LA 2572 Viva Confetti (Geometry.Circle)

https://icpcarchive.ecs.baylor.edu/index.php?option=com_onlinejudge&Itemid=8&page=problem_stats&problemid=573&category=

　　一道很好的几何题。要求是给出一些圆，按顺序覆盖上去，问哪些圆是可以被看见的。

　　看刘汝佳的书，开始的时候不明白“每个可见部分都是由一些‘小圆弧’围城的”这样又怎么样，直到我看了他的代码以后才懂。其实意思就是，因为每个可见部分都是由小圆弧围成，所以，如果我们将小圆弧中点（不会是圆与圆的交点）相对于弧的位置向里或向外稍微移动一下，然后找到有哪个圆最后覆盖在在那上面，那么这个圆必定能被看见。

　　刚接触几何，表示还真想不到可以这样做，所以就觉得这个做法实在太妙了！

代码如下：

#include <cstdio>
#include <cstring>
#include <cmath>
#include <set>
#include <vector>
#include <iostream>
#include <algorithm>

using namespace std;

struct Point {
    double x, y;
    Point() {}
    Point(double x, double y) : x(x), y(y) {}
} ;
template<class T> T sqr(T x) { return x * x;}
inline double ptDis(Point a, Point b) { return sqrt(sqr(a.x - b.x) + sqr(a.y - b.y));}

// basic calculations
typedef Point Vec;
Vec operator + (Vec a, Vec b) { return Vec(a.x + b.x, a.y + b.y);}
Vec operator - (Vec a, Vec b) { return Vec(a.x - b.x, a.y - b.y);}
Vec operator * (Vec a, double p) { return Vec(a.x * p, a.y * p);}
Vec operator / (Vec a, double p) { return Vec(a.x / p, a.y / p);}

const double EPS = 5e-13;
const double PI = acos(-1.0);
inline int sgn(double x) { return fabs(x) < EPS ? 0 : (x < 0 ? -1 : 1);}
bool operator < (Point a, Point b) { return a.x < b.x || (a.x == b.x && a.y < b.y);}
bool operator == (Point a, Point b) { return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0;}

inline double dotDet(Vec a, Vec b) { return a.x * b.x + a.y * b.y;}
inline double vecLen(Vec x) { return sqrt(sqr(x.x) + sqr(x.y));}
inline double angle(Vec v) { return atan2(v.y, v.x);}
inline double angle(Vec a, Vec b) { return acos(dotDet(a, b) / vecLen(a) / vecLen(b));}
inline double crossDet(Vec a, Vec b) { return a.x * b.y - a.y * b.x;}
inline double triArea(Point a, Point b, Point c) { return fabs(crossDet(b - a, c - a));}
inline Vec vecUnit(Vec x) { return x / vecLen(x);}
inline Vec rotate(Vec x, double rad) { return Vec(x.x * cos(rad) - x.y * sin(rad), x.x * sin(rad) + x.y * cos(rad));}
Vec normal(Vec x) {
    double len = vecLen(x);
    return Vec(- x.y / len, x.x / len);
}

struct Line {
    Point s, t;
    Line() {}
    Line(Point s, Point t) : s(s), t(t) {}
    Point point(double x) {
        return s + (t - s) * x;
    }
    Line move(double x) {
        Vec nor = normal(t - s);
        nor = nor / vecLen(nor) * x;
        return Line(s + nor, t + nor);
    }
    Vec vec() { return t - s;}
} ;
typedef Line Seg;

inline bool onSeg(Point x, Point a, Point b) { return sgn(crossDet(a - x, b - x)) == 0 && sgn(dotDet(a - x, b - x)) < 0;}
inline bool onSeg(Point x, Seg s) { return onSeg(x, s.s, s.t);}

// 0 : not intersect
// 1 : proper intersect
// 2 : improper intersect
int segIntersect(Point a, Point c, Point b, Point d) {
    Vec v1 = b - a, v2 = c - b, v3 = d - c, v4 = a - d;
    int a_bc = sgn(crossDet(v1, v2));
    int b_cd = sgn(crossDet(v2, v3));
    int c_da = sgn(crossDet(v3, v4));
    int d_ab = sgn(crossDet(v4, v1));
    if (a_bc * c_da > 0 && b_cd * d_ab > 0) return 1;
    if (onSeg(b, a, c) && c_da) return 2;
    if (onSeg(c, b, d) && d_ab) return 2;
    if (onSeg(d, c, a) && a_bc) return 2;
    if (onSeg(a, d, b) && b_cd) return 2;
    return 0;
}
inline int segIntersect(Seg a, Seg b) { return segIntersect(a.s, a.t, b.s, b.t);}

// point of the intersection of 2 lines
Point lineIntersect(Point P, Vec v, Point Q, Vec w) {
    Vec u = P - Q;
    double t = crossDet(w, u) / crossDet(v, w);
    return P + v * t;
}
inline Point lineIntersect(Line a, Line b) { return lineIntersect(a.s, a.t - a.s, b.s, b.t - b.s);}

// Warning: This is a DIRECTED Distance!!!
double pt2Line(Point x, Point a, Point b) {
    Vec v1 = b - a, v2 = x - a;
    return crossDet(v1, v2) / vecLen(v1);
}
inline double pt2Line(Point x, Line L) { return pt2Line(x, L.s, L.t);}

double pt2Seg(Point x, Point a, Point b) {
    if (a == b) return vecLen(x - a);
    Vec v1 = b - a, v2 = x - a, v3 = x - b;
    if (sgn(dotDet(v1, v2)) < 0) return vecLen(v2);
    if (sgn(dotDet(v1, v3)) > 0) return vecLen(v3);
    return fabs(crossDet(v1, v2)) / vecLen(v1);
}
inline double pt2Seg(Point x, Seg s) { return pt2Seg(x, s.s, s.t);}

struct Poly {
    vector<Point> pt;
    Poly() {}
    Poly(vector<Point> pt) : pt(pt) {}
    double area() {
        double ret = 0.0;
        int sz = pt.size();
        for (int i = 1; i < sz; i++) {
            ret += crossDet(pt[i], pt[i - 1]);
        }
        return fabs(ret / 2.0);
    }
} ;

struct Circle {
    Point c;
    double r;
    Circle() {}
    Circle(Point c, double r) : c(c), r(r) {}
    Point point(double a) {
        return Point(c.x + cos(a) * r, c.y + sin(a) * r);
    }
} ;

int lineCircleIntersect(Line L, Circle C, double &t1, double &t2, vector<Point> &sol) {
    double a = L.s.x, b = L.t.x - C.c.x, c = L.s.y, d = L.t.y - C.c.y;
    double e = sqr(a) + sqr(c), f = 2 * (a * b + c * d), g = sqr(b) + sqr(d) - sqr(C.r);
    double delta = sqr(f) - 4.0 * e * g;
    if (sgn(delta) < 0) return 0;
    if (sgn(delta) == 0) {
        t1 = t2 = -f / (2.0 * e);
        sol.push_back(L.point(t1));
        return 1;
    }
    t1 = (-f - sqrt(delta)) / (2.0 * e);
    sol.push_back(L.point(t1));
    t2 = (-f + sqrt(delta)) / (2.0 * e);
    sol.push_back(L.point(t2));
    return 2;
}

int lineCircleIntersect(Line L, Circle C, vector<Point> &sol) {
    Vec dir = L.t - L.s, nor = normal(dir);
    Point mid = lineIntersect(C.c, nor, L.s, dir);
    double len = sqr(C.r) - sqr(ptDis(C.c, mid));
    if (sgn(len) < 0) return 0;
    if (sgn(len) == 0) {
        sol.push_back(mid);
        return 1;
    }
    Vec dis = vecUnit(dir);
    len = sqrt(len);
    sol.push_back(mid + dis * len);
    sol.push_back(mid - dis * len);
    return 2;
}

// -1 : coincide
int circleCircleIntersect(Circle C1, Circle C2, vector<Point> &sol) {
    double d = vecLen(C1.c - C2.c);
    if (sgn(d) == 0) {
        if (sgn(C1.r - C2.r) == 0) {
            return -1;
        }
        return 0;
    }
    if (sgn(C1.r + C2.r - d) < 0) return 0;
    if (sgn(fabs(C1.r - C2.r) - d) > 0) return 0;
    double a = angle(C2.c - C1.c);
    double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2.0 * C1.r * d));
    Point p1 = C1.point(a - da), p2 = C1.point(a + da);
    sol.push_back(p1);
    if (p1 == p2) return 1;
    sol.push_back(p2);
    return 2;
}

void circleCircleIntersect(Circle C1, Circle C2, vector<double> &sol) {
    double d = vecLen(C1.c - C2.c);
    if (sgn(d) == 0) return ;
    if (sgn(C1.r + C2.r - d) < 0) return ;
    if (sgn(fabs(C1.r - C2.r) - d) > 0) return ;
    double a = angle(C2.c - C1.c);
    double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2.0 * C1.r * d));
    sol.push_back(a - da);
    sol.push_back(a + da);
}

int tangent(Point p, Circle C, vector<Vec> &sol) {
    Vec u = C.c - p;
    double dist = vecLen(u);
    if (dist < C.r) return 0;
    if (sgn(dist - C.r) == 0) {
        sol.push_back(rotate(u, PI / 2.0));
        return 1;
    }
    double ang = asin(C.r / dist);
    sol.push_back(rotate(u, -ang));
    sol.push_back(rotate(u, ang));
    return 2;
}

// ptA : points of tangency on circle A
// ptB : points of tangency on circle B
int tangent(Circle A, Circle B, vector<Point> &ptA, vector<Point> &ptB) {
    if (A.r < B.r) {
        swap(A, B);
        swap(ptA, ptB);
    }
    int d2 = sqr(A.c.x - B.c.x) + sqr(A.c.y - B.c.y);
    int rdiff = A.r - B.r, rsum = A.r + B.r;
    if (d2 < sqr(rdiff)) return 0;
    double base = atan2(B.c.y - A.c.y, B.c.x - A.c.x);
    if (d2 == 0 && A.r == B.r) return -1;
    if (d2 == sqr(rdiff)) {
        ptA.push_back(A.point(base));
        ptB.push_back(B.point(base));
        return 1;
    }
    double ang = acos((A.r - B.r) / sqrt(d2));
    ptA.push_back(A.point(base + ang));
    ptB.push_back(B.point(base + ang));
    ptA.push_back(A.point(base - ang));
    ptB.push_back(B.point(base - ang));
    if (d2 == sqr(rsum)) {
        ptA.push_back(A.point(base));
        ptB.push_back(B.point(PI + base));
    } else if (d2 > sqr(rsum)) {
        ang = acos((A.r + B.r) / sqrt(d2));
        ptA.push_back(A.point(base + ang));
        ptB.push_back(B.point(PI + base + ang));
        ptA.push_back(A.point(base - ang));
        ptB.push_back(B.point(PI + base - ang));
    }
    return (int) ptA.size();
}

/****************** template above *******************/

#define DEC(i, a, b) for (int i = (a); i >= (b); i--)
#define REP(i, n) for (int i = 0; i < (n); i++)
#define _clr(x) memset(x, 0, sizeof(x))
#define PB push_back
Circle c[111];

int topCircle(Point x, int n) {
    DEC(i, n - 1, 0) {
        if (ptDis(c[i].c, x) < c[i].r) return i;
    }
    return -1;
}

int main() {
//    freopen("in", "r", stdin);
    int n;
    while (cin >> n && n) {
        REP(i, n) cin >> c[i].c.x >> c[i].c.y >> c[i].r;
        set<int> vis;
        vis.clear();
        REP(i, n) {
            vector<double> pos;
            pos.clear();
            pos.PB(0.0);
            pos.PB(2.0 * PI);
            REP(j, n) {
                circleCircleIntersect(c[i], c[j], pos);
            }
            sort(pos.begin(), pos.end());
            int sz = pos.size() - 1;
            REP(j, sz) {
                double mid = (pos[j] + pos[j + 1]) / 2.0;
                for (int d = -1; d <= 1; d += 2) {
                    double nr = c[i].r + d * EPS;
                    int t = topCircle(Point(c[i].c.x + nr * cos(mid), c[i].c.y + nr * sin(mid)), n);
                    if (~t) vis.insert(t);
                }
            }
        }
        cout << vis.size() << endl;
    }
    return 0;
}

——written by Lyon