{"id":150,"date":"2025-01-12T12:53:19","date_gmt":"2025-01-12T04:53:19","guid":{"rendered":"http:\/\/150.230.63.147\/blog\/?p=150"},"modified":"2025-01-12T12:53:19","modified_gmt":"2025-01-12T04:53:19","slug":"%e5%a6%82%e4%bd%95%e8%a8%88%e7%ae%97n%e5%80%8b%e5%9c%93%e7%9a%84%e8%81%af%e9%9b%86%e9%9d%a2%e7%a9%8d","status":"publish","type":"post","link":"https:\/\/blog.seekrealthing.net\/blog\/?p=150","title":{"rendered":"\u5982\u4f55\u8a08\u7b97n\u500b\u5713\u7684\u806f\u96c6\u9762\u7a4d"},"content":{"rendered":"\n

\u5982\u4f55\u8a08\u7b97n\u500b\u5713\u7684\u806f\u96c6\u9762\u7a4d<\/h1>\n

\u524d\u8a00<\/h3>\n

\u4e00\u822c\u4eba\u7b2c\u4e00\u6b21\u9047\u5230\u9019\u500b\u554f\u984c\uff0c\u53ef\u80fd\u6703\u60f3\u8981\u60f3\u8fa6\u6cd5\u7528\u6392\u5bb9\u539f\u7406\uff0c\u627e\u5713\u4e4b\u9593\u4ea4\u758a\u7684\u51f8\u5305\u4e4b\u985e\u7684\u2026\u3002
\n\u7136\u800c\u6211\u53ea\u8981\u8209\u4e00\u500b\u4f8b\u5b50\uff0c\u4f60\u5c31\u6703\u767c\u73fe\u6211\u5011\u5c31\u7b97\u628a\u51f8\u5305\u627e\u51fa\u4f86\u4e86\uff0c\u6211\u5011\u4e5f\u975e\u5e38\u96e3\u77e5\u9053\u627e\u5230\u7684\u51f8\u5305\u5230\u5e95\u662f\u8981\u6e1b\u6389\u9084\u662f\u8981\u52a0\u4e0a\u9019\u500b\u9762\u7a4d\u3002
\n\"\"
\n\u4f86\u6e90<\/a>
\n\u6211\u5011\u6703\u767c\u73fe\u5305\u542b\u4e2d\u7a7a\u7684\u90a3\u500b\u56db\u908a\u5f62\uff0c\u6211\u5011\u4e0d\u5bb9\u6613\u77e5\u9053\u5230\u5e95\u90a3\u584a\u8981\u4e0d\u8981\u52a0\u4e0a\u3002<\/p>\n

\u60f3\u6cd5<\/h3>\n

\u6211\u5011\u53ef\u4ee5\u5229\u7528\u7dda\u7a4d\u5206\uff0c\u5c0d\u5713\u7684\u806f\u96c6\u7684\u908a\u754c\u9006\u6642\u9418\u7a4d\u5206\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u9762\u7a4d\u3002
\n\u9996\u5148\u6211\u5011\u9700\u8981\u5f97\u5230\u908a\u754c\uff0c\u4e5f\u5c31\u662f\u6c92\u6709\u88ab\u5225\u7684\u5713\u5305\u542b\u7684\u5713\u5f27\u3002\u8981\u5f97\u5230\u9019\u500b\u6211\u5011\u53ef\u4ee5\u5c0d\u65bc\u67d0\u500b\u5713$c_i$\u53bb\u904d\u6b77\u6240\u6709\u5225\u7684\u5713$c_j,i\\neq j$\uff0c\u628a\u4ea4\u758a\u5f97\u5230\u7684\u5713\u5f27\u8a18\u4e0b\u4f86\uff0c\u90a3\u9ebc\u6c92\u88ab\u8a18\u4e0b\u4f86\u7684\u5713\u5f27\u5c31\u662f\u6211\u5011\u8981\u7a4d\u5206\u7684\u90e8\u5206\u4e86\u3002
\n\u5c0d\u65bc\u5713$a,b$\uff0c\u6211\u5011\u53ea\u9700\u8981\u8003\u616e\u4e09\u7a2e\u72c0\u6cc1($d=|a.center-b.center|$):<\/p>\n

    \n
  1. same(a.radius+b.radius,d)<\/li>\n
  2. d<=abs(a.radius-b.radius)+eps<\/li>\n
  3. d<abs(a.radius+b.radius)-eps<\/li>\n<\/ol>\n

    \u63a5\u8457\u6211\u5011\u53ef\u4ee5\u5229\u7528$d,a.radius,b.radius$\u548c\u9918\u5f26\u5b9a\u7406\u4f86\u8a08\u7b97\u5713\u5f27\u7684\u89d2\u5ea6\u7bc4\u570d(\u4ee5\u5e73\u884c$x$\u8ef8\u70ba\u89d2\u5ea6$0$)\u3002<\/p>\n

    \u63a5\u8457\u6211\u5011\u4f86\u770b\u5c0d\u65bc\u4e00\u500b\u5713$c_i$\uff0c\u5713\u5fc3\u5728$(x_0,y_0)$\uff0c\u5c0d\u65bc\u5713\u5f27$[0,\\theta]$\u7a4d\u5206\u6240\u5f97\u5230\u7684\u503c:
    \n$\\int_\\Gamma \ud835\udc65\ud835\udc51\ud835\udc66\u2212\ud835\udc66\ud835\udc51\ud835\udc65=\\int_\\Gamma (\ud835\udc65_0+\ud835\udc5f\\cos\\theta)\\frac{dy}{d\\theta}\u2212(\ud835\udc66_0+\ud835\udc5f\\sin\\theta)\\frac{dx}{d\\theta} d\\theta$
    \n$=\\int_\\Gamma (\ud835\udc65_0+\ud835\udc5f\\cos\\theta)(r\\cos\\theta)\u2212(\ud835\udc66_0+\ud835\udc5f\\sin\\theta)(-r\\sin\\theta) d\\theta$
    \n$=\ud835\udc5f[\ud835\udc65_0\\sin\\theta\u2212\ud835\udc66_0\\cos\\theta+r\\theta]_0^\\theta=\ud835\udc5f[\ud835\udc65_0\\sin\\theta\u2212\ud835\udc66_0\\cos\\theta+\ud835\udc66_0+r\\theta]$<\/p>\n

    \u6700\u5f8c\u6211\u5011\u76f4\u63a5\u4e0acode\uff0c\u53ef\u4ee5\u76f4\u63a5\u770bCoverSegment\u548cCircleUnionArea\u9019\u5169\u500bfunction\u3002<\/p>\n

    \u7a0b\u5f0f\u78bc:<\/h3>\n
    const db eps=1e-8,pi=acos(-1);  \nbool same(db a,db b){return abs(a-b)<eps;}  \ndb sq(db x){return x*x;}  \nstruct P{  \n  db x,y;  \n  P():x(0),y(0){}  \n  P(db x,db y):x(x),y(y){}  \n  P operator+(P b){return P(x+b.x,y+b.y);}  \n  P operator-(P b){return P(x-b.x,y-b.y);}  \n  P operator*(db b){return P(x*b,y*b);}  \n  P operator\/(db b){return P(x\/b,y\/b);}  \n  db operator*(P b){return x*b.x+y*b.y;}  \n  db operator^(P b){return x*b.y-y*b.x;}  \n  db abs(){return hypot(x,y);}  \n  P unit(){return *this\/abs();}  \n  P spin(db o){  \n    db c=cos(o),s=sin(o);  \n    return P(c*x-s*y,s*x+c*y);  \n  }  \n  db angle(){return atan2(y,x);}  \n};  \nstruct C{  \n  P c;  \n  db r;  \n  C(P c=P(0,0),db r=0):c(c),r(r){}  \n};  \nvector<pair<db,db>> CoverSegment(C &a,C &b){  \n  db d=(a.c-b.c).abs();  \n  vector<pair<db,db>> res;  \n  if(same(a.r+b.r,d));  \n  else if(d<=abs(a.r-b.r)+eps){  \n    if(a.r<b.r)res.pb({0,2*pi});  \n  }else if(d<abs(a.r+b.r)-eps){  \n    db o=acos((sq(a.r)+sq(d)-sq(b.r))\/(2*a.r*d)),z=(b.c-a.c).angle();  \n    if(z<0)z+=2*pi;  \n    db l=z-o,r=z+o;  \n    if(l<0)l+=2*pi;if(r>2*pi)r-=2*pi;  \n    if(l>r)res.pb({l,2*pi}),res.pb({0,r});  \n    else res.pb({l,r});  \n  }return res;  \n}  \ndb CircleUnionArea(vector<C> c){\/\/circles should be distinct  \n  db res=0;  \n  rep(i,0,SZ(c)){  \n    db w=0;vector<pair<db,db>> s={{2*pi,9}},z;  \n    rep(j,0,SZ(c))if(i!=j){  \n      z=CoverSegment(c[i],c[j]);  \n      for(auto &e:z)s.pb(e);  \n    }sort(all(s));  \n    auto f=[&](db t){return c[i].r*(c[i].c.x*sin(t)-c[i].c.y*cos(t)+c[i].c.y+c[i].r*t);};  \n    for(auto &e:s){  \n      if(e.fi>w)res+=f(e.fi)-f(w);  \n      w=max(w,e.se);  \n    }  \n  }return res\/2;  \n}  \nconst int _n=110;  \ndb t,n,r,x,y;  \nvector<C> a;  \nvector<PII> ps;  \nmain(void) {ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);  \n  cin>>n>>r;rep(i,0,n){cin>>x>>y;ps.pb({x,y});}  \n  int nn=unique(all(ps))-ps.begin();rep(i,0,nn)a.pb(C({ps[i].fi,ps[i].se},r));  \n  cout<<fixed<<setprecision(2)<<round(CircleUnionArea(a)*100)\/100<<'\\n';  \n  return 0;  \n}  \n<\/code><\/pre>\n
    \u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeCodepad\u770b
    \n    Codepad<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[3],"tags":[22],"class_list":["post-150","post","type-post","status-publish","format-standard","hentry","category-ojcode","tag-22"],"jetpack_publicize_connections":[],"yoast_head":"\n\u5982\u4f55\u8a08\u7b97n\u500b\u5713\u7684\u806f\u96c6\u9762\u7a4d - Logic Formatting<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blog.seekrealthing.net\/blog\/?p=150\" \/>\n<meta property=\"og:locale\" content=\"zh_TW\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u5982\u4f55\u8a08\u7b97n\u500b\u5713\u7684\u806f\u96c6\u9762\u7a4d - 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