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ZOJ 1450 Minimal Circle 点集的最小圆覆盖
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From: http://blog.csdn.net/zmx354/article/details/17076267
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給定一個點集，求出能覆蓋點集內所有點的半徑最小的圓。包含點在圓上的情況。個人感覺算是比較麻煩的計算幾何模板了。
在網上看了很多解題，大多數都摘抄自《求一個包含點集所有點的最小圓的算法》這篇論文。
論文中提出的算法一共分一下四步：
第 1 步. 在點集中任取 3 點 A , B , C .
第 2 步. 作一個包 含 A , B , C 三點的最小 圓. 圓周可 能通過這 3 點( 如圖 1 所示) , 也 可能只通過
其中兩點, 但包含第 3 點. 后一種情況圓周上的兩點一定是 位于圓的一條直徑的兩端.
第 3 步. 在點集中找 出距離第 2 步所建圓圓心最遠的點 D . 若 D 點已在圓內或圓周上, 則該
圓即為所求的圓, 算法結束. 否則, 執行第 4 步.
第 4 步. 在 A , B , C , D 中 選 3 個點, 使 由它們生成的一個 包含這 4 點的圓為最 小. 這 3 點成
為新的 A , B 和 C , 返回執 行第 2 步 .
若在第 4 步生成的圓的圓周只通過 A , B , C , D 中的兩點, 則圓周上的兩點取成新的 A 和 B ,
從另兩點中任取一點作為新的 C .
但是我讀的時候感覺其中最關鍵的第四步說的最為模糊了。
下面是我自己的一些關于求四邊形即四個點的最小圓覆蓋的一些想法。
1. 若此四邊形中存在大于等于180度的內角時，則可看作求三角形的最小覆蓋圓（鈍角三角形為以長邊做直徑的圓，否則為該三角形外接圓）。
2. 若存在兩個銳角，兩個非銳角，則必為兩個銳角頂點連線為直徑的圓。
3. 若有三個銳角，一個非銳角，則必為三個銳角頂點組成的三角形的外接圓。
4. 若有四個直角，則任選其中三點組成三角形求其外接圓即可。
解決了第四步，剩下的就是按照上述算法求解了。
下面是AC代碼，里面還有一些其他的模板，一直想攢一套完整的模板，所以就沒刪，求最小點集覆蓋圓的函數為 Cal_Min_Circle(P *p,int n)。
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#include?<iostream>??#include?<cstring>??#include?<cstdlib>??#include?<cstdio>??#include?<queue>??#include?<cmath>??#include?<algorithm>????#define?LL?long?long??#define?PI?(acos(-1.0))??#define?EPS?(1e-10)????using?namespace?std;????struct?P??{??????double?x,y,a;??}p[1100],cp[1100];????struct?L??{??????P?p1,p2;??}?line[50];????double?X_Mul(P?a1,P?a2,P?b1,P?b2)??{??????P?v1?=?{a2.x-a1.x,a2.y-a1.y},v2?=?{b2.x-b1.x,b2.y-b1.y};??????return?v1.x*v2.y?-?v1.y*v2.x;??}????double?Cal_Point_Dis(P?p1,P?p2)??{??????return?sqrt((p1.x-p2.x)*(p1.x-p2.x)?+?(p1.y-p2.y)*(p1.y-p2.y));??}????double?Cal_Point_Dis_Pow_2(P?p1,P?p2)??{??????return?(p1.x-p2.x)*(p1.x-p2.x)?+?(p1.y-p2.y)*(p1.y-p2.y);??}????P?Cal_Segment_Cross_Point(P?a1,P?a2,P?b1,P?b2)??{??????double?t?=?X_Mul(b1,b2,b1,a1)?/?X_Mul(a1,a2,b1,b2);??????P?cp?=?{a1.x+(a2.x-a1.x)*t,a1.y+(a2.y-a1.y)*t};??????return?cp;??}????//規范相交??bool?Is_Seg_Cross_Without_Point(P?a1,P?a2,P?b1,P?b2)??{??????double?xm1?=?X_Mul(a1,a2,a1,b1);??????double?xm2?=?X_Mul(a1,a2,a1,b2);??????double?xm3?=?X_Mul(b1,b2,b1,a1);??????double?xm4?=?X_Mul(b1,b2,b1,a2);????????return?(xm1*xm2?<?(-EPS)?&&?xm3*xm4?<?(-EPS));??}????//向量ab與X軸正方向的夾角??double?Cal_Angle(P?t,P?p)??{??????return?((t.x-p.x)*(t.x-p.x)?+?1?-?(t.x-p.x-1)*(t.x-p.x-1))/(2*sqrt((t.x-p.x)*(t.x-p.x)?+?(t.y-p.y)*(t.y-p.y)));??}????//計算?∠b2.a.b1?的大小??double?Cal_Angle_Three_Point(P?a,P?b1,P?b2)??{??????double?l1?=?Cal_Point_Dis_Pow_2(b1,a);??????double?l2?=?Cal_Point_Dis_Pow_2(b2,a);??????double?l3?=?Cal_Point_Dis_Pow_2(b1,b2);????????return?acos(?(l1+l2-l3)/(2*sqrt(l1*l2))?);??}????bool?cmp_angle(P?p1,P?p2)??{??????return?(p1.a?<?p2.a?||?(fabs(p1.a-p2.a)?<?EPS?&&?p1.y?<?p2.y)?||(fabs(p1.a-p2.a)?<?EPS?&&?fabs(p1.y-p2.y)?<?EPS?&&?p1.x?<?p2.x)???);??}????//p為點集??應該保證至少有三點不共線??//n?為點集大小??//返回值為凸包上點集上的大小??//凸包上的點存儲在cp中??int?Creat_Convex_Hull(P?*p,int?n)??{??????int?i,top;??????P?re;?//re?為建立極角排序時的參考點??下左點????????for(re?=?p[0],i?=?1;?i?<?n;?++i)??????{??????????if(re.y?>?p[i].y?||?(fabs(re.y-p[i].y)?<?EPS?&&?re.x?>?p[i].x))??????????????re?=?p[i];??????}????????for(i?=?0;?i?<?n;?++i)??????{??????????if(p[i].x?==?re.x?&&?p[i].y?==?re.y)??????????{??????????????p[i].a?=?-2;??????????}??????????else?p[i].a?=?Cal_Angle(re,p[i]);??????}????????sort(p,p+n,cmp_angle);????????top?=0?;??????cp[top++]?=?p[0];??????cp[top++]?=?p[1];????????for(i?=?2?;?i?<?n;)??????{??????????if(top?<?2?||?X_Mul(cp[top-2],cp[top-1],cp[top-2],p[i])?>?EPS)??????????{??????????????cp[top++]?=?p[i++];??????????}??????????????????????else??????????{??????????????--top;??????????}??????}????????return?top;??}????bool?Is_Seg_Parallel(P?a1,P?a2,P?b1,P?b2)??{??????double?xm1?=?X_Mul(a1,a2,a1,b1);??????double?xm2?=?X_Mul(a1,a2,a1,b2);????????return?(fabs(xm1)?<?EPS?&&?fabs(xm2)?<?EPS);??}????bool?Point_In_Seg(P?m,P?a1,P?a2)??{??????double?l1?=?Cal_Point_Dis(m,a1);??????double?l2?=?Cal_Point_Dis(m,a2);??????double?l3?=?Cal_Point_Dis(a1,a2);????????return?(fabs(l1+l2-l3)?<?EPS);??}????//計算三角形外接圓圓心??P?Cal_Triangle_Circumcircle_Center(P?a,P?b,P?c)??{????????P?mp1?=?{(b.x+a.x)/2,(b.y+a.y)/2},mp2?=?{(b.x+c.x)/2,(b.y+c.y)/2};??????P?v1?=?{a.y-b.y,b.x-a.x},v2?=?{c.y-b.y,b.x-c.x};????????P?p1?=?{mp1.x+v1.x,mp1.y+v1.y},p2?=?{mp2.x+v2.x,mp2.y+v2.y};????????return?Cal_Segment_Cross_Point(mp1,p1,mp2,p2);??}????bool?Is_Acute_Triangle(P?p1,P?p2,P?p3)??{??????//三點共線??????if(fabs(X_Mul(p1,p2,p1,p3))?<?EPS)??????{??????????return?false;??????}????????double?a?=?Cal_Angle_Three_Point(p1,p2,p3);??????if(a?>?PI?||?fabs(a-PI)?<?EPS)??????{??????????return?false;??????}??????a?=?Cal_Angle_Three_Point(p2,p1,p3);??????if(a?>?PI?||?fabs(a-PI)?<?EPS)??????{??????????return?false;??????}??????a?=?Cal_Angle_Three_Point(p3,p1,p2);??????if(a?>?PI?||?fabs(a-PI)?<?EPS)??????{??????????return?false;??????}??????return?true;??}????bool?Is_In_Circle(P?center,double?rad,P?p)??{??????double?dis?=?Cal_Point_Dis(center,p);??????return?(dis?<?rad?||?fabs(dis-rad)?<?EPS);??}????P?Cal_Seg_Mid_Point(P?p2,P?p1)??{??????P?mp?=?{(p2.x+p1.x)/2,(p1.y+p2.y)/2};??????return?mp;??}????void?Cal_Min_Circle(P?*p,int?n)??{??????if(n?==?0)??????????return?;??????if(n?==?1)??????{??????????printf("%.2lf?%.2lf?%.2lf\n",p[0].x,p[0].y,0.00);??????????return?;??????}??????if(n?==?2)??????{??????????printf("%.2lf?%.2lf?%.2lf\n",(p[1].x+p[0].x)/2,(p[0].y+p[1].y)/2,Cal_Point_Dis(p[0],p[1])/2);??????????return?;??????}????????P?center?=?Cal_Seg_Mid_Point(p[0],p[1]),tc1;????????double?dis,temp,rad?=?Cal_Point_Dis(p[0],center),tra1;??????int?i,site,a?=?0,b?=?1,c?=?2,d,s1,s2,tp;????????for(dis?=?-1,site?=?0,i?=?0;?i?<?n;?++i)??????{??????????temp?=?Cal_Point_Dis(center,p[i]);??????????if(temp?>?dis)??????????{??????????????dis?=?temp;??????????????site?=?i;??????????}??????}????????while(?Is_In_Circle(center,rad,p[site])?==?false?)??????{??????????d?=?site;?????????//?printf("a?=?%d?b?=?%d?c?=?%d?d?=?%d\n",a,b,c,d);????????????//printf("x?=?%.2lf??y?=?%.2lf\n",center.x,center.y);???????????//?printf("rad?=?%.2lf\n\n",rad);?????????//?getchar();????????????double?l1?=?Cal_Point_Dis(p[a],p[d]);??????????double?l2?=?Cal_Point_Dis(p[b],p[d]);??????????double?l3?=?Cal_Point_Dis(p[c],p[d]);????????????if((l1?>?l2?||?fabs(l1-l2)?<?EPS)?&&?(l1?>?l3?||?fabs(l1-l3)?<?EPS))??????????{??????????????s1?=?a,s2?=?d,tp?=?b;??????????}??????????else?if((l2?>?l1?||?fabs(l1-l2)?<?EPS)?&&?(l2?>?l3?||?fabs(l2-l3)?<?EPS))??????????{??????????????s1?=?b,s2?=?d,tp?=?c;??????????}??????????else??????????{??????????????s1?=?c,s2?=?d,tp?=?a;??????????}????????????center?=?Cal_Seg_Mid_Point(p[s1],p[s2]);??????????rad?=?Cal_Point_Dis(center,p[s1]);????????????if(?Is_In_Circle(center,rad,p[a])?==?false?||?Is_In_Circle(center,rad,p[b])?==?false?||?Is_In_Circle(center,rad,p[c])?==?false?)??????????{??????????????center?=?Cal_Triangle_Circumcircle_Center(p[a],p[c],p[d]);??????????????rad?=?Cal_Point_Dis(p[d],center);??????????????s1?=?a,s2?=?c,tp?=?d;????????????????if(Is_In_Circle(center,rad,p[b])?==?false)??????????????{??????????????????center?=?Cal_Triangle_Circumcircle_Center(p[a],p[b],p[d]);??????????????????rad?=?Cal_Point_Dis(p[d],center);??????????????????s1?=?a,s2?=?b,tp?=?d;??????????????}??????????????else??????????????{??????????????????tc1?=?Cal_Triangle_Circumcircle_Center(p[a],p[b],p[d]);??????????????????tra1?=?Cal_Point_Dis(p[d],tc1);????????????????????if(tra1?<?rad?&&?Is_In_Circle(tc1,tra1,p[c]))??????????????????{??????????????????????rad?=?tra1,center?=?tc1;??????????????????????s1?=?a,s2?=?b,tp?=?d;??????????????????}??????????????}????????????????if(Is_In_Circle(center,rad,p[c])?==?false)??????????????{??????????????????center?=?Cal_Triangle_Circumcircle_Center(p[c],p[b],p[d]);??????????????????rad?=?Cal_Point_Dis(center,p[d]);??????????????????s1?=?c,s2?=?b,tp?=?d;??????????????}??????????????else??????????????{??????????????????tc1?=?Cal_Triangle_Circumcircle_Center(p[c],p[b],p[d]);??????????????????tra1?=?Cal_Point_Dis(p[d],tc1);??????????????????if(tra1?<?rad?&&?Is_In_Circle(tc1,tra1,p[a]))??????????????????{??????????????????????rad?=?tra1,center?=?tc1;??????????????????????s1?=?b,s2?=?c,tp?=?d;??????????????????}??????????????}??????????}????????????a?=?s1,b?=?s2,c?=?tp;????????????for(dis?=?-1,site?=?0,i?=?0;i?<?n;?++i)??????????{??????????????temp?=?Cal_Point_Dis(center,p[i]);??????????????if(temp?>?dis)??????????????{??????????????????dis?=?temp;??????????????????site?=?i;??????????????}??????????}??????}??????printf("%.2f?%.2f?%.2f\n",center.x,center.y,rad);??}????int?main()??{??????int?i,n;??????while(scanf("%d",&n)?&&?n)??????{??????????for(i?=?0;i?<?n;?++i)??????????{??????????????scanf("%lf?%lf",&p[i].x,&p[i].y);??????????}??????????Cal_Min_Circle(p,n);??????}??}?
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