2021“MINIEYE杯”中国大学生算法设计超级联赛（8）1008. Square Card（计算几何）

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Problem Description

Eric is playing a game on an infinite plane.

On the plane, there is a circular area with radius r1 called the Scoring Area. Every time Eric will throw a square card with side length a into the plane, then the card will start rotating around its centre. If at one moment the card is strictly inside the Scoring Area, the current play will be scored.

There is another circular area with radius r2 called the Bonus Area. If the play is scored in the current situation, the square card will continue rotating and if at one moment it is strictly inside the Bonus Area, he will get an extra bonus.

Eric isn't good at this game, so we can briefly consider that he will throw the card to any postion with an equal probability.

Now Eric wants to know what the ratio of the possibility of being scored and getting the bonus simultaneously to the possibility of being scored is.

The coordinates of the centre of the Scoring Area are (x1,y1).
The coordinates of the centre of the Bonus Area are (x2,y2).

Input

The first line contains a number T(1≤T≤20), the number of testcases.

For each testcase, there are three lines.
In the first line there are three integers r1,x1,y1(−1000≤x1,y1≤1000,0<r1≤1000), the radius and the postion of the Scoring Area.
In the second line there are three integers r2,x2,y2(−1000≤x2,y2≤1000,0<r2≤1000), the radius and the postion of the Bonus Area.
In the third line there is an integer a(0<a≤1000), the side length of the square card.

It is guaranteed that the Scoring Area is big enough thus there is always a chance for the play to be scored.

Output

For each test case, output a decimal p(0≤p≤1) in one line, the ratio of the possibility of being scored and getting the bonus simultaneously to the possibility of being scored.

Your answer should be rounded to 6 digits after the decimal point.

Sample Input

1
5 1 2
3 2 1
1

Sample Output

0.301720

高中几何题。对于每个圆，把正方形的两个相邻顶点放在圆的边界上，另外两个顶点保证在圆内，此时正方形的中心到圆心设为rr1，则以(x1, y1)为圆心，rr1为半径的新的圆就是scoring area的可行范围，bonus area同理。计算可得\(rr1 = \sqrt{r_1^2-(\frac{a}{2})^2} - \frac{a}{2}\)，rr2同理。题目要求的实际上是条件概率，由概率统计知识可知答案就是两个可行范围的交比上第一个圆的可行范围。两圆面积交可以用kuangbin的板子（省去题解中的特判）。注意这个题没有spj。

#include<bits/stdc++.h>
#include<cmath>
using namespace std;


const double eps = 1e-8;
const double pi = acos(-1.0);
const int maxp = 1010;
//`Compares a double to zero`
int sgn(double x){
	if(fabs(x) < eps)return 0;
	if(x < 0)return -1;
	else return 1;
}
//square of a double
inline double sqr(double x){return x*x;}
/*
 * Point
 * Point()               - Empty constructor
 * Point(double _x,double _y)  - constructor
 * input()             - double input
 * output()            - %.2f output
 * operator ==         - compares x and y
 * operator <          - compares first by x, then by y
 * operator -          - return new Point after subtracting curresponging x and y
 * operator ^          - cross product of 2d points
 * operator *          - dot product
 * len()               - gives length from origin
 * len2()              - gives square of length from origin
 * distance(Point p)   - gives distance from p
 * operator + Point b  - returns new Point after adding curresponging x and y
 * operator * double k - returns new Point after multiplieing x and y by k
 * operator / double k - returns new Point after divideing x and y by k
 * rad(Point a,Point b)- returns the angle of Point a and Point b from this Point
 * trunc(double r)     - return Point that if truncated the distance from center to r
 * rotleft()           - returns 90 degree ccw rotated point
 * rotright()          - returns 90 degree cw rotated point
 * rotate(Point p,double angle) - returns Point after rotateing the Point centering at p by angle radian ccw
 */
struct Point{
	double x,y;
	Point(){}
	Point(double _x,double _y){
		x = _x;
		y = _y;
	}
	void input(){
		scanf("%lf%lf",&x,&y);
	}
	void output(){
		printf("%.2f %.2f\n",x,y);
	}
	bool operator == (Point b)const{
		return sgn(x-b.x) == 0 && sgn(y-b.y) == 0;
	}
	bool operator < (Point b)const{
		return sgn(x-b.x)== 0?sgn(y-b.y)<0:x<b.x;
	}
	Point operator -(const Point &b)const{
		return Point(x-b.x,y-b.y);
	}
	//叉积
	double operator ^(const Point &b)const{
		return x*b.y - y*b.x;
	}
	//点积
	double operator *(const Point &b)const{
		return x*b.x + y*b.y;
	}
	//返回长度
	double len(){
		return hypot(x,y);//库函数
	}
	//返回长度的平方
	double len2(){
		return x*x + y*y;
	}
	//返回两点的距离
	double distance(Point p){
		return hypot(x-p.x,y-p.y);
	}
	Point operator +(const Point &b)const{
		return Point(x+b.x,y+b.y);
	}
	Point operator *(const double &k)const{
		return Point(x*k,y*k);
	}
	Point operator /(const double &k)const{
		return Point(x/k,y/k);
	}
	//`计算pa  和  pb 的夹角`
	//`就是求这个点看a,b 所成的夹角`
	//`测试 LightOJ1203`
	double rad(Point a,Point b){
		Point p = *this;
		return fabs(atan2( fabs((a-p)^(b-p)),(a-p)*(b-p) ));
	}
	//`化为长度为r的向量`
	Point trunc(double r){
		double l = len();
		if(!sgn(l))return *this;
		r /= l;
		return Point(x*r,y*r);
	}
	//`逆时针旋转90度`
	Point rotleft(){
		return Point(-y,x);
	}
	//`顺时针旋转90度`
	Point rotright(){
		return Point(y,-x);
	}
	//`绕着p点逆时针旋转angle`
	Point rotate(Point p,double angle){
		Point v = (*this) - p;
		double c = cos(angle), s = sin(angle);
		return Point(p.x + v.x*c - v.y*s,p.y + v.x*s + v.y*c);
	}
};
/*
 * Stores two points
 * Line()                         - Empty constructor
 * Line(Point _s,Point _e)        - Line through _s and _e
 * operator ==                    - checks if two points are same
 * Line(Point p,double angle)     - one end p , another end at angle degree
 * Line(double a,double b,double c) - Line of equation ax + by + c = 0
 * input()                        - inputs s and e
 * adjust()                       - orders in such a way that s < e
 * length()                       - distance of se
 * angle()                        - return 0 <= angle < pi
 * relation(Point p)              - 3 if point is on line
 *                                  1 if point on the left of line
 *                                  2 if point on the right of line
 * pointonseg(double p)           - return true if point on segment
 * parallel(Line v)               - return true if they are parallel
 * segcrossseg(Line v)            - returns 0 if does not intersect
 *                                  returns 1 if non-standard intersection
 *                                  returns 2 if intersects
 * linecrossseg(Line v)           - line and seg
 * linecrossline(Line v)          - 0 if parallel
 *                                  1 if coincides
 *                                  2 if intersects
 * crosspoint(Line v)             - returns intersection point
 * dispointtoline(Point p)        - distance from point p to the line
 * dispointtoseg(Point p)         - distance from p to the segment
 * dissegtoseg(Line v)            - distance of two segment
 * lineprog(Point p)              - returns projected point p on se line
 * symmetrypoint(Point p)         - returns reflection point of p over se
 *
 */
struct Line{
	Point s,e;
	Line(){}
	Line(Point _s,Point _e){
		s = _s;
		e = _e;
	}
	bool operator ==(Line v){
		return (s == v.s)&&(e == v.e);
	}
	//`根据一个点和倾斜角angle确定直线,0<=angle<pi`
	Line(Point p,double angle){
		s = p;
		if(sgn(angle-pi/2) == 0){
			e = (s + Point(0,1));
		}
		else{
			e = (s + Point(1,tan(angle)));
		}
	}
	//ax+by+c=0
	Line(double a,double b,double c){
		if(sgn(a) == 0){
			s = Point(0,-c/b);
			e = Point(1,-c/b);
		}
		else if(sgn(b) == 0){
			s = Point(-c/a,0);
			e = Point(-c/a,1);
		}
		else{
			s = Point(0,-c/b);
			e = Point(1,(-c-a)/b);
		}
	}
	void input(){
		s.input();
		e.input();
	}
	void adjust(){
		if(e < s)swap(s,e);
	}
	//求线段长度
	double length(){
		return s.distance(e);
	}
	//`返回直线倾斜角 0<=angle<pi`
	double angle(){
		double k = atan2(e.y-s.y,e.x-s.x);
		if(sgn(k) < 0)k += pi;
		if(sgn(k-pi) == 0)k -= pi;
		return k;
	}
	//`点和直线关系`
	//`1  在左侧`
	//`2  在右侧`
	//`3  在直线上`
	int relation(Point p){
		int c = sgn((p-s)^(e-s));
		if(c < 0)return 1;
		else if(c > 0)return 2;
		else return 3;
	}
	// 点在线段上的判断
	bool pointonseg(Point p){
		return sgn((p-s)^(e-s)) == 0 && sgn((p-s)*(p-e)) <= 0;
	}
	//`两向量平行(对应直线平行或重合)`
	bool parallel(Line v){
		return sgn((e-s)^(v.e-v.s)) == 0;
	}
	//`两线段相交判断`
	//`2 规范相交`
	//`1 非规范相交`
	//`0 不相交`
	int segcrossseg(Line v){
		int d1 = sgn((e-s)^(v.s-s));
		int d2 = sgn((e-s)^(v.e-s));
		int d3 = sgn((v.e-v.s)^(s-v.s));
		int d4 = sgn((v.e-v.s)^(e-v.s));
		if( (d1^d2)==-2 && (d3^d4)==-2 )return 2;
		return (d1==0 && sgn((v.s-s)*(v.s-e))<=0) ||
			(d2==0 && sgn((v.e-s)*(v.e-e))<=0) ||
			(d3==0 && sgn((s-v.s)*(s-v.e))<=0) ||
			(d4==0 && sgn((e-v.s)*(e-v.e))<=0);
	}
	//`直线和线段相交判断`
	//`-*this line   -v seg`
	//`2 规范相交`
	//`1 非规范相交`
	//`0 不相交`
	int linecrossseg(Line v){
		int d1 = sgn((e-s)^(v.s-s));
		int d2 = sgn((e-s)^(v.e-s));
		if((d1^d2)==-2) return 2;
		return (d1==0||d2==0);
	}
	//`两直线关系`
	//`0 平行`
	//`1 重合`
	//`2 相交`
	int linecrossline(Line v){
		if((*this).parallel(v))
			return v.relation(s)==3;
		return 2;
	}
	//`求两直线的交点`
	//`要保证两直线不平行或重合`
	Point crosspoint(Line v){
		double a1 = (v.e-v.s)^(s-v.s);
		double a2 = (v.e-v.s)^(e-v.s);
		return Point((s.x*a2-e.x*a1)/(a2-a1),(s.y*a2-e.y*a1)/(a2-a1));
	}
	//点到直线的距离
	double dispointtoline(Point p){
		return fabs((p-s)^(e-s))/length();
	}
	//点到线段的距离
	double dispointtoseg(Point p){
		if(sgn((p-s)*(e-s))<0 || sgn((p-e)*(s-e))<0)
			return min(p.distance(s),p.distance(e));
		return dispointtoline(p);
	}
	//`返回线段到线段的距离`
	//`前提是两线段不相交，相交距离就是0了`
	double dissegtoseg(Line v){
		return min(min(dispointtoseg(v.s),dispointtoseg(v.e)),min(v.dispointtoseg(s),v.dispointtoseg(e)));
	}
	//`返回点p在直线上的投影`
	Point lineprog(Point p){
		return s + ( ((e-s)*((e-s)*(p-s)))/((e-s).len2()) );
	}
	//`返回点p关于直线的对称点`
	Point symmetrypoint(Point p){
		Point q = lineprog(p);
		return Point(2*q.x-p.x,2*q.y-p.y);
	}
};
struct circle{
	Point p;//圆心
	double r;//半径
	circle(){}
	circle(Point _p,double _r){
		p = _p;
		r = _r;
	}
	circle(double x,double y,double _r){
		p = Point(x,y);
		r = _r;
	}
	//`三角形的外接圆`
	//`需要Point的+ /  rotate()  以及Line的crosspoint()`
	//`利用两条边的中垂线得到圆心`
	//`测试：UVA12304`
	circle(Point a,Point b,Point c){
		Line u = Line((a+b)/2,((a+b)/2)+((b-a).rotleft()));
		Line v = Line((b+c)/2,((b+c)/2)+((c-b).rotleft()));
		p = u.crosspoint(v);
		r = p.distance(a);
	}
	//`三角形的内切圆`
	//`参数bool t没有作用，只是为了和上面外接圆函数区别`
	//`测试：UVA12304`
	circle(Point a,Point b,Point c,bool t){
		Line u,v;
		double m = atan2(b.y-a.y,b.x-a.x), n = atan2(c.y-a.y,c.x-a.x);
		u.s = a;
		u.e = u.s + Point(cos((n+m)/2),sin((n+m)/2));
		v.s = b;
		m = atan2(a.y-b.y,a.x-b.x) , n = atan2(c.y-b.y,c.x-b.x);
		v.e = v.s + Point(cos((n+m)/2),sin((n+m)/2));
		p = u.crosspoint(v);
		r = Line(a,b).dispointtoseg(p);
	}
	//输入
	void input(){
		p.input();
		scanf("%lf",&r);
	}
	//输出
	void output(){
		printf("%.2lf %.2lf %.2lf\n",p.x,p.y,r);
	}
	bool operator == (circle v){
		return (p==v.p) && sgn(r-v.r)==0;
	}
	bool operator < (circle v)const{
		return ((p<v.p)||((p==v.p)&&sgn(r-v.r)<0));
	}
	//面积
	double area(){
		return pi*r*r;
	}
	//周长
	double circumference(){
		return 2*pi*r;
	}
	//`点和圆的关系`
	//`0 圆外`
	//`1 圆上`
	//`2 圆内`
	int relation(Point b){
		double dst = b.distance(p);
		if(sgn(dst-r) < 0)return 2;
		else if(sgn(dst-r)==0)return 1;
		return 0;
	}
	//`线段和圆的关系`
	//`比较的是圆心到线段的距离和半径的关系`
	int relationseg(Line v){
		double dst = v.dispointtoseg(p);
		if(sgn(dst-r) < 0)return 2;
		else if(sgn(dst-r) == 0)return 1;
		return 0;
	}
	//`直线和圆的关系`
	//`比较的是圆心到直线的距离和半径的关系`
	int relationline(Line v){
		double dst = v.dispointtoline(p);
		if(sgn(dst-r) < 0)return 2;
		else if(sgn(dst-r) == 0)return 1;
		return 0;
	}
	//`两圆的关系`
	//`5 相离`
	//`4 外切`
	//`3 相交`
	//`2 内切`
	//`1 内含`
	//`需要Point的distance`
	//`测试：UVA12304`
	int relationcircle(circle v){
		double d = p.distance(v.p);
		if(sgn(d-r-v.r) > 0)return 5;
		if(sgn(d-r-v.r) == 0)return 4;
		double l = fabs(r-v.r);
		if(sgn(d-r-v.r)<0 && sgn(d-l)>0)return 3;
		if(sgn(d-l)==0)return 2;
		if(sgn(d-l)<0)return 1;
	}
	//`求两个圆的交点，返回0表示没有交点，返回1是一个交点，2是两个交点`
	//`需要relationcircle`
	//`测试：UVA12304`
	int pointcrosscircle(circle v,Point &p1,Point &p2){
		int rel = relationcircle(v);
		if(rel == 1 || rel == 5)return 0;
		double d = p.distance(v.p);
		double l = (d*d+r*r-v.r*v.r)/(2*d);
		double h = sqrt(r*r-l*l);
		Point tmp = p + (v.p-p).trunc(l);
		p1 = tmp + ((v.p-p).rotleft().trunc(h));
		p2 = tmp + ((v.p-p).rotright().trunc(h));
		if(rel == 2 || rel == 4)
			return 1;
		return 2;
	}
	//`求直线和圆的交点，返回交点个数`
	int pointcrossline(Line v,Point &p1,Point &p2){
		if(!(*this).relationline(v))return 0;
		Point a = v.lineprog(p);
		double d = v.dispointtoline(p);
		d = sqrt(r*r-d*d);
		if(sgn(d) == 0){
			p1 = a;
			p2 = a;
			return 1;
		}
		p1 = a + (v.e-v.s).trunc(d);
		p2 = a - (v.e-v.s).trunc(d);
		return 2;
	}
	//`得到过a,b两点，半径为r1的两个圆`
	int gercircle(Point a,Point b,double r1,circle &c1,circle &c2){
		circle x(a,r1),y(b,r1);
		int t = x.pointcrosscircle(y,c1.p,c2.p);
		if(!t)return 0;
		c1.r = c2.r = r1;
		return t;
	}
	//`得到与直线u相切，过点q,半径为r1的圆`
	//`测试：UVA12304`
	int getcircle(Line u,Point q,double r1,circle &c1,circle &c2){
		double dis = u.dispointtoline(q);
		if(sgn(dis-r1*2)>0)return 0;
		if(sgn(dis) == 0){
			c1.p = q + ((u.e-u.s).rotleft().trunc(r1));
			c2.p = q + ((u.e-u.s).rotright().trunc(r1));
			c1.r = c2.r = r1;
			return 2;
		}
		Line u1 = Line((u.s + (u.e-u.s).rotleft().trunc(r1)),(u.e + (u.e-u.s).rotleft().trunc(r1)));
		Line u2 = Line((u.s + (u.e-u.s).rotright().trunc(r1)),(u.e + (u.e-u.s).rotright().trunc(r1)));
		circle cc = circle(q,r1);
		Point p1,p2;
		if(!cc.pointcrossline(u1,p1,p2))cc.pointcrossline(u2,p1,p2);
		c1 = circle(p1,r1);
		if(p1 == p2){
			c2 = c1;
			return 1;
		}
		c2 = circle(p2,r1);
		return 2;
	}
	//`同时与直线u,v相切，半径为r1的圆`
	//`测试：UVA12304`
	int getcircle(Line u,Line v,double r1,circle &c1,circle &c2,circle &c3,circle &c4){
		if(u.parallel(v))return 0;//两直线平行
		Line u1 = Line(u.s + (u.e-u.s).rotleft().trunc(r1),u.e + (u.e-u.s).rotleft().trunc(r1));
		Line u2 = Line(u.s + (u.e-u.s).rotright().trunc(r1),u.e + (u.e-u.s).rotright().trunc(r1));
		Line v1 = Line(v.s + (v.e-v.s).rotleft().trunc(r1),v.e + (v.e-v.s).rotleft().trunc(r1));
		Line v2 = Line(v.s + (v.e-v.s).rotright().trunc(r1),v.e + (v.e-v.s).rotright().trunc(r1));
		c1.r = c2.r = c3.r = c4.r = r1;
		c1.p = u1.crosspoint(v1);
		c2.p = u1.crosspoint(v2);
		c3.p = u2.crosspoint(v1);
		c4.p = u2.crosspoint(v2);
		return 4;
	}
	//`同时与不相交圆cx,cy相切，半径为r1的圆`
	//`测试：UVA12304`
	int getcircle(circle cx,circle cy,double r1,circle &c1,circle &c2){
		circle x(cx.p,r1+cx.r),y(cy.p,r1+cy.r);
		int t = x.pointcrosscircle(y,c1.p,c2.p);
		if(!t)return 0;
		c1.r = c2.r = r1;
		return t;
	}
 
	//`过一点作圆的切线(先判断点和圆的关系)`
	//`测试：UVA12304`
	int tangentline(Point q,Line &u,Line &v){
		int x = relation(q);
		if(x == 2)return 0;
		if(x == 1){
			u = Line(q,q + (q-p).rotleft());
			v = u;
			return 1;
		}
		double d = p.distance(q);
		double l = r*r/d;
		double h = sqrt(r*r-l*l);
		u = Line(q,p + ((q-p).trunc(l) + (q-p).rotleft().trunc(h)));
		v = Line(q,p + ((q-p).trunc(l) + (q-p).rotright().trunc(h)));
		return 2;
	}
	//`求两圆相交的面积`
	double areacircle(circle v){
		int rel = relationcircle(v);
		if(rel >= 4)return 0.0;
		if(rel <= 2)return min(area(),v.area());
		double d = p.distance(v.p);
		double hf = (r+v.r+d)/2.0;
		double ss = 2*sqrt(hf*(hf-r)*(hf-v.r)*(hf-d));
		double a1 = acos((r*r+d*d-v.r*v.r)/(2.0*r*d));
		a1 = a1*r*r;
		double a2 = acos((v.r*v.r+d*d-r*r)/(2.0*v.r*d));
		a2 = a2*v.r*v.r;
		return a1+a2-ss;
	}
	//`求两圆相交的面积(精度更高)(需要long double)`
	double areacircle2(circle v)
	{
		double a=hypot(p.x-v.p.x,p.y-v.p.y),b=r,c=v.r;
		double s1=pi*r*r,s2=pi*v.r*v.r;
		if(sgn(a-b-c)>=0)
			return 0;
		if(sgn(a+min(b,c)-max(b,c))<=0)
			return min(s1,s2);
		else
		{
			double cta1=2*acos((a*a+b*b-c*c)/(2*a*b));
			double cta2=2*acos((a*a+c*c-b*b)/(2*a*c));
			return cta1/(2*pi)*s1-0.5*sin(cta1)*b*b+cta2/(2*pi)*s2-0.5*sin(cta2)*c*c;
		}
	}
	//`求圆和三角形pab的相交面积`
	//`测试：POJ3675 HDU3982 HDU2892`
	double areatriangle(Point a,Point b){
		if(sgn((p-a)^(p-b)) == 0)return 0.0;
		Point q[5];
		int len = 0;
		q[len++] = a;
		Line l(a,b);
		Point p1,p2;
		if(pointcrossline(l,q[1],q[2])==2){
			if(sgn((a-q[1])*(b-q[1]))<0)q[len++] = q[1];
			if(sgn((a-q[2])*(b-q[2]))<0)q[len++] = q[2];
		}
		q[len++] = b;
		if(len == 4 && sgn((q[0]-q[1])*(q[2]-q[1]))>0)swap(q[1],q[2]);
		double res = 0;
		for(int i = 0;i < len-1;i++){
			if(relation(q[i])==0||relation(q[i+1])==0){
				double arg = p.rad(q[i],q[i+1]);
				res += r*r*arg/2.0;
			}
			else{
				res += fabs((q[i]-p)^(q[i+1]-p))/2.0;
			}
		}
		return res;
	}
};
int main() {
	int t;
	cin >> t;
	while(t--) {
		double r1, x1, y1, r2, x2, y2, a;
		cin >> r1 >> x1 >> y1;
		cin >> r2 >> x2 >> y2;
		cin >> a;
		double rr1 = sqrt(r1 * r1 - a * a / 4) - a / 2.0, rr2 = sqrt(r2 * r2 - a * a / 4) - a / 2.0;
		circle c1(x1, y1, rr1);
		circle c2(x2, y2, rr2);
		
		double s1 = c1.areacircle(c2);
		double s2 = pi * rr1 * rr1;
		printf("%.6lf\n", s1 / s2);
	}

	return 0;
}

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