题意:一个凸多边形,在里面放两个半径为r的圆,问两个圆的最大距离的坐标是多少
题解:两个圆要满足在多边形里面,所有的圆形可能位置就是这个多边形向内r形成的多边形,可以用半平面交找到这个多边形,(线段平移用到法向量),再找出这个多边形最远的两个点,O(n*n)枚举即可,这里套用别人模板
#include <bits/stdc++.h> #define LL long long #define PI 3.1415926535897932384626 #define maxn 1000 #define EXIT exit(0); #define DEBUG puts("Here is a BUG"); #define CLEAR(name, init) memset(name, init, sizeof(name)) const double eps = 1e-6; const int MAXN = (int)1e9 + 5; using namespace std; #define Vector Point #define ChongHe 0 #define NeiHan 1 #define NeiQie 2 #define INTERSECTING 3 #define WaiQie 4 #define XiangLi 5 int dcmp(double x) { return fabs(x) < eps ? 0 : (x < 0 ? -1 : 1); } struct Point { double x, y; Point(const Point& rhs): x(rhs.x), y(rhs.y) { } //拷贝构造函数 Point(double x = 0.0, double y = 0.0): x(x), y(y) { } //构造函数 friend istream& operator >> (istream& in, Point& P) { return in >> P.x >> P.y; } friend ostream& operator << (ostream& out, const Point& P) { return out << P.x << ' ' << P.y; } friend Vector operator + (const Vector& A, const Vector& B) { return Vector(A.x+B.x, A.y+B.y); } friend Vector operator - (const Point& A, const Point& B) { return Vector(A.x-B.x, A.y-B.y); } friend Vector operator * (const Vector& A, const double& p) { return Vector(A.x*p, A.y*p); } friend Vector operator / (const Vector& A, const double& p) { return Vector(A.x/p, A.y/p); } friend bool operator == (const Point& A, const Point& B) { return dcmp(A.x-B.x) == 0 && dcmp(A.y-B.y) == 0; } friend bool operator < (const Point& A, const Point& B) { return A.x < B.x || (A.x == B.x && A.y < B.y); } void in(void) { scanf("%lf%lf", &x, &y); } void out(void) { printf("%lf %lf", x, y); } }; struct Line { Point P; //直线上一点 Vector dir; //方向向量(半平面交中该向量左侧表示相应的半平面) double ang; //极角，即从x正半轴旋转到向量dir所需要的角（弧度） Line() { } //构造函数 Line(const Line& L): P(L.P), dir(L.dir), ang(L.ang) { } Line(const Point& P, const Vector& dir): P(P), dir(dir) { ang = atan2(dir.y, dir.x); } bool operator < (const Line& L) const { //极角排序 return ang < L.ang; } Point point(double t) { return P + dir*t; } }; typedef vector<Point> Polygon; struct Circle { Point c; //圆心 double r; //半径 Circle() { } Circle(const Circle& rhs): c(rhs.c), r(rhs.r) { } Circle(const Point& c, const double& r): c(c), r(r) { } Point point(double ang) const { return Point(c.x + cos(ang)*r, c.y + sin(ang)*r); } //圆心角所对应的点 double area(void) const { return PI * r * r; } }; double Dot(const Vector& A, const Vector& B) { return A.x*B.x + A.y*B.y; } //点积 double Length(const Vector& A){ return sqrt(Dot(A, A)); } double Angle(const Vector& A, const Vector& B) { return acos(Dot(A, B)/Length(A)/Length(B)); } //向量夹角 double Cross(const Vector& A, const Vector& B) { return A.x*B.y - A.y*B.x; } //叉积 double Area(const Point& A, const Point& B, const Point& C) { return fabs(Cross(B-A, C-A)); } //三边构成三角形的判定 bool check_length(double a, double b, double c) { return dcmp(a+b-c) > 0 && dcmp(fabs(a-b)-c) < 0; } bool isTriangle(double a, double b, double c) { return check_length(a, b, c) && check_length(a, c, b) && check_length(b, c, a); } //平行四边形的判定（保证四边形顶点按顺序给出） bool isParallelogram(Polygon p) { if (dcmp(Length(p[0]-p[1]) - Length(p[2]-p[3])) || dcmp(Length(p[0]-p[3]) - Length(p[2]-p[1]))) return false; Line a = Line(p[0], p[1]-p[0]); Line b = Line(p[1], p[2]-p[1]); Line c = Line(p[3], p[2]-p[3]); Line d = Line(p[0], p[3]-p[0]); return dcmp(a.ang - c.ang) == 0 && dcmp(b.ang - d.ang) == 0; } //梯形的判定 bool isTrapezium(Polygon p) { Line a = Line(p[0], p[1]-p[0]); Line b = Line(p[1], p[2]-p[1]); Line c = Line(p[3], p[2]-p[3]); Line d = Line(p[0], p[3]-p[0]); return (dcmp(a.ang - c.ang) == 0 && dcmp(b.ang - d.ang)) || (dcmp(a.ang - c.ang) && dcmp(b.ang - d.ang) == 0); } //菱形的判定 bool isRhombus(Polygon p) { if (!isParallelogram(p)) return false; return dcmp(Length(p[1]-p[0]) - Length(p[2]-p[1])) == 0; } //矩形的判定 bool isRectangle(Polygon p) { if (!isParallelogram(p)) return false; return dcmp(Length(p[2]-p[0]) - Length(p[3]-p[1])) == 0; } //正方形的判定 bool isSquare(Polygon p) { return isRectangle(p) && isRhombus(p); } //三点共线的判定 bool isCollinear(Point A, Point B, Point C) { return dcmp(Cross(B-A, C-B)) == 0; } //向量绕起点旋转 Vector Rotate(const Vector& A, const double& rad) { return Vector(A.x*cos(rad)-A.y*sin(rad), A.x*sin(rad)+A.y*cos(rad)); } //向量的单位法线(调用前请确保A 不是零向量) Vector Normal(const Vector& A) { double len = Length(A); return Vector(-A.y / len, A.x / len); } //两直线交点(用前确保两直线有唯一交点，当且仅当Cross(A.dir, B.dir)非0) Point GetLineIntersection(const Line& A, const Line& B) { Vector u = A.P - B.P; double t = Cross(B.dir, u) / Cross(A.dir, B.dir); return A.P + A.dir*t; } //点到直线距离 double DistanceToLine(const Point& P, const Line& L) { Vector v1 = L.dir, v2 = P - L.P; return fabs(Cross(v1, v2)) / Length(v1); } //点到线段距离 double DistanceToSegment(const Point& P, const Point& A, const Point& B) { if (A == B) return Length(P - A); Vector v1 = B - A, v2 = P - A, v3 = P - B; if (dcmp(Dot(v1, v2)) < 0) return Length(v2); if (dcmp(Dot(v1, v3)) > 0) return Length(v3); return fabs(Cross(v1, v2)) / Length(v1); } //点在直线上的投影 Point GetLineProjection(const Point& P, const Line& L) { return L.P + L.dir*(Dot(L.dir, P - L.P)/Dot(L.dir, L.dir)); } //点在线段上的判定 bool isOnSegment(const Point& P, const Point& A, const Point& B) { //若允许点与端点重合，可关闭下面的注释 //if (P == A || P == B) return true; // return dcmp(Cross(A-P, B-P)) == 0 && dcmp(Dot(A-P, B-P)) < 0; return dcmp(Length(P-A) + Length(B-P) - Length(A-B)) == 0; } //线段相交判定 bool SegmentProperIntersection(const Point& a1, const Point& a2, const Point& b1, const Point& b2) { //若允许在端点处相交，可适当关闭下面的注释 //if (isOnSegment(a1, b1, b2) || isOnSegment(a2, b1, b2) || isOnSegment(b1, a1, a2) || isOnSegment(b2, a1, a2)) return true; double c1 = Cross(a2-a1, b1-a1), c2 = Cross(a2-a1, b2-a1); double c3 = Cross(b2-b1, a1-b1), c4 = Cross(b2-b1, a2-b1); return dcmp(c1)*dcmp(c2) < 0 && dcmp(c3)*dcmp(c4) < 0; } //多边形的有向面积 double PolygonArea(Polygon po) { int n = po.size(); double area = 0.0; for(int i = 1; i < n-1; i++) { area += Cross(po[i]-po[0], po[i+1]-po[0]); } return area * 0.5; } //点在多边形内的判定(多边形顶点需按逆时针排列) bool isInPolygon(const Point& p, const Polygon& poly) { int n = poly.size(); for(int i = 0; i < n; i++) { //若允许点在多边形边上，可关闭下行注释 // if (isOnSegment(p, poly[(i+1)%n], poly[i])) return true; if (Cross(poly[(i+1)%n]-poly[i], p-poly[i]) < 0) return false; } return true; } //过定点作圆的切线 int getTangents(const Point& P, const Circle& C, std::vector<Line>& L) { Vector u = C.c - P; double dis = Length(u); if (dcmp(dis - C.r) < 0) return 0; if (dcmp(dis - C.r) == 0) { L.push_back(Line(P, Rotate(u, PI / 2.0))); return 1; } double ang = asin(C.r / dis); L.push_back(Line(P, Rotate(u, ang))); L.push_back(Line(P, Rotate(u, -ang))); return 2; } //直线和圆的交点 int GetLineCircleIntersection(Line& L, const Circle& C, vector<Point>& sol) { double t1, t2; double a = L.dir.x, b = L.P.x - C.c.x, c = L.dir.y, d = L.P.y - C.c.y; double e = a*a + c*c, f = 2.0*(a*b + c*d), g = b*b + d*d - C.r*C.r; double delta = f*f - 4*e*g; //判别式 if (dcmp(delta) < 0) return 0; //相离 if (dcmp(delta) == 0) { //相切 t1 = t2 = -f / (2 * e); sol.push_back(L.point(t1)); return 1; } t1 = (-f - sqrt(delta)) / (2.0 * e); sol.push_back(L.point(t1)); // 相交 t2 = (-f + sqrt(delta)) / (2.0 * e); sol.push_back(L.point(t2)); return 2; } //两圆位置关系判定 int GetCircleLocationRelation(const Circle& A, const Circle& B) { double d = Length(A.c-B.c); double sum = A.r + B.r; double sub = fabs(A.r - B.r); if (dcmp(d) == 0) return dcmp(sub) != 0; if (dcmp(d - sum) > 0) return XiangLi; if (dcmp(d - sum) == 0) return WaiQie; if (dcmp(d - sub) > 0 && dcmp(d - sum) < 0) return INTERSECTING; if (dcmp(d - sub) == 0) return NeiQie; if (dcmp(d - sub) < 0) return NeiHan; } //两圆相交的面积 double GetCircleIntersectionArea(const Circle& A, const Circle& B) { int rel = GetCircleLocationRelation(A, B); if (rel < INTERSECTING) return min(A.area(), B.area()); if (rel > INTERSECTING) return 0; double dis = Length(A.c - B.c); double ang1 = acos((A.r*A.r + dis*dis - B.r*B.r) / (2.0*A.r*dis)); double ang2 = acos((B.r*B.r + dis*dis - A.r*A.r) / (2.0*B.r*dis)); return ang1*A.r*A.r + ang2*B.r*B.r - A.r*dis*sin(ang1); } //凸包(Andrew算法) //如果不希望在凸包的边上有输入点，把两个 <= 改成 < //如果不介意点集被修改，可以改成传递引用 Polygon ConvexHull(vector<Point> p) { //预处理，删除重复点 sort(p.begin(), p.end()); p.erase(unique(p.begin(), p.end()), p.end()); int n = p.size(), m = 0; Polygon res(n+1); for(int i = 0; i < n; i++) { while(m > 1 && Cross(res[m-1]-res[m-2], p[i]-res[m-2]) <= 0) m--; res[m++] = p[i]; } int k = m; for(int i = n-2; i >= 0; i--) { while(m > k && Cross(res[m-1]-res[m-2], p[i]-res[m-2]) <= 0) m--; res[m++] = p[i]; } m -= n > 1; res.resize(m); return res; } //点P在有向直线L左边的判定(线上算) bool isOnLeft(const Line& L, const Point& P) { return Cross(L.dir, P-L.P) >= 0; } //半平面交主过程 //如果不介意点集被修改，可以改成传递引用 Polygon HalfPlaneIntersection(vector<Line> L) {//有可能退化为点或者线段(修改isOnLeft部分) int n = L.size(); int head, rear; //双端队列的第一个元素和最后一个元素的下标 vector<Point> p(n); //p[i]为q[i]和q[i+1]的交点 vector<Line> q(n); //双端队列 Polygon ans; sort(L.begin(), L.end()); //按极角排序 q[head=rear=0] = L[0]; //双端队列初始化为只有一个半平面L[0] for(int i = 1; i < n; i++) { while(head < rear && !isOnLeft(L[i], p[rear-1])) rear--; while(head < rear && !isOnLeft(L[i], p[head])) head++; q[++rear] = L[i]; if (fabs(Cross(q[rear].dir, q[rear-1].dir)) < eps) { //两向量平行且同向，取内侧的一个 rear--; if (isOnLeft(q[rear], L[i].P)) q[rear] = L[i]; } if (head < rear) p[rear-1] = GetLineIntersection(q[rear-1], q[rear]); } while(head < rear && !isOnLeft(q[head], p[rear-1])) rear--; //删除无用平面 if (rear - head <= 1) return ans; //空集 p[rear] = GetLineIntersection(q[rear], q[head]); //计算首尾两个半平面的交点 for(int i = head; i <= rear; i++) //从deque复制到输出中 ans.push_back(p[i]); return ans; } Point a[maxn]; Vector e; vector<Line> L; Polygon p; int main() { int n; double r; scanf("%d%lf", &n, &r); for(int i=0;i<n;i++) cin>>a[i]; for(int i=0;i<n;i++){ e = Normal(a[i]-a[(i+1)%n]); L.push_back(Line(a[(i+1)%n]+e*r, a[i]-a[(i+1)%n])); } p = HalfPlaneIntersection(L); int ans1 = 0, ans2 = 0, num = p.size(); for(int i=0;i<num;i++){ for(int j=i;j<num;j++){ if(Length(p[i]-p[j]) > Length(p[ans1]-p[ans2])) ans1 = i, ans2 = j; } } printf("%f %f %f %f ", p[ans1].x, p[ans1].y, p[ans2].x, p[ans2].y); return 0; } /* 4 1 0 0 0 2 3 2 3 0 */