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#include "geo/cross_point.hpp"
#pragma once
#include "geo/base.hpp"
// 平行でないことを仮定
template <typename REAL, typename T>
Point<REAL> cross_point(const Line<T> L1, const Line<T> L2) {
T det = L1.a * L2.b - L1.b * L2.a;
assert(det != 0);
REAL x = -REAL(L1.c) * L2.b + REAL(L1.b) * L2.c;
REAL y = -REAL(L1.a) * L2.c + REAL(L1.c) * L2.a;
return Point<REAL>(x / det, y / det);
}
// 浮動小数点数はエラー
// 0: 交点なし
// 1: 一意な交点
// 2:2 つ以上の交点(整数型を利用して厳密にやる)
template <typename T>
int count_cross(Segment<T> S1, Segment<T> S2, bool include_ends) {
static_assert(!std::is_floating_point<T>::value);
Line<T> L1 = S1.to_Line();
Line<T> L2 = S2.to_Line();
if (L1.is_parallel(L2)) {
if (L1.eval(S2.A) != 0) return 0;
// 4 点とも同一直線上にある
T a1 = S1.A.x, b1 = S1.B.x;
T a2 = S2.A.x, b2 = S2.B.x;
if (a1 == b1) {
a1 = S1.A.y, b1 = S1.B.y;
a2 = S2.A.y, b2 = S2.B.y;
}
if (a1 > b1) swap(a1, b1);
if (a2 > b2) swap(a2, b2);
T a = max(a1, a2);
T b = min(b1, b2);
if (a < b) return 2;
if (a > b) return 0;
return (include_ends ? 1 : 0);
}
// 平行でない場合
T a1 = L2.eval(S1.A), b1 = L2.eval(S1.B);
T a2 = L1.eval(S2.A), b2 = L1.eval(S2.B);
if (a1 > b1) swap(a1, b1);
if (a2 > b2) swap(a2, b2);
bool ok1 = 0, ok2 = 0;
if (include_ends) {
ok1 = (a1 <= T(0)) && (T(0) <= b1);
ok2 = (a2 <= T(0)) && (T(0) <= b2);
} else {
ok1 = (a1 < T(0)) && (T(0) < b1);
ok2 = (a2 < T(0)) && (T(0) < b2);
}
return (ok1 && ok2 ? 1 : 0);
}
// 4 次式まで登場している、オーバーフロー注意!
// https://codeforces.com/contest/607/problem/E
template <typename REAL, typename T>
vc<Point<REAL>> cross_point(const Circle<T> C, const Line<T> L) {
T a = L.a, b = L.b, c = L.a * (C.O.x) + L.b * (C.O.y) + L.c;
T r = C.r;
bool SW = 0;
T abs_a = (a < 0 ? -a : a);
T abs_b = (b < 0 ? -b : b);
if (abs_a < abs_b) {
swap(a, b);
SW = 1;
}
// ax+by+c=0, x^2+y^2=r^2
T D = 4 * c * c * b * b - 4 * (a * a + b * b) * (c * c - a * a * r * r);
if (D < 0) return {};
REAL sqD = sqrtl(D);
REAL y1 = (-2 * b * c + sqD) / (2 * (a * a + b * b));
REAL y2 = (-2 * b * c - sqD) / (2 * (a * a + b * b));
REAL x1 = (-b * y1 - c) / a;
REAL x2 = (-b * y2 - c) / a;
if (SW) swap(x1, y1), swap(x2, y2);
x1 += C.O.x, x2 += C.O.x;
y1 += C.O.y, y2 += C.O.y;
if (D == 0) return {Point<REAL>(x1, y1)};
return {Point<REAL>(x1, y1), Point<REAL>(x2, y2)};
}
// https://codeforces.com/contest/2/problem/C
template <typename REAL, typename T>
tuple<bool, Point<T>, Point<T>> cross_point_circle(Circle<T> C1, Circle<T> C2) {
using P = Point<T>;
P O{0, 0};
P A = C1.O, B = C2.O;
if (A == B) return {false, O, O};
T d = (B - A).norm();
REAL cos_val = (C1.r * C1.r + d * d - C2.r * C2.r) / (2 * C1.r * d);
if (cos_val < -1 || 1 < cos_val) return {false, O, O};
REAL t = acos(cos_val);
REAL u = (B - A).angle();
P X = A + P{C1.r * cos(u + t), C1.r * sin(u + t)};
P Y = A + P{C1.r * cos(u - t), C1.r * sin(u - t)};
return {true, X, Y};
}
#line 2 "geo/cross_point.hpp"
#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+=(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+(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(const Point& other) const { return x * other.x + y * other.y; }
T det(const Point& other) const { 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};
}
Point rot90(bool ccw) { return (ccw ? Point{-y, x} : Point{y, -x}); }
};
#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, typename U>
REAL dist(Point<T> A, Point<U> B) {
REAL dx = REAL(A.x) - REAL(B.x);
REAL dy = REAL(A.y) - REAL(B.y);
return sqrt(dx * dx + dy * dy);
}
// 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) {
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;
}
};
#line 4 "geo/cross_point.hpp"
// 平行でないことを仮定
template <typename REAL, typename T>
Point<REAL> cross_point(const Line<T> L1, const Line<T> L2) {
T det = L1.a * L2.b - L1.b * L2.a;
assert(det != 0);
REAL x = -REAL(L1.c) * L2.b + REAL(L1.b) * L2.c;
REAL y = -REAL(L1.a) * L2.c + REAL(L1.c) * L2.a;
return Point<REAL>(x / det, y / det);
}
// 浮動小数点数はエラー
// 0: 交点なし
// 1: 一意な交点
// 2:2 つ以上の交点(整数型を利用して厳密にやる)
template <typename T>
int count_cross(Segment<T> S1, Segment<T> S2, bool include_ends) {
static_assert(!std::is_floating_point<T>::value);
Line<T> L1 = S1.to_Line();
Line<T> L2 = S2.to_Line();
if (L1.is_parallel(L2)) {
if (L1.eval(S2.A) != 0) return 0;
// 4 点とも同一直線上にある
T a1 = S1.A.x, b1 = S1.B.x;
T a2 = S2.A.x, b2 = S2.B.x;
if (a1 == b1) {
a1 = S1.A.y, b1 = S1.B.y;
a2 = S2.A.y, b2 = S2.B.y;
}
if (a1 > b1) swap(a1, b1);
if (a2 > b2) swap(a2, b2);
T a = max(a1, a2);
T b = min(b1, b2);
if (a < b) return 2;
if (a > b) return 0;
return (include_ends ? 1 : 0);
}
// 平行でない場合
T a1 = L2.eval(S1.A), b1 = L2.eval(S1.B);
T a2 = L1.eval(S2.A), b2 = L1.eval(S2.B);
if (a1 > b1) swap(a1, b1);
if (a2 > b2) swap(a2, b2);
bool ok1 = 0, ok2 = 0;
if (include_ends) {
ok1 = (a1 <= T(0)) && (T(0) <= b1);
ok2 = (a2 <= T(0)) && (T(0) <= b2);
} else {
ok1 = (a1 < T(0)) && (T(0) < b1);
ok2 = (a2 < T(0)) && (T(0) < b2);
}
return (ok1 && ok2 ? 1 : 0);
}
// 4 次式まで登場している、オーバーフロー注意!
// https://codeforces.com/contest/607/problem/E
template <typename REAL, typename T>
vc<Point<REAL>> cross_point(const Circle<T> C, const Line<T> L) {
T a = L.a, b = L.b, c = L.a * (C.O.x) + L.b * (C.O.y) + L.c;
T r = C.r;
bool SW = 0;
T abs_a = (a < 0 ? -a : a);
T abs_b = (b < 0 ? -b : b);
if (abs_a < abs_b) {
swap(a, b);
SW = 1;
}
// ax+by+c=0, x^2+y^2=r^2
T D = 4 * c * c * b * b - 4 * (a * a + b * b) * (c * c - a * a * r * r);
if (D < 0) return {};
REAL sqD = sqrtl(D);
REAL y1 = (-2 * b * c + sqD) / (2 * (a * a + b * b));
REAL y2 = (-2 * b * c - sqD) / (2 * (a * a + b * b));
REAL x1 = (-b * y1 - c) / a;
REAL x2 = (-b * y2 - c) / a;
if (SW) swap(x1, y1), swap(x2, y2);
x1 += C.O.x, x2 += C.O.x;
y1 += C.O.y, y2 += C.O.y;
if (D == 0) return {Point<REAL>(x1, y1)};
return {Point<REAL>(x1, y1), Point<REAL>(x2, y2)};
}
// https://codeforces.com/contest/2/problem/C
template <typename REAL, typename T>
tuple<bool, Point<T>, Point<T>> cross_point_circle(Circle<T> C1, Circle<T> C2) {
using P = Point<T>;
P O{0, 0};
P A = C1.O, B = C2.O;
if (A == B) return {false, O, O};
T d = (B - A).norm();
REAL cos_val = (C1.r * C1.r + d * d - C2.r * C2.r) / (2 * C1.r * d);
if (cos_val < -1 || 1 < cos_val) return {false, O, O};
REAL t = acos(cos_val);
REAL u = (B - A).angle();
P X = A + P{C1.r * cos(u + t), C1.r * sin(u + t)};
P Y = A + P{C1.r * cos(u - t), C1.r * sin(u - t)};
return {true, X, Y};
}