value); double c = cos(theta), s = sin(theta); return Point{c * x - s * y, s * x + c * y}; } }; #ifdef FASTIO template <typename T> void rd(Point<T>& p) { fastio::rd(p.x), fastio::rd(p.y); } template <typename T> void wt(Point<T>& p) { fastio::wt(p.x); fastio::wt(' '); fastio::wt(p.y); } #endif // A -> B -> C と進むときに、左に曲がるならば +1、右に曲がるならば -1 template <typename T> int ccw(Point<T> A, Point<T> B, Point<T> C) { T x = (B - A).det(C - A); if (x > 0) return 1; if (x < 0) return -1; return 0; } template <typename REAL, typename T> REAL dist(Point<T> A, Point<T> B) { A = A - B; T p = A.dot(A); return sqrt(REAL(p)); } // ax+by+c template <typename T> struct Line { T a, b, c; Line(T a, T b, T c) : a(a), b(b), c(c) {} Line(Point<T> A, Point<T> B) { a = A.y - B.y, b = B.x - A.x, c = A.x * B.y - A.y * B.x; } Line(T x1, T y1, T x2, T y2) : Line(Point<T>(x1, y1), Point<T>(x2, y2)) {} template <typename U> U eval(Point<U> P) { return a * P.x + b * P.y + c; } template <typename U> T eval(U x, U y) { return a * x + b * y + c; } // 同じ直線が同じ a,b,c で表現されるようにする void normalize() { static_assert(is_same_v<T, int> || is_same_v<T, long long>); T g = gcd(gcd(abs(a), abs(b)), abs(c)); a /= g, b /= g, c /= g; if (b < 0) { a = -a, b = -b, c = -c; } if (b == 0 && a < 0) { a = -a, b = -b, c = -c; } } bool is_parallel(Line other) { return a * other.b - b * other.a == 0; } bool is_orthogonal(Line other) { return a * other.a + b * other.b == 0; } }; template <typename T> struct Segment { Point<T> A, B; Segment(Point<T> A, Point<T> B) : A(A), B(B) {} Segment(T x1, T y1, T x2, T y2) : Segment(Point<T>(x1, y1), Point<T>(x2, y2)) {} bool contain(Point<T> C) { static_assert(is_integral<T>::value); T det = (C - A).det(B - A); if (det != 0) return 0; return (C - A).dot(B - A) >= 0 && (C - B).dot(A - B) >= 0; } Line<T> to_Line() { return Line(A, B); } }; template <typename REAL> struct Circle { Point<REAL> O; REAL r; Circle(Point<REAL> O, REAL r) : O(O), r(r) {} Circle(REAL x, REAL y, REAL r) : O(x, y), r(r) {} template <typename T> bool contain(Point<T> p) { REAL dx = p.x - O.x, dy = p.y - O.y; return dx * dx + dy * dy <= r * r; } }; template <typename T> struct Polygon { vc<Point<T>> points; T a; template <typename A, typename B> Polygon(vc<pair<A, B>> pairs) { for (auto&& [a, b]: pairs) points.eb(Point<T>(a, b)); build(); } Polygon(vc<Point<T>> points) : points(points) { build(); } int size() { return len(points); } template <typename REAL> REAL area() { return a * 0.5; } template <enable_if_t<is_integral<T>::value, int> = 0> T area_2() { return a; } bool is_convex() { FOR(j, len(points)) { int i = (j == 0 ? len(points) - 1 : j - 1); int k = (j == len(points) - 1 ? 0 : j + 1); if ((points[j] - points[i]).det(points[k] - points[j]) < 0) return false; } return true; } private: void build() { a = 0; FOR(i, len(points)) { int j = (i + 1 == len(points) ? 0 : i + 1); a += points[i].det(points[j]); } if (a < 0) { a = -a; reverse(all(points)); } } }; #line 2 "geo/convex_hull.hpp" #line 4 "geo/convex_hull.hpp" template <typename T> vector<int> ConvexHull(vector<pair<T, T>>& XY, string mode = "full", bool inclusive = false, bool sorted = false) { assert(mode == "full" || mode == "lower" || mode == "upper"); ll N = XY.size(); if (N == 1) return {0}; if (N == 2) { if (XY[0] < XY[1]) return {0, 1}; if (XY[1] < XY[0]) return {1, 0}; if (inclusive) return {0, 1}; return {0}; } vc<int> I = argsort(XY); auto check = [&](ll i, ll j, ll k) -> bool { auto xi = XY[i].fi, yi = XY[i].se; auto xj = XY[j].fi, yj = XY[j].se; auto xk = XY[k].fi, yk = XY[k].se; auto dx1 = xj - xi, dy1 = yj - yi; auto dx2 = xk - xj, dy2 = yk - yj; T det = dx1 * dy2 - dy1 * dx2; return (inclusive ? det >= 0 : det > 0); }; auto calc = [&]() { vector<int> P; for (auto&& k: I) { while (P.size() > 1) { auto i = P[P.size() - 2]; auto j = P[P.size() - 1]; if (check(i, j, k)) break; P.pop_back(); } P.eb(k); } return P; }; vc<int> P; if (mode == "full" || mode == "lower") { vc<int> Q = calc(); P.insert(P.end(), all(Q)); } if (mode == "full" || mode == "upper") { if (!P.empty()) P.pop_back(); reverse(all(I)); vc<int> Q = calc(); P.insert(P.end(), all(Q)); } if (mode == "upper") reverse(all(P)); if (len(P) >= 2 && P[0] == P.back()) P.pop_back(); return P; } template <typename T> vector<int> ConvexHull(vector<Point<T>>& XY, string mode = "full", bool inclusive = false, bool sorted = false) { assert(mode == "full" || mode == "lower" || mode == "upper"); ll N = XY.size(); if (N == 1) return {0}; if (N == 2) { if (XY[0] < XY[1]) return {0, 1}; if (XY[1] < XY[0]) return {1, 0}; if (inclusive) return {0, 1}; return {0}; } vc<int> I = argsort(XY); auto check = [&](ll i, ll j, ll k) -> bool { auto xi = XY[i].x, yi = XY[i].y; auto xj = XY[j].x, yj = XY[j].y; auto xk = XY[k].x, yk = XY[k].y; auto dx1 = xj - xi, dy1 = yj - yi; auto dx2 = xk - xj, dy2 = yk - yj; T det = dx1 * dy2 - dy1 * dx2; return (inclusive ? det >= 0 : det > 0); }; auto calc = [&]() { vector<int> P; for (auto&& k: I) { while (P.size() > 1) { auto i = P[P.size() - 2]; auto j = P[P.size() - 1]; if (check(i, j, k)) break; P.pop_back(); } P.eb(k); } return P; }; vc<int> P; if (mode == "full" || mode == "lower") { vc<int> Q = calc(); P.insert(P.end(), all(Q)); } if (mode == "full" || mode == "upper") { if (!P.empty()) P.pop_back(); reverse(all(I)); vc<int> Q = calc(); P.insert(P.end(), all(Q)); } if (mode == "upper") reverse(all(P)); if (len(P) >= 2 && P[0] == P.back()) P.pop_back(); return P; } #line 3 "geo/convex_polygon.hpp" // ほとんどテストされていないのであやしい // n=2 は現状サポートしていない // 同一直線上に複数の点があると正しく動かない説がある template <typename T> struct ConvexPolygon { using P = Point<T>; int n; vc<P> point; ConvexPolygon(vc<P> point_, bool is_conv) : n(len(point_)), point(point_) { if (!is_conv) { vc<int> I = ConvexHull<T>(point_, "full"); point = rearrange(point_, I); } // assert(n >= 3); // counter clockwise になおす if (n >= 3) { if ((point[1] - point[0]).det(point[2] - point[0]) < 0) { reverse(all(point)); } } } // 比較関数 comp(i,j) template <typename F> int periodic_min_comp(F comp) { int L = 0, M = n, R = n + n; while (1) { if (R - L == 2) break; int L1 = (L + M) / 2, R1 = (M + R + 1) / 2; if (comp(L1, M)) { R = M, M = L1; } elif (comp(R1, M)) { L = M, M = R1; } else { L = L1, R = R1; } } return M % n; } int nxt_idx(int i) { return (i + 1 == n ? 0 : i + 1); } int prev_idx(int i) { return (i == 0 ? n - 1 : i - 1); } // 中:1, 境界:0, 外:-1 int side(P p) { int L = 1, R = n - 1; T a = (point[L] - point[0]).det(p - point[0]); T b = (point[R] - point[0]).det(p - point[0]); if (a < 0 || b > 0) return -1; // p は 0 から見て [L,R] 方向 while (R - L >= 2) { int M = (L + R) / 2; T c = (point[M] - point[0]).det(p - point[0]); if (c < 0) R = M, b = c; else L = M, a = c; } T c = (point[R] - point[L]).det(p - point[L]); T x = min({a, -b, c}); if (x < 0) return -1; if (x > 0) return 1; return 0; } pair<int, T> min_dot(P p) { int idx = periodic_min_comp([&](int i, int j) -> bool { return point[i % n].dot(p) < point[j % n].dot(p); }); return {idx, point[idx].dot(p)}; } pair<int, T> max_dot(P p) { int idx = periodic_min_comp([&](int i, int j) -> bool { return point[i % n].dot(p) > point[j % n].dot(p); }); return {idx, point[idx].dot(p)}; } // pair<int, int> visible_range(P p) {} }; #line 2 "geo/angle_sort.hpp" #line 4 "geo/angle_sort.hpp" // 偏角ソートに対する argsort template <typename T> vector<int> angle_sort(vector<Point<T>>& P) { vector<int> lower, origin, upper; const Point<T> O = {0, 0}; FOR(i, len(P)) { if (P[i] == O) origin.eb(i); elif ((P[i].y < 0) || (P[i].y == 0 && P[i].x > 0)) lower.eb(i); else upper.eb(i); } sort(all(lower), [&](auto& i, auto& j) { return P[i].det(P[j]) > 0; }); sort(all(upper), [&](auto& i, auto& j) { return P[i].det(P[j]) > 0; }); auto& I = lower; I.insert(I.end(), all(origin)); I.insert(I.end(), all(upper)); return I; } // 偏角ソートに対する argsort template <typename T> vector<int> angle_sort(vector<pair<T, T>>& P) { vc<Point<T>> tmp(len(P)); FOR(i, len(P)) tmp[i] = Point<T>(P[i]); return angle_sort<T>(tmp); } #line 4 "geo/minkowski_sum.hpp" // https://codeforces.com/contest/87/problem/E template <typename T> ConvexPolygon<T> minkowski_sum(ConvexPolygon<T> A, ConvexPolygon<T> B) { using P = Point<T>; vc<P> F; P p(0, 0); FOR(2) { swap(A, B); vc<P> point = A.point; int n = len(point); FOR(i, n) { int j = (i + 1) % n; F.eb(point[j] - point[i]); } p = p + MIN(point); } auto I = angle_sort(F); int n = len(I); F = rearrange(F, I); vc<P> point(n); FOR(i, n - 1) point[i + 1] = point[i] + F[i]; P add = p - MIN(point); for (auto& x: point) x = x + add; I = ConvexHull(point); point = rearrange(point, I); return ConvexPolygon<T>(point, true); }