题意：判断简单多边形内是否可以放一个半径为R的圆
思路：如果这个多边形是正多边形，令r(x,y)为圆心在(x,y)处多边形内最大圆的半径，不难发现，f(x,y)越靠近正多边形的中心，r越大，所以可以利用模拟退火法来逼近最优点。对于一般的多边形，由于可能存在多个这样的"局部最优点"，所以可以选不同的点作为起点进行多若干次模拟退火即可。
模拟退火的过程：每次由原状态S生成一个新状态T，如果T比S优，那么接受这一次转移，否则以一定概率P接受这次转移，因为这样可能会跳过局部最优解而得到全局最优解。
PS:步长每次改变的系数一般设为0.8~0.9，eps不能设太高。
#pragma comment(linker, "/STACK:10240000") #include <bits/stdc++.h> using namespace std; #define X first #define Y second #define pb push_back #define mp make_pair #define all(a) (a).begin(), (a).end() #define fillchar(a, x) memset(a, x, sizeof(a)) typedef long long ll; typedef pair<int, int> pii; namespace Debug { void print(){cout<<endl;}template<typename T> void print(const T t){cout<<t<<endl;}template<typename F,typename...R> void print(const F f,const R...r){cout<<f<<" ";print(r...);}template<typename T> void print(T*p, T*q){int d=p<q?1:-1;while(p!=q){cout<<*p<<", ";p+=d;}cout<<endl;} } template<typename T>bool umax(T&a, const T&b){return b<=a?false:(a=b,true);} template<typename T>bool umin(T&a, const T&b){return b>=a?false:(a=b,true);} /* -------------------------------------------------------------------------------- */ const double eps = 1e-4;/** 设置比较精度 **/ struct Real { double x; double get() { return x; } int read() { return scanf("%lf", &x); } Real(const double &x) { this->x = x; } Real() {} Real abs() { return x > 0? x : -x; } Real operator + (const Real &that) const { return Real(x + that.x);} Real operator - (const Real &that) const { return Real(x - that.x);} Real operator * (const Real &that) const { return Real(x * that.x);} Real operator / (const Real &that) const { return Real(x / that.x);} Real operator - () const { return Real(-x); } Real operator += (const Real &that) { return Real(x += that.x); } Real operator -= (const Real &that) { return Real(x -= that.x); } Real operator *= (const Real &that) { return Real(x *= that.x); } Real operator /= (const Real &that) { return Real(x /= that.x); } bool operator < (const Real &that) const { return x - that.x <= -eps; } bool operator > (const Real &that) const { return x - that.x >= eps; } bool operator == (const Real &that) const { return x - that.x > -eps && x - that.x < eps; } bool operator <= (const Real &that) const { return x - that.x < eps; } bool operator >= (const Real &that) const { return x - that.x > -eps; } friend ostream& operator << (ostream &out, const Real &val) { out << val.x; return out; } friend istream& operator >> (istream &in, Real &val) { in >> val.x; return in; } }; struct Point { Real x, y; int read() { return scanf("%lf%lf", &x.x, &y.x); } Point(const Real &x, const Real &y) { this->x = x; this->y = y; } Point() {} Point operator + (const Point &that) const { return Point(this->x + that.x, this->y + that.y); } Point operator - (const Point &that) const { return Point(this->x - that.x, this->y - that.y); } Real operator * (const Point &that) const { return x * that.x + y * that.y; } Point operator * (const Real &that) const { return Point(x * that, y * that); } Point operator += (const Point &that) { return Point(this->x += that.x, this->y += that.y); } Point operator -= (const Point &that) { return Point(this->x -= that.x, this->y -= that.y); } Point operator *= (const Real &that) { return Point(x *= that, y *= that); } bool operator == (const Point &that) const { return x == that.x && y == that.y; } Real cross(const Point &that) const { return x * that.y - y * that.x; } Real dist() { return sqrt((x * x + y * y).get()); } }; typedef Point Vector; struct Segment { Point a, b; Segment(const Point &a, const Point &b) { this->a = a; this->b = b; } Segment() {} bool intersect(const Segment &that) const { Point c = that.a, d = that.b; Vector ab = b - a, cd = d - c, ac = c - a, ad = d - a, ca = a - c, cb = b - c; return ab.cross(ac) * ab.cross(ad) < 0 && cd.cross(ca) * cd.cross(cb) < 0; } Point getLineIntersection(const Segment &that) const { Vector u = a - that.a, v = b - a, w = that.b - that.a; Real t = w.cross(u) / v.cross(w); return a + v * t; } Real Distance(Point P) { Point A = a, B = b; if (A == B) return (P - A).dist(); Vector v1 = B - A, v2 = P - A, v3 = P - B; if (v1 * v2 < 0) return v2.dist(); if (v1 * v3 > 0) return v3.dist(); return v1.cross(v2).abs() / v1.dist(); } }; const int maxn = 55; double PI = acos(-1.0); Point p[maxn]; int n; Real getAngel(Point o, Point a, Point b) { a -= o; b -= o; Real ans = acos((a * b / a.dist() / b.dist()).get()); return a.cross(b) <= 0? ans : -ans; } bool inPolygon(Point o) { Real total = 0; for (int i = 0; i < n; i ++) { total += getAngel(o, p[i], p[(i + 1) % n]); } return total.abs() > PI; } Real getR(Point o) { Real ans = 1e9; for (int i = 0; i < n; i ++) { Segment seg(p[i], p[(i + 1) % n]); umin(ans, seg.Distance(o)); } return ans; } int main() { #ifndef ONLINE_JUDGE freopen("in.txt", "r", stdin); //freopen("out.txt", "w", stdout); #endif // ONLINE_JUDGE while (cin >> n, n) { p[0].read(); Real maxx = p[0].x, minx = p[0].x, maxy = p[0].y, miny = p[0].y; for (int i = 1; i < n; i ++) { p[i].read(); umax(maxx, p[i].x); umin(minx, p[i].x); umax(maxy, p[i].y); umin(miny, p[i].y); } Real R; R.read(); Point a(minx, miny), b(maxx, maxy); bool ok = false; for (int i = 0; !ok && i < n; i ++) { Real deta = (b - a).dist() / 4; Point O = (p[i] + p[(i + 1) % n]) * 0.5; int cnt = 0; while (!ok && deta > 0 && cnt < 100) { for (int j = 0; ; j ++) { double randnum = rand(); Point newp(O.x + deta * sin(randnum), O.y + deta * cos(randnum)); if (!inPolygon(newp)) continue; Real buf = getR(newp); if (buf > getR(O) || j > 4) { /** 这里考虑了概率因素 **/ if (buf >= R) ok = true; O = newp; break; } } deta *= 0.8; cnt ++; } } puts(ok? "Yes" : "No"); } }