// verification-helper: PROBLEM https://onlinejudge.u-aizu.ac.jp/courses/library/4/CGL/7/CGL_7_G
// verification-helper: ERROR 1e-5
#include "src/geometry/geometry.hpp"
#include <algorithm>
#include <iomanip>
#include <iostream>
#include <vector>
using namespace std;
int main() {
cout << fixed << setprecision(10);
Circle c1{}, c2{};
cin >> c1.c >> c1.r >> c2.c >> c2.r;
auto a = tangent_cc(c1, c2);
vector<Point> ps;
ps.reserve(a.size());
for (auto e : a) ps.push_back(get_cross_point_cl(c1, e)[0]);
sort(ps.begin(), ps.end());
for (auto e : ps) cout << e.x << " " << e.y << endl;
return 0;
}
#line 1 "test/geometry/geometry/aoj_cgl_7_g.test.cpp"
// verification-helper: PROBLEM https://onlinejudge.u-aizu.ac.jp/courses/library/4/CGL/7/CGL_7_G
// verification-helper: ERROR 1e-5
#line 1 "src/geometry/geometry.hpp"
#include <algorithm>
#include <cmath>
#include <iostream>
#include <iterator>
#include <limits>
#include <set>
#include <vector>
double constexpr EPS = 1e-10;
double constexpr PI = 3.14159265358979323846;
inline bool equals(double a, double b) { return std::abs(a - b) < EPS; }
static int const COUNTER_CLOCKWISE = 1;
static int const CLOCKWISE = -1;
static int const ONLINE_BACK = 2;
static int const ONLINE_FRONT = -2;
static int const ON_SEGMENT = 0;
struct Point {
double x, y;
Point() = default;
Point(double x, double y) : x(x), y(y) {}
Point operator+(Point const &p) const { return {x + p.x, y + p.y}; }
Point operator-(Point const &p) const { return {x - p.x, y - p.y}; }
Point operator*(double const &k) const { return {x * k, y * k}; }
Point operator/(double const &k) const { return {x / k, y / k}; }
friend std::istream &operator>>(std::istream &is, Point &p) {
is >> p.x >> p.y;
return is;
}
bool operator==(Point const &p) const {
return (std::abs(x - p.x) < EPS && std::abs(y - p.y) < EPS);
}
bool operator<(Point const &p) const { return (x != p.x ? x < p.x : y < p.y); }
[[nodiscard]] double norm() const { return x * x + y * y; }
[[nodiscard]] double abs() const { return std::sqrt(norm()); }
};
using Vector = Point;
inline double dot(Vector a, Vector b) { return a.x * b.x + a.y * b.y; }
inline double cross(Vector a, Vector b) { return a.x * b.y - a.y * b.x; }
inline bool is_parallel(Vector a, Vector b) { return equals(cross(a, b), 0.0); }
inline bool is_orthogonal(Vector a, Vector b) { return equals(dot(a, b), 0.0); }
struct EndPoint {
Point p;
int seg, st;
EndPoint() = default;
EndPoint(Point p, int seg, int st) : p(p), seg(seg), st(st) {}
bool operator<(EndPoint const &ep) const {
if (p.y == ep.p.y) return st < ep.st;
return p.y < ep.p.y;
}
};
struct Segment {
Point p1, p2;
Segment() = default;
Segment(Point p1, Point p2) : p1(p1), p2(p2) {}
friend std::istream &operator>>(std::istream &is, Segment &s) {
is >> s.p1 >> s.p2;
return is;
}
};
using Line = Segment;
inline Point project(Segment s, Point p) {
Vector base = s.p2 - s.p1;
double r = dot(p - s.p1, base) / base.norm();
return s.p1 + base * r;
}
inline Point reflect(Segment s, Point p) { return p + (project(s, p) - p) * 2.0; }
struct Circle {
Point c;
double r;
Circle() = default;
Circle(Point c, double r) : c(c), r(r) {}
};
using Polygon = std::vector<Point>;
inline int ccw(Point p0, Point p1, Point p2) {
Vector a = p1 - p0, b = p2 - p0;
if (cross(a, b) > EPS) return COUNTER_CLOCKWISE;
if (cross(a, b) < -EPS) return CLOCKWISE;
if (dot(a, b) < -EPS) return ONLINE_BACK;
if (a.norm() < b.norm()) return ONLINE_FRONT;
return ON_SEGMENT;
}
inline bool intersect_ss(Point p1, Point p2, Point p3, Point p4) {
return (ccw(p1, p2, p3) * ccw(p1, p2, p4) <= 0 &&
ccw(p3, p4, p1) * ccw(p3, p4, p2) <= 0);
}
inline bool intersect_ss(Segment s1, Segment s2) {
return intersect_ss(s1.p1, s1.p2, s2.p1, s2.p2);
}
inline int intersect_cs(Circle c, Segment s) {
if ((project(s, c.c) - c.c).norm() - c.r * c.r > EPS) return 0;
double d1 = (c.c - s.p1).abs(), d2 = (c.c - s.p2).abs();
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;
Point h = project(s, c.c);
if (dot(s.p1 - h, s.p2 - h) < 0) return 2;
return 0;
}
inline int intersect_cc(Circle c1, Circle c2) {
if (c1.r < c2.r) std::swap(c1, c2);
double d = (c1.c - c2.c).abs();
double r = c1.r + c2.r;
if (equals(d, r)) return 3;
if (d > r) return 4;
if (equals(d + c2.r, c1.r)) return 1;
if (d + c2.r < c1.r) return 0;
return 2;
}
inline double get_distance_lp(Line l, Point p) {
return std::abs(cross(l.p2 - l.p1, p - l.p1) / (l.p2 - l.p1).abs());
}
inline double get_distance_sp(Segment s, Point p) {
if (dot(s.p2 - s.p1, p - s.p1) < 0.0) return (p - s.p1).abs();
if (dot(s.p1 - s.p2, p - s.p2) < 0.0) return (p - s.p2).abs();
return get_distance_lp(s, p);
}
inline double get_distance_ss(Segment s1, Segment s2) {
if (intersect_ss(s1, s2)) return 0.0;
return std::min({get_distance_sp(s1, s2.p1),
get_distance_sp(s1, s2.p2),
get_distance_sp(s2, s1.p1),
get_distance_sp(s2, s1.p2)});
}
inline Point get_cross_point_ll(Line l1, Line l2) {
double a = cross(l1.p2 - l1.p1, l2.p2 - l2.p1);
double b = cross(l1.p2 - l1.p1, l1.p2 - l2.p1);
if (std::abs(a) < EPS && std::abs(b) < EPS) return l2.p1;
return l2.p1 + (l2.p2 - l2.p1) * (b / a);
}
inline Point get_cross_point_ss(Segment s1, Segment s2) {
Vector base = s2.p2 - s2.p1;
double d1 = std::abs(cross(base, s1.p1 - s2.p1));
double d2 = std::abs(cross(base, s1.p2 - s2.p1));
return s1.p1 + (s1.p2 - s1.p1) * (d1 / (d1 + d2));
}
inline std::vector<Point> get_cross_point_cl(Circle c, Line l) {
std::vector<Point> ps;
Vector pr = project(l, c.c);
Vector e = (l.p2 - l.p1) / (l.p2 - l.p1).abs();
if (equals(get_distance_lp(l, c.c), c.r)) return std::vector<Point>{pr, pr};
double base = std::sqrt(c.r * c.r - (pr - c.c).norm());
ps.push_back(pr + e * base);
ps.push_back(pr - e * base);
return ps;
}
inline std::vector<Point> get_cross_point_cs(Circle c, Segment s) {
Line l(s);
std::vector<Point> ps = get_cross_point_cl(c, l);
if (intersect_cs(c, s) == 2) return ps;
if (dot(l.p1 - ps[0], l.p2 - ps[0]) < 0) ps[1] = ps[0];
else ps[0] = ps[1];
return ps;
}
inline double arg(Vector p) { return atan2(p.y, p.x); }
inline Point polar(double r, double a) { return {cos(a) * r, sin(a) * r}; }
inline std::vector<Point> get_cross_point_cc(Circle c1, Circle c2) {
double d = (c1.c - c2.c).abs();
double a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));
double t = arg(c2.c - c1.c);
std::vector<Point> ps;
ps.push_back(c1.c + polar(c1.r, t + a));
ps.push_back(c1.c + polar(c1.r, t - a));
return ps;
}
inline std::vector<Point> tangent_cp(Circle c, Point p) {
return get_cross_point_cc(c, Circle(p, std::sqrt((c.c - p).norm() - c.r * c.r)));
}
inline std::vector<Line> tangent_cc(Circle c1, Circle c2) {
std::vector<Line> ls;
if (c1.r < c2.r) std::swap(c1, c2);
double g = (c1.c - c2.c).abs();
if (equals(g, 0)) return ls;
Point u = (c2.c - c1.c) / g;
Point v = Point(-u.y, u.x);
for (int s = 1; s >= -1; s -= 2) {
double h = (c1.r + s * c2.r) / g;
if (equals(1, h * h)) ls.emplace_back(c1.c + u * c1.r, c1.c + (u + v) * c1.r);
else if (1 - h * h > 0) {
Point uu = u * h, vv = v * std::sqrt(1 - h * h);
ls.emplace_back(c1.c + (uu + vv) * c1.r, c2.c - (uu + vv) * c2.r * s);
ls.emplace_back(c1.c + (uu - vv) * c1.r, c2.c - (uu - vv) * c2.r * s);
}
}
return ls;
}
inline Circle get_inscribed_circle(Point p1, Point p2, Point p3) {
Circle ca{};
double a = (p2 - p3).abs(), b = (p3 - p1).abs(), c = (p1 - p2).abs();
ca.c = (p1 * a + p2 * b + p3 * c) / (a + b + c);
ca.r = get_distance_lp(Line(p1, p2), ca.c);
return ca;
}
inline Circle get_circumscribed_circle(Point p1, Point p2, Point p3) {
Circle ca{};
Point m = (p1 + p2) / 2, n = (p2 + p3) / 2;
ca.c = get_cross_point_ll(Line(m, m + Point((p2 - p1).y, (p1 - p2).x)),
Line(n, n + Point((p3 - p2).y, (p2 - p3).x)));
ca.r = (ca.c - p1).abs();
return ca;
}
// IN:2,ON:1,OUT:0
inline int contains(Polygon g, Point p) {
int const N = g.size();
bool x = false;
for (int i = 0; i < N; ++i) {
Point a = g[i] - p, b = g[(i + 1) % N] - p;
if (std::abs(cross(a, b)) < EPS && dot(a, b) < EPS) return 1;
if (a.y > b.y) std::swap(a, b);
if (a.y < EPS && EPS < b.y && cross(a, b) > EPS) x = !x;
}
return (x ? 2 : 0);
}
inline bool is_convex(Polygon p) {
int const N = p.size();
for (int i = 0; i < N; ++i)
if (ccw(p[(i - 1 + N) % N], p[i], p[(i + 1) % N]) == CLOCKWISE) return false;
return true;
}
inline Polygon convex_hull(Polygon p) {
int const N = p.size();
std::sort(p.begin(), p.end(), [](Point const &a, Point const &b) {
return (a.y != b.y ? a.y < b.y : a.x < b.x);
});
Polygon a(2UL * N);
int k = 0;
for (int i = 0; i < N; ++i) {
while (k > 1 && cross(a[k - 1] - a[k - 2], p[i] - a[k - 1]) < 0) k--;
a[k++] = p[i];
}
for (int i = N - 2, t = k; i >= 0; --i) {
while (k > t && cross(a[k - 1] - a[k - 2], p[i] - a[k - 1]) < 0) k--;
a[k++] = p[i];
}
a.resize(k - 1);
return a;
}
inline double area(Polygon p) {
double res = 0;
for (int i = 0; i < (int)p.size(); ++i)
res += cross(p[i], p[(i + 1) % p.size()]) / 2.0;
return res;
}
inline double area(Circle c1, Circle c2) {
if (c1.r < c2.r) std::swap(c1, c2);
int num = intersect_cc(c1, c2);
if (num >= 3) return 0;
if (num <= 1) return c2.r * c2.r * PI;
double d = (c1.c - c2.c).abs();
double res = 0;
for (int i = 0; i < 2; ++i) {
double th = 2 * acos((d * d + c1.r * c1.r - c2.r * c2.r) / (2 * d * c1.r));
res += (th - sin(th)) * c1.r * c1.r / 2;
std::swap(c1, c2);
}
return res;
}
inline double area(Polygon p, Circle c) {
if (p.size() < 3) return 0;
auto dfs = [&](auto self, Circle c, Point a, Point b) {
Vector va = c.c - a, vb = c.c - b;
double f = cross(va, vb), res = 0;
if (equals(f, 0.0)) return res;
if (std::max(va.abs(), vb.abs()) < c.r + EPS) return f;
Vector d(dot(va, vb), cross(va, vb));
if (get_distance_sp(Segment(a, b), c.c) > c.r - EPS) return c.r * c.r * arg(d);
auto u = get_cross_point_cs(c, Segment(a, b));
if (u.empty()) return res;
if (u.size() > 1 && dot(u[1] - u[0], a - u[0]) > 0) std::swap(u[0], u[1]);
u.emplace(u.begin(), a);
u.emplace_back(b);
for (int i = 1; i < (int)u.size(); ++i) res += self(self, c, u[i - 1], u[i]);
return res;
};
double res = 0;
for (int i = 0; i < (int)p.size(); ++i)
res += dfs(dfs, c, p[i], p[(i + 1) % p.size()]);
return res / 2;
}
inline double convex_diameter(Polygon p) {
int const N = p.size();
if (N == 2) return (p[0] - p[1]).abs();
int i = 0, j = 0;
for (int k = 0; k < N; ++k) {
if (p[i] < p[k]) i = k;
if (!(p[j] < p[k])) j = k;
}
double res = 0;
int ti = i, tj = j;
while (i != tj || j != ti) {
res = std::max(res, (p[i] - p[j]).abs());
if (cross(p[(i + 1) % N] - p[i], p[(j + 1) % N] - p[j]) < 0.0) i = (i + 1) % N;
else j = (j + 1) % N;
}
return res;
}
inline Polygon convex_cut(Polygon p, Line l) {
Polygon q;
for (int i = 0; i < (int)p.size(); ++i) {
Point a = p[i], b = p[(i + 1) % p.size()];
if (ccw(l.p1, l.p2, a) != CLOCKWISE) q.push_back(a);
if (ccw(l.p1, l.p2, a) * ccw(l.p1, l.p2, b) < 0)
q.push_back(get_cross_point_ll(Line(a, b), l));
}
return q;
}
inline double closest_pair(std::vector<Point> ps) {
std::sort(ps.begin(), ps.end());
std::vector<Point> a(ps.size());
auto solve = [&](auto self, int l, int r) {
if (r - l < 2) return 1e18;
int mid = (l + r) >> 1;
double x = ps[mid].x;
double d = std::min(self(self, l, mid), self(self, mid, r));
inplace_merge(ps.begin() + l,
ps.begin() + mid,
ps.begin() + r,
[](Point const &a, Point const &b) { return a.y < b.y; });
int ptr = 0;
for (int i = l; i < r; ++i) {
if (std::abs(ps[i].x - x) >= d) continue;
for (int j = 0; j < ptr; ++j) {
Point luz = ps[i] - a[ptr - j - 1];
if (luz.y >= d) break;
d = std::min(d, luz.abs());
}
a[ptr++] = ps[i];
}
return d;
};
return solve(solve, 0, ps.size());
}
inline int manhattan_intersection(std::vector<Segment> ss) {
int constexpr INF = std::numeric_limits<int>::max();
int constexpr BOTTOM = 0, LEFT = 1, RIGHT = 2, TOP = 3;
int const N = ss.size();
std::vector<EndPoint> ep;
for (int i = 0; i < N; ++i) {
if (ss[i].p1.y == ss[i].p2.y) {
if (ss[i].p1.x > ss[i].p2.x) std::swap(ss[i].p1, ss[i].p2);
ep.emplace_back(ss[i].p1, i, LEFT);
ep.emplace_back(ss[i].p2, i, RIGHT);
} else {
if (ss[i].p1.y > ss[i].p2.y) std::swap(ss[i].p1, ss[i].p2);
ep.emplace_back(ss[i].p1, i, BOTTOM);
ep.emplace_back(ss[i].p2, i, TOP);
}
}
std::sort(ep.begin(), ep.end());
std::set<int> st;
st.insert(INF);
int cnt = 0;
for (int i = 0; i < 2 * N; ++i) {
if (ep[i].st == TOP) st.erase(ep[i].p.x);
else if (ep[i].st == BOTTOM) st.insert(ep[i].p.x);
else if (ep[i].st == LEFT) {
auto b = st.lower_bound(ss[ep[i].seg].p1.x);
auto e = st.upper_bound(ss[ep[i].seg].p2.x);
cnt += std::distance(b, e);
}
}
return cnt;
}
#line 5 "test/geometry/geometry/aoj_cgl_7_g.test.cpp"
#line 7 "test/geometry/geometry/aoj_cgl_7_g.test.cpp"
#include <iomanip>
#line 10 "test/geometry/geometry/aoj_cgl_7_g.test.cpp"
using namespace std;
int main() {
cout << fixed << setprecision(10);
Circle c1{}, c2{};
cin >> c1.c >> c1.r >> c2.c >> c2.r;
auto a = tangent_cc(c1, c2);
vector<Point> ps;
ps.reserve(a.size());
for (auto e : a) ps.push_back(get_cross_point_cl(c1, e)[0]);
sort(ps.begin(), ps.end());
for (auto e : ps) cout << e.x << " " << e.y << endl;
return 0;
}
test/geometry/geometry/aoj_cgl_7_g.test.cpp