多边形与圆相交面积计算

POJ 3675 裸的题了。直接上模板就行了。注意的是，求出的是有向面积，有可能是负数。

#include <iostream>
#include <cstdio>
#include <cmath>
#include <vector>
#include <cstring>
#include <algorithm>
#include <string>
#include <set>
#include <functional>
#include <numeric>
#include <sstream>
#include <stack>

#define CL(arr, val) memset(arr, val, sizeof(arr))
#define REP(i, n)for((i) = 0; (i) < (n); ++(i))
#define FOR(i, l, h)for((i) = (l); (i) <= (h); ++(i))
#define FORD(i, h, l)for((i) = (h); (i) >= (l); --(i))
#define L(x)(x) << 1
#define R(x)(x) << 1 | 1
#define MID(l, r)(l + r) >> 1
#define Min(x, y)x < y ? x : y
#define Max(x, y)x < y ? y : x
#define E(x)(1 << (x))
#define iabs(x) (x) < 0 ? -(x) : (x)
#define OUT(x)printf("%I64d
", x)
#define lowbit(x)(x)&(-x)
#define Read()freopen("data.in", "r", stdin)
#define Write()freopen("data.out", "w", stdout);

const double eps = 1e-8;
typedef long long LL;
const int inf = ~0u>>2;

using namespace std;

inline double max (double a, double b) { if (a > b) return a; else return b; }
inline double min (double a, double b) { if (a < b) return a; else return b; }
inline int fi (double a){
    if (a > eps) return 1;
    else if (a >= -eps) return 0;
    else return -1;
}
class Vector{
public:
    double x, y;
    Vector (void) {}
    Vector (double x0, double y0) : x(x0), y(y0) {}
    double operator * (const Vector& a) const { return x * a.y - y * a.x; }
    double operator % (const Vector& a) const { return x * a.x + y * a.y; }
    Vector verti (void) const { return Vector(-y, x); }
    double length (void) const { return sqrt(x * x + y * y); }
    Vector adjust (double len) {
        double ol = len / length();
        return Vector(x * ol, y * ol);
    }
    Vector oppose (void) { return Vector(-x, -y); }
};
class point{
public:
    double x, y;
    point (void) {}
    point (double x0, double y0) : x(x0), y(y0) {}
    Vector operator - (const point& a) const { return Vector(x - a.x, y - a.y); }
    point operator + (const Vector& a) const { return point(x + a.x, y + a.y); }
};
class segment{
public:
    point a, b;
    segment (void) {}
    segment (point a0, point b0) : a(a0), b(b0) {}
    point intersect (const segment& s) const {
        Vector v1 = s.a - a, v2 = s.b - a, v3 = s.b - b, v4 = s.a - b;
        double s1 = v1 * v2, s2 = v3 * v4;
        double se = s1 + s2;
        s1 /= se, s2 /= se;
        return point(a.x * s2 + b.x * s1, a.y * s2 + b.y * s1);
    }
    point pverti (const point& p) const {
        Vector t = (b - a).verti();
        segment uv(p, p + t);
        return intersect(uv);
    }
    bool on_segment (const point& p) const {
        if (fi(min(a.x, b.x) - p.x) <= 0 && fi(p.x - max(a.x, b.x)) <= 0 &&
            fi(min(a.y, b.y) - p.y) <= 0 && fi(p.y - max(a.y, b.y)) <= 0) return true;
        else return false;
    }
};

double radius;
point polygon[110];

double kuras_area (point a, point b, double cx, double cy){
    point ori(cx, cy);
    int sgn = fi((b - ori) * (a - ori));
    double da = (a - ori).length(), db = (b - ori).length();
    int ra = fi(da - radius), rb = fi(db - radius);
    double angle = acos(((b - ori) % (a - ori)) / (da * db));
    segment t(a, b); point h, u; Vector seg;
    double ans, dlt, mov, tangle;

    if (fi(da) == 0 || fi(db) == 0) return 0;
    else if (sgn == 0) return 0;
    else if (ra <= 0 && rb <= 0) return fabs((b - ori) * (a - ori)) / 2 * sgn;
    else if (ra >= 0 && rb >= 0){
        h = t.pverti(ori);
        dlt = (h - ori).length();
        if (!t.on_segment(h) || fi(dlt - radius) >= 0)
            return radius * radius * (angle / 2) * sgn;
        else{
            ans = radius * radius * (angle / 2);
            tangle = acos(dlt / radius);
            ans -= radius * radius * tangle;
            ans += radius * sin(tangle) * dlt;
            return ans * sgn;
        }
    }
    else{
        h = t.pverti(ori);
        dlt = (h - ori).length();
        seg = b - a;
        mov = sqrt(radius * radius - dlt * dlt);
        seg = seg.adjust(mov);
        if (t.on_segment(h + seg)) u = h + seg;
        else u = h + seg.oppose();
        if (ra == 1) swap(a, b);
        ans = fabs((a - ori) * (u - ori)) / 2;
        tangle = acos(((u - ori) % (b - ori)) / ((u - ori).length() * (b - ori).length()));
        ans += radius * radius * (tangle / 2);
        return ans * sgn;
    }
}

const double pi = acos(-1.0);

int main (){
	int n;
	while(scanf("%lf",&radius)!=EOF){
		scanf("%d",&n);
		for(int i=0;i<n;i++){
			scanf("%lf%lf",&polygon[i].x,&polygon[i].y);
		}
		double ans=0;
		for(int i=0;i<n;i++){
			ans+=kuras_area(polygon[i],polygon[(i+1)%n],0,0);
		}
		printf("%.2lf
",fabs(ans));
	}
}

　　

HDU 3982。很明显，求出切割的半平面交之后就可以得出一个多边形。然后求多边形与圆面积相交部分即可。

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
using namespace std;

const double eps = 1e-6;
typedef long long LL;
const int inf = ~0u>>2;
const int MAXN=4222; 

inline double max (double a, double b) { if (a > b) return a; else return b; }
inline double min (double a, double b) { if (a < b) return a; else return b; }
inline int fi (double a){
    if (a > eps) return 1;
    else if (a >= -eps) return 0;
    else return -1;
}

struct Point {
	double x,y;
	Point(double xx,double yy){x=xx; y=yy;}
	Point(){}
}convex[MAXN],cherry;

class Vector{
public:
    double x, y;
    Vector (void) {}
    Vector (double x0, double y0) : x(x0), y(y0) {}
    double operator * (const Vector& a) const { return x * a.y - y * a.x; }		//向量叉乘 
    double operator % (const Vector& a) const { return x * a.x + y * a.y; }		//向量点乘
    Vector verti (void) const { return Vector(-y, x); }
    double length (void) const { return sqrt(x * x + y * y); }
    Vector adjust (double len) {
        double ol = len / length();
        return Vector(x * ol, y * ol);
    }
    Vector oppose (void) { return Vector(-x, -y); }
};
class point{
public:
    double x, y;
    point (void) {}
    point (double x0, double y0) : x(x0), y(y0) {}
    Vector operator - (const point& a) const { return Vector(x - a.x, y - a.y); }
    point operator + (const Vector& a) const { return point(x + a.x, y + a.y); }
    void _copy(Point t){
    	x=t.x; y=t.y;
    }
};
class segment{
public:
    point a, b;
    segment (void) {}
    segment (point a0, point b0) : a(a0), b(b0) {}
    point intersect (const segment& s) const {
        Vector v1 = s.a - a, v2 = s.b - a, v3 = s.b - b, v4 = s.a - b;
        double s1 = v1 * v2, s2 = v3 * v4;
        double se = s1 + s2;
        s1 /= se, s2 /= se;
        return point(a.x * s2 + b.x * s1, a.y * s2 + b.y * s1);
    }
    point pverti (const point& p) const {
        Vector t = (b - a).verti();
        segment uv(p, p + t);
        return intersect(uv);
    }
    bool on_segment (const point& p) const {
        if (fi(min(a.x, b.x) - p.x) <= 0 && fi(p.x - max(a.x, b.x)) <= 0 &&
            fi(min(a.y, b.y) - p.y) <= 0 && fi(p.y - max(a.y, b.y)) <= 0) return true;
        else return false;
    }
};

double radius;
point polygon[MAXN];//多边形 

double kuras_area (point a, point b, double cx, double cy){
    point ori(cx, cy);
    int sgn = fi((b - ori) * (a - ori));
    double da = (a - ori).length(), db = (b - ori).length();
    int ra = fi(da - radius), rb = fi(db - radius);
    double angle = acos(((b - ori) % (a - ori)) / (da * db));
    segment t(a, b); point h, u; Vector seg;
    double ans, dlt, mov, tangle;

    if (fi(da) == 0 || fi(db) == 0) return 0;
    else if (sgn == 0) return 0;
    else if (ra <= 0 && rb <= 0) return fabs((b - ori) * (a - ori)) / 2 * sgn;
    else if (ra >= 0 && rb >= 0){
        h = t.pverti(ori);
        dlt = (h - ori).length();
        if (!t.on_segment(h) || fi(dlt - radius) >= 0)
            return radius * radius * (angle / 2) * sgn;
        else{
            ans = radius * radius * (angle / 2);
            tangle = acos(dlt / radius);
            ans -= radius * radius * tangle;
            ans += radius * sin(tangle) * dlt;
            return ans * sgn;
        }
    }
    else{
        h = t.pverti(ori);
        dlt = (h - ori).length();
        seg = b - a;
        mov = sqrt(radius * radius - dlt * dlt);
        seg = seg.adjust(mov);
        if (t.on_segment(h + seg)) u = h + seg;
        else u = h + seg.oppose();
        if (ra == 1) swap(a, b);
        ans = fabs((a - ori) * (u - ori)) / 2;
        tangle = acos(((u - ori) % (b - ori)) / ((u - ori).length() * (b - ori).length()));
        ans += radius * radius * (tangle / 2);
        return ans * sgn;
    }
}

const double pi = acos(-1.0);

struct 	Line {
	Point a,b;
	double ang;
}line[MAXN/2],st[MAXN/2];
int n,cnt;

double operator *(const Point x,const Point y){
	return x.x*y.y-x.y*y.x;
}
Point operator -(Point x,const Point y){
	x.x-=y.x; x.y-=y.y;
	return x;
}
Point operator * (const Line &x,const Line &y){	//交点 
	double a1=(y.b-x.a)*(y.a-x.a),a2=(y.a-x.b)*(y.b-x.b);
	Point r;
	r.x=(x.a.x*a2+x.b.x*a1)/(a2+a1);
	r.y=(x.a.y*a2+x.b.y*a1)/(a2+a1);
	return r;
}

bool operator ==(const Point &a,const Point &b){
	return fabs(a.x-b.x)<eps&&fabs(a.y-b.y)<eps;
}

bool operator <(const Line &x,const Line &y){
	if(fabs(x.ang-y.ang)<eps)
	return (y.b-x.a)*(x.b-y.a)>eps;
	return x.ang <y.ang;
}

bool JudgeOut(const Line &x,const Point &p){
	return (p-x.a)*(x.b-x.a)>eps;
}

bool Parellel(const Line &x,const Line &y){
	return fabs((x.b-x.a)*(y.b-y.a))<eps;
}

void HplaneI(){
	sort(line,line+n);
	int top=1,bot=0,tmp=1;
	for(int i=1;i<n;i++){
		if(line[i].ang-line[i-1].ang>eps) line[tmp++]=line[i];
	}
	n=tmp;
	st[0]=line[0],st[1]=line[1];
	for(int i=2;i<n;i++){
		if(Parellel(st[top],st[top-1])||Parellel(st[bot],st[bot+1]))return ;
		while(bot<top&&JudgeOut(line[i],st[top]*st[top-1])) top--;
		while(bot<top&&JudgeOut(line[i],st[bot]*st[bot+1])) bot++;
		st[++top]=line[i];
	}
	while(bot<top&&JudgeOut(st[bot],st[top]*st[top-1])) top--;
	while(bot<top&&JudgeOut(st[top],st[bot]*st[bot+1])) bot++;
	if(top<=bot+1) return ;
	st[++top]=st[bot]; cnt=0;
	for(int i=bot;i<top;i++){
		polygon[cnt++]._copy(st[i]*st[i+1]);
	}
}

int main(){
	int T,icase=0;
	scanf("%d",&T);
	while(T--){
		scanf("%lf%d",&radius,&n);
		for(int i=0;i<n;i++){
			scanf("%lf%lf%lf%lf",&line[i].a.x,&line[i].a.y,&line[i].b.x,&line[i].b.y);
		}
		scanf("%lf%lf",&cherry.x,&cherry.y);
		for(int i=0;i<n;i++){
			if((line[i].a-cherry)*(line[i].b-cherry)<0){
				swap(line[i].a,line[i].b);
			}
			line[i].ang=atan2(line[i].b.y-line[i].a.y,line[i].b.x-line[i].a.x);
		}
		line[n].a=Point(-radius,-radius); line[n].b=Point(radius,-radius); line[n++].ang=0;
		line[n].a=Point(radius,-radius); line[n].b=Point(radius,radius); line[n++].ang=pi/2;
		line[n].a=Point(radius,radius); line[n].b=Point(-radius,radius); line[n++].ang=pi;
		line[n].a=Point(-radius,radius); line[n].b=Point(-radius,-radius); line[n++].ang=-pi/2;
		cnt=0;
		HplaneI();
		double ans=0;

		for(int i=0;i<cnt;i++){
	//		printf("%lf %lf 
",polygon[i].x,polygon[i].y);
			ans+=kuras_area(polygon[i],polygon[(i+1)%cnt],0,0);
		}
		printf("Case %d: ",++icase);
		printf("%.5lf",fabs(fabs(ans)-pi*radius*radius)<eps?100:fabs(ans)/(pi*radius*radius)*100);
		puts("%");
	}
	return 0;
}