POJ 3675 Telescope 简单多边形和圆的面积交

这道题得控制好精度，不然会贡献WA  QAQ

还是那个规则：

int sgn(double x){

    if(x > eps)    return 1;

    else if(x < - eps)  return -1;

    else    return 0;

}

思路：把简单多边形的每一个点和原点连线，就把这个多边形和圆的交变成了多个三角形与圆的交，根据有向面积的思路，加加减减就可以得到公共面积。

贴上代码了～

#include <cstdio>

#include <cstring>

#include <iostream>

#include <algorithm>

#include <cmath>

using namespace std;

#define sqr(x) ((x) * (x))



const int MAXN = 55;

const double EPS = 1e-8;

const double PI = acos(-1.0);//3.14159265358979323846

const double INF = 1;

const double eps = 1e-8;



int sgn(double x){

    if(x > eps)    return 1;

    else if(x < - eps)  return -1;

    else    return 0;

}

struct Point {

    double x, y, ag;

    Point() {}

    Point(double x, double y): x(x), y(y) {}

    void read() {

        scanf("%lf%lf", &x, &y);

    }

    bool operator == (const Point &rhs) const {

        return sgn(x - rhs.x) == 0 && sgn(y - rhs.y) == 0;

    }

    bool operator < (const Point &rhs) const {

        if(y != rhs.y) return y < rhs.y;

        return x < rhs.x;

    }

    Point operator + (const Point &rhs) const {

        return Point(x + rhs.x, y + rhs.y);

    }

    Point operator - (const Point &rhs) const {

        return Point(x - rhs.x, y - rhs.y);

    }

    Point operator * (const double &b) const {

        return Point(x * b, y * b);

    }

    Point operator / (const double &b) const {

        return Point(x / b, y / b);

    }

    double operator * (const Point &rhs) const {

        return x * rhs.x + y * rhs.y;

    }

    double length() {

        return sqrt(x * x + y * y);

    }

    double angle() {

        return atan2(y, x);

    }

    Point unit() {

        return *this / length();

    }

    void makeAg() {

        ag = atan2(y, x);

    }

    void print() {

        printf("%.10f %.10f\n", x, y);

    }

};

typedef Point Vector;



double dist(const Point &a, const Point &b) {

    return (a - b).length();

}



double cross(const Point &a, const Point &b) {

    return a.x * b.y - a.y * b.x;

}

//ret >= 0 means turn right

double cross(const Point &sp, const Point &ed, const Point &op) {

    return cross(sp - op, ed - op);

}



double area(const Point& a, const Point &b, const Point &c) {

    return fabs(cross(a - c, b - c)) / 2;

}

//counter-clockwise

Point rotate(const Point &p, double angle, const Point &o = Point(0, 0)) {

    Point t = p - o;

    double x = t.x * cos(angle) - t.y * sin(angle);

    double y = t.y * cos(angle) + t.x * sin(angle);

    return Point(x, y) + o;

}



double cosIncludeAngle(const Point &a, const Point &b, const Point &o) {

    Point p1 = a - o, p2 = b - o;

    return (p1 * p2) / (p1.length() * p2.length());

}



double includedAngle(const Point &a, const Point &b, const Point &o) {

    return acos(cosIncludeAngle(a, b, o));

    /*

    double ret = abs((a - o).angle() - (b - o).angle());

    if(sgn(ret - PI) > 0) ret = 2 * PI - ret;

    return ret;

    */

}



struct Seg {

    Point st, ed;

    double ag;

    Seg() {}

    Seg(Point st, Point ed): st(st), ed(ed) {}

    void read() {

        st.read(); ed.read();

    }

    void makeAg() {

        ag = atan2(ed.y - st.y, ed.x - st.x);

    }

};

typedef Seg Line;



//ax + by + c > 0

Line buildLine(double a, double b, double c) {

    if(sgn(a) == 0 && sgn(b) == 0) return Line(Point(sgn(c) > 0 ? -1 : 1, INF), Point(0, INF));

    if(sgn(a) == 0) return Line(Point(sgn(b), -c/b), Point(0, -c/b));

    if(sgn(b) == 0) return Line(Point(-c/a, 0), Point(-c/a, sgn(a)));

    if(b < 0) return Line(Point(0, -c/b), Point(1, -(a + c) / b));

    else return Line(Point(1, -(a + c) / b), Point(0, -c/b));

}



void moveRight(Line &v, double r) {

    double dx = v.ed.x - v.st.x, dy = v.ed.y - v.st.y;

    dx = dx / dist(v.st, v.ed) * r;

    dy = dy / dist(v.st, v.ed) * r;

    v.st.x += dy; v.ed.x += dy;

    v.st.y -= dx; v.ed.y -= dx;

}



bool isOnSeg(const Seg &s, const Point &p) {

    return (p == s.st || p == s.ed) ||

        (((p.x - s.st.x) * (p.x - s.ed.x) < 0 ||

          (p.y - s.st.y) * (p.y - s.ed.y) < 0) &&

         sgn(cross(s.ed, p, s.st)) == 0);

}



bool isInSegRec(const Seg &s, const Point &p) {

    return sgn(min(s.st.x, s.ed.x) - p.x) <= 0 && sgn(p.x - max(s.st.x, s.ed.x)) <= 0

        && sgn(min(s.st.y, s.ed.y) - p.y) <= 0 && sgn(p.y - max(s.st.y, s.ed.y)) <= 0;

}



bool isIntersected(const Point &s1, const Point &e1, const Point &s2, const Point &e2) {

    return (max(s1.x, e1.x) >= min(s2.x, e2.x)) &&

        (max(s2.x, e2.x) >= min(s1.x, e1.x)) &&

        (max(s1.y, e1.y) >= min(s2.y, e2.y)) &&

        (max(s2.y, e2.y) >= min(s1.y, e1.y)) &&

        (cross(s2, e1, s1) * cross(e1, e2, s1) >= 0) &&

        (cross(s1, e2, s2) * cross(e2, e1, s2) >= 0);

}



bool isIntersected(const Seg &a, const Seg &b) {

    return isIntersected(a.st, a.ed, b.st, b.ed);

}



bool isParallel(const Seg &a, const Seg &b) {

    return sgn(cross(a.ed - a.st, b.ed - b.st)) == 0;

}



//return Ax + By + C =0 's A, B, C

void Coefficient(const Line &L, double &A, double &B, double &C) {

    A = L.ed.y - L.st.y;

    B = L.st.x - L.ed.x;

    C = L.ed.x * L.st.y - L.st.x * L.ed.y;

}

//point of intersection

Point operator * (const Line &a, const Line &b) {

    double A1, B1, C1;

    double A2, B2, C2;

    Coefficient(a, A1, B1, C1);

    Coefficient(b, A2, B2, C2);

    Point I;

    I.x = - (B2 * C1 - B1 * C2) / (A1 * B2 - A2 * B1);

    I.y =   (A2 * C1 - A1 * C2) / (A1 * B2 - A2 * B1);

    return I;

}



bool isEqual(const Line &a, const Line &b) {

    double A1, B1, C1;

    double A2, B2, C2;

    Coefficient(a, A1, B1, C1);

    Coefficient(b, A2, B2, C2);

    return sgn(A1 * B2 - A2 * B1) == 0 && sgn(A1 * C2 - A2 * C1) == 0 && sgn(B1 * C2 - B2 * C1) == 0;

}



double Point_to_Line(const Point &p, const Line &L) {

    return fabs(cross(p, L.st, L.ed)/dist(L.st, L.ed));

}



double Point_to_Seg(const Point &p, const Seg &L) {

    if(sgn((L.ed - L.st) * (p - L.st)) < 0) return dist(p, L.st);

    if(sgn((L.st - L.ed) * (p - L.ed)) < 0) return dist(p, L.ed);

    return Point_to_Line(p, L);

}



double Seg_to_Seg(const Seg &a, const Seg &b) {

    double ans1 = min(Point_to_Seg(a.st, b), Point_to_Seg(a.ed, b));

    double ans2 = min(Point_to_Seg(b.st, a), Point_to_Seg(b.ed, a));

    return min(ans1, ans2);

}



struct Circle {

    Point c;

    double r;

    Circle() {}

    Circle(Point c, double r): c(c), r(r) {}

    void read() {

        c.read();

        scanf("%lf", &r);

    }

    double area() const {

        return PI * r * r;

    }

    bool contain(const Circle &rhs) const {

        return sgn(dist(c, rhs.c) + rhs.r - r) <= 0;

    }

    bool contain(const Point &p) const {

        return sgn(dist(c, p) - r) <= 0;

    }

    bool intersect(const Circle &rhs) const {

        return sgn(dist(c, rhs.c) - r - rhs.r) < 0;

    }

    bool tangency(const Circle &rhs) const {

        return sgn(dist(c, rhs.c) - r - rhs.r) == 0;

    }

    Point pos(double angle) const {

        Point p = Point(c.x + r, c.y);

        return rotate(p, angle, c);

    }

};



double CommonArea(const Circle &A, const Circle &B) {

    double area = 0.0;

    const Circle & M = (A.r > B.r) ? A : B;

    const Circle & N = (A.r > B.r) ? B : A;

    double D = dist(M.c, N.c);

    if((D < M.r + N.r) && (D > M.r - N.r)) {

        double cosM = (M.r * M.r + D * D - N.r * N.r) / (2.0 * M.r * D);

        double cosN = (N.r * N.r + D * D - M.r * M.r) / (2.0 * N.r * D);

        double alpha = 2 * acos(cosM);

        double beta = 2 * acos(cosN);

        double TM = 0.5 * M.r * M.r * (alpha - sin(alpha));

        double TN = 0.5 * N.r * N.r * (beta - sin(beta));

        area = TM + TN;

    }

    else if(D <= M.r - N.r) {

        area = N.area();

    }

    return area;

}



int intersection(const Seg &s, const Circle &cir, Point &p1, Point &p2) {

    double angle = cosIncludeAngle(s.ed, cir.c, s.st);

    //double angle1 = cos(includedAngle(s.ed, cir.c, s.st));

    double B = dist(cir.c, s.st);

    double a = 1, b = -2 * B * angle, c = sqr(B) - sqr(cir.r);

    double delta = sqr(b) - 4 * a * c;

    if(sgn(delta) < 0) return 0;

    if(sgn(delta) == 0) delta = 0;

    double x1 = (-b - sqrt(delta)) / (2 * a), x2 = (-b + sqrt(delta)) / (2 * a);

    Vector v = (s.ed - s.st).unit();

    p1 = s.st + v * x1;

    p2 = s.st + v * x2;

    return 1 + sgn(delta);

}



double CommonArea(const Circle &cir, Point p1, Point p2) {

    if(p1 == cir.c || p2 == cir.c) return 0;

    if(cir.contain(p1) && cir.contain(p2)) {

        return area(cir.c, p1, p2);

    } else if(!cir.contain(p1) && !cir.contain(p2)) {

        Point q1, q2;

        int t = intersection(Line(p1, p2), cir, q1, q2);

        if(t == 0) {

            double angle = includedAngle(p1, p2, cir.c);

            return 0.5 * sqr(cir.r) * angle;

        } else {

            double angle1 = includedAngle(p1, p2, cir.c);

            double angle2 = includedAngle(q1, q2, cir.c);

            if(isInSegRec(Seg(p1, p2), q1))return 0.5 * sqr(cir.r) * (angle1 - angle2 + sin(angle2));

            else return 0.5 * sqr(cir.r) * angle1;

        }

    } else {

        if(cir.contain(p2)) swap(p1, p2);

        Point q1, q2;

        intersection(Line(p1, p2), cir, q1, q2);

        double angle = includedAngle(q2, p2, cir.c);

        double a = area(cir.c, p1, q2);

        double b = 0.5 * sqr(cir.r) * angle;

        return a + b;

    }

}



struct Triangle {

    Point p[3];

    Triangle() {}

    Triangle(Point *t) {

        for(int i = 0; i < 3; ++i) p[i] = t[i];

    }

    void read() {

        for(int i = 0; i < 3; ++i) p[i].read();

    }

    double area() const {

        return ::area(p[0], p[1], p[2]);

    }

    Point& operator[] (int i) {

        return p[i];

    }

};



double CommonArea(Triangle tir, const Circle &cir) {

    double ret = 0;

    ret += sgn(cross(tir[0], cir.c, tir[1])) * CommonArea(cir, tir[0], tir[1]);

    ret += sgn(cross(tir[1], cir.c, tir[2])) * CommonArea(cir, tir[1], tir[2]);

    ret += sgn(cross(tir[2], cir.c, tir[0])) * CommonArea(cir, tir[2], tir[0]);

    return abs(ret);

}



struct Poly {

    int n;

    Point p[MAXN];//p[n] = p[0]

    void init(Point *pp, int nn) {

        n = nn;

        for(int i = 0; i < n; ++i) p[i] = pp[i];

        p[n] = p[0];

    }

    double area() {

        if(n < 3) return 0;

        double s = p[0].y * (p[n - 1].x - p[1].x);

        for(int i = 1; i < n; ++i)

            s += p[i].y * (p[i - 1].x - p[i + 1].x);

        return s / 2;

    }

};

//the convex hull is clockwise

void Graham_scan(Point *p, int n, int *stk, int &top) {//stk[0] = stk[top]

    sort(p, p + n);

    top = 1;

    stk[0] = 0; stk[1] = 1;

    for(int i = 2; i < n; ++i) {

        while(top && cross(p[i], p[stk[top]], p[stk[top - 1]]) <= 0) --top;

        stk[++top] = i;

    }

    int len = top;

    stk[++top] = n - 2;

    for(int i = n - 3; i >= 0; --i) {

        while(top != len && cross(p[i], p[stk[top]], p[stk[top - 1]]) <= 0) --top;

        stk[++top] = i;

    }

}

//use for half_planes_cross

bool cmpAg(const Line &a, const Line &b) {

    if(sgn(a.ag - b.ag) == 0)

        return sgn(cross(b.ed, a.st, b.st)) < 0;

    return a.ag < b.ag;

}

//clockwise, plane is on the right

bool half_planes_cross(Line *v, int vn, Poly &res, Line *deq) {

    int i, n;

    sort(v, v + vn, cmpAg);

    for(i = n = 1; i < vn; ++i) {

        if(sgn(v[i].ag - v[i-1].ag) == 0) continue;

        v[n++] = v[i];

    }

    int head = 0, tail = 1;

    deq[0] = v[0], deq[1] = v[1];

    for(i = 2; i < n; ++i) {

        if(isParallel(deq[tail - 1], deq[tail]) || isParallel(deq[head], deq[head + 1]))

            return false;

        while(head < tail && sgn(cross(v[i].ed, deq[tail - 1] * deq[tail], v[i].st)) > 0)

            --tail;

        while(head < tail && sgn(cross(v[i].ed, deq[head] * deq[head + 1], v[i].st)) > 0)

            ++head;

        deq[++tail] = v[i];

    }

    while(head < tail && sgn(cross(deq[head].ed, deq[tail - 1] * deq[tail], deq[head].st)) > 0)

        --tail;

    while(head < tail && sgn(cross(deq[tail].ed, deq[head] * deq[head + 1], deq[tail].st)) > 0)

        ++head;

    if(tail <= head + 1) return false;

    res.n = 0;

    for(i = head; i < tail; ++i)

        res.p[res.n++] = deq[i] * deq[i + 1];

    res.p[res.n++] = deq[head] * deq[tail];

    res.n = unique(res.p, res.p + res.n) - res.p;

    res.p[res.n] = res.p[0];

    return true;

}



//ix and jx is the points whose distance is return, res.p[n - 1] = res.p[0], res must be clockwise

double dia_rotating_calipers(Poly &res, int &ix, int &jx) {

    double dia = 0;

    int q = 1;

    for(int i = 0; i < res.n - 1; ++i) {

        while(sgn(cross(res.p[i], res.p[q + 1], res.p[i + 1]) - cross(res.p[i], res.p[q], res.p[i + 1])) > 0)

            q = (q + 1) % (res.n - 1);

        if(sgn(dist(res.p[i], res.p[q]) - dia) > 0) {

            dia = dist(res.p[i], res.p[q]);

            ix = i; jx = q;

        }

        if(sgn(dist(res.p[i + 1], res.p[q]) - dia) > 0) {

            dia = dist(res.p[i + 1], res.p[q]);

            ix = i + 1; jx = q;

        }

    }

    return dia;

}

//a and b must be clockwise, find the minimum distance between two convex hull

double half_rotating_calipers(Poly &a, Poly &b) {

    int sa = 0, sb = 0;

    for(int i = 0; i < a.n; ++i) if(sgn(a.p[i].y - a.p[sa].y) < 0) sa = i;

    for(int i = 0; i < b.n; ++i) if(sgn(b.p[i].y - b.p[sb].y) < 0) sb = i;

    double tmp, ans = dist(a.p[0], b.p[0]);

    for(int i = 0; i < a.n; ++i) {

        while(sgn(tmp = cross(a.p[sa], a.p[sa + 1], b.p[sb + 1]) - cross(a.p[sa], a.p[sa + 1], b.p[sb])) > 0)

            sb = (sb + 1) % (b.n - 1);

        if(sgn(tmp) < 0) ans = min(ans, Point_to_Seg(b.p[sb], Seg(a.p[sa], a.p[sa + 1])));

        else ans = min(ans, Seg_to_Seg(Seg(a.p[sa], a.p[sa + 1]), Seg(b.p[sb], b.p[sb + 1])));

        sa = (sa + 1) % (a.n - 1);

    }

    return ans;

}



double rotating_calipers(Poly &a, Poly &b) {

    return min(half_rotating_calipers(a, b), half_rotating_calipers(b, a));

}



/*******************************************************************************************/



Point p[MAXN];

Circle cir;

double r;

int n;



int main() {

    while(scanf("%lf", &r) != EOF) {

        scanf("%d", &n);

        for(int i = 0; i < n; ++i) p[i].read();

        p[n] = p[0];

        cir = Circle(Point(0, 0), r);

        double ans = 0;

        for(int i = 0; i < n; ++i)

            ans += sgn(cross(p[i], Point(0, 0), p[i + 1])) * CommonArea(cir, p[i], p[i + 1]);

        printf("%.2f\n", fabs(ans));

    }

}