C.cpp

#include <bits/stdc++.h>

#define FOR(i,a,b) for(int i=(a),_b=(b); i<=_b; i++)
#define FORD(i,a,b) for(int i=(a),_b=(b); i>=_b; i--)
#define REP(i,a) for(int i=0,_a=(a); i<_a; i++)
#define EACH(it,a) for(__typeof(a.begin()) it = a.begin(); it != a.end(); ++it)

#define DEBUG(x) { cout << #x << " = "; cout << (x) << endl; }
#define PR(a,n) { cout << #a << " = "; FOR(_,1,n) cout << a[_] << ' '; cout << endl; }
#define PR0(a,n) { cout << #a << " = "; REP(_,n) cout << a[_] << ' '; cout << endl; }

#define sqr(x) ((x) * (x))
#define ll long long
using namespace std;

#define EPS 1e-9

inline int cmp(double a, double b) {
    return (a < b - EPS) ? -1 : ((a > b + EPS) ? 1 : 0);
}

struct Point {
    double x, y;
    Point(double x = 0.0, double y = 0.0) : x(x), y(y) {}

    Point operator + (Point a) { return Point(x+a.x, y+a.y); }
    Point operator - (Point a) { return Point(x-a.x, y-a.y); }
    Point operator * (double k) { return Point(x*k, y*k); }
    Point operator / (double k) { return Point(x/k, y/k); }

    double operator * (Point a) { return x*a.x + y*a.y; } // dot product
    double operator % (Point a) { return x*a.y - y*a.x; } // cross product

    int cmp(Point q) const { if (int t = ::cmp(x,q.x)) return t; return ::cmp(y,q.y); }

    #define Comp(x) bool operator x (Point q) const { return cmp(q) x 0; }
    Comp(>) Comp(<) Comp(==) Comp(>=) Comp(<=) Comp(!=)
    #undef Comp

    Point conj() { return Point(x, -y); }
    double norm() { return x*x + y*y; }

    Point normTo(double need) {
        return Point(x, y) / norm() * need;
    }

    // Note: There are 2 ways for implementing len():
    // 1. sqrt(norm()) --> fast, but inaccurate (produce some values that are of order X^2)
    // 2. hypot(x, y) --> slow, but much more accurate
    double len() { return sqrt(norm()); }

    Point rotate(double alpha) {
        double cosa = cos(alpha), sina = sin(alpha);
        return Point(x * cosa - y * sina, x * sina + y * cosa);
    }
};

struct Line {
    double a, b, c;
    Point A, B; // Added for polygon intersect line. Do not rely on assumption that these are valid

    Line(double a, double b, double c) : a(a), b(b), c(c) {} 

    Line(Point A, Point B) : A(A), B(B) {
        a = B.y - A.y;
        b = A.x - B.x;
        c = - (a * A.x + b * A.y);
    }
    Line(Point P, double m) {
        a = -m; b = 1;
        c = -((a * P.x) + (b * P.y));
    }
    double f(Point A) {
        return a*A.x + b*A.y + c;
    }
};

struct Circle : Point {
    double r;
    Circle(double x = 0, double y = 0, double r = 0) : Point(x, y), r(r) {}
    Circle(Point p, double r) : Point(p), r(r) {}
    
    bool contains(Point p) { return (*this - p).len() <= r + EPS; }
};

vector<Point> intersection(Line l, Circle cir) {
    double r = cir.r, a = l.a, b = l.b, c = l.c + l.a*cir.x + l.b*cir.y;
    vector<Point> res;

    double x0 = -a*c/(a*a+b*b),  y0 = -b*c/(a*a+b*b);
    if (c*c > r*r*(a*a+b*b)+EPS) return res;
    else if (fabs(c*c - r*r*(a*a+b*b)) < EPS) {
        res.push_back(Point(x0, y0) + Point(cir.x, cir.y));
        return res;
    }
    else {
        double d = r*r - c*c/(a*a+b*b);
        double mult = sqrt (d / (a*a+b*b));
        double ax,ay,bx,by;
        ax = x0 + b * mult;
        bx = x0 - b * mult;
        ay = y0 - a * mult;
        by = y0 + a * mult;

        res.push_back(Point(ax, ay) + Point(cir.x, cir.y));
        res.push_back(Point(bx, by) + Point(cir.x, cir.y));
        return res;
    }
}

bool areIntersect(Circle u, Circle v) {
    if (cmp((u - v).len(), u.r + v.r) > 0) return false;
    if (cmp((u - v).len() + v.r, u.r) < 0) return false;
    if (cmp((u - v).len() + u.r, v.r) < 0) return false;
    return true;
}

vector<Point> circleIntersect(Circle u, Circle v) {
    vector<Point> res;
    if (!areIntersect(u, v)) return res;

    double d = (u - v).len();
    double alpha = acos((u.r * u.r + d*d - v.r * v.r) / 2.0 / u.r / d);

    Point p1 = (v - u).rotate(alpha);
    Point p2 = (v - u).rotate(-alpha);
    p1 = p1 / p1.len() * u.r;
    p2 = p2 / p2.len() * u.r;

    res.push_back(p1 + u);
    res.push_back(p2 + u);

    return res;
}


// V - E + F = 2
int n;
Circle a[111];
bool visited[111];

void dfs(int u) {
    visited[u] = true;
    FOR(v,1,n) if (!visited[v] && areIntersect(a[u], a[v])) {
        dfs(v);
    }
}

int main() {
    ios :: sync_with_stdio(false);
    freopen("circles.in", "r", stdin);
    freopen("circles.out", "w", stdout);
    set<Point> all;
    cin >> n;
    FOR(i,1,n) cin >> a[i].x >> a[i].y >> a[i].r;

    FOR(i,1,n) FOR(j,i+1,n) {
        auto res = circleIntersect(a[i], a[j]);
        for(auto p : res) all.insert(p);
    }
    int V = all.size();
//    DEBUG(V);

    int E = 0;
    FOR(i,1,n) {
        for(auto p : all) {
            double dist = (a[i] - p).len();
            if (fabs(a[i].r - dist) < EPS) ++E;
        }
    }
//    DEBUG(E);

    memset(visited, false, sizeof visited);
    int F = 0;
    FOR(i,1,n) if (!visited[i]) {
        dfs(i);
        ++F;
    }
//    DEBUG(F);
    cout << 1 - V + E + F << endl;
}