浙江大学ACM模板汇编.docx
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浙江大学ACM模板汇编
ZhejiangUniversity
ICPCTeam
RoutineLibrary
byWishingBone(Dec.2002)
LastUpdate(Nov.2004)byRiveria
1、几何
1.1注意
1.注意舍入方式(0.5的舍入方向);防止输出-0.
2.几何题注意多测试不对称数据.
3.整数几何注意xmult和dmult是否会出界;
符点几何注意eps的使用.
4.避免使用斜率;注意除数是否会为0.
5.公式一定要化简后再代入.
6.判断同一个2*PI域内两角度差应该是
abs(a1-a2)pi+pi-beta;
相等应该是
abs(a1-a2)pi+pi-eps;
7.需要的话尽量使用atan2,注意:
atan2(0,0)=0,
atan2(1,0)=pi/2,atan2(-1,0)=-pi/2,atan2(0,1)=0,atan2(0,-1)=pi.
8.crossproduct=|u|*|v|*sin(a)
dotproduct=|u|*|v|*cos(a)
9.(P1-P0)x(P2-P0)结果的意义:
正:
在顺时针(0,pi)内
负:
在逆时针(0,pi)内
0:
,共线,夹角为0或pi
10.误差限缺省使用1e-8!
1.2几何公式
三角形:
1.半周长P=(a+b+c)/2
2.面积S=aHa/2=absin(C)/2=sqrt(P(P-a)(P-b)(P-c))
3.中线Ma=sqrt(2(b^2+c^2)-a^2)/2=sqrt(b^2+c^2+2bccos(A))/2
4.角平分线Ta=sqrt(bc((b+c)^2-a^2))/(b+c)=2bccos(A/2)/(b+c)
5.高线Ha=bsin(C)=csin(B)=sqrt(b^2-((a^2+b^2-c^2)/(2a))^2)
6.内切圆半径r=S/P=asin(B/2)sin(C/2)/sin((B+C)/2)
=4Rsin(A/2)sin(B/2)sin(C/2)=sqrt((P-a)(P-b)(P-c)/P)
=Ptan(A/2)tan(B/2)tan(C/2)
7.外接圆半径R=abc/(4S)=a/(2sin(A))=b/(2sin(B))=c/(2sin(C))
四边形:
D1,D2为对角线,M对角线中点连线,A为对角线夹角
1.a^2+b^2+c^2+d^2=D1^2+D2^2+4M^2
2.S=D1D2sin(A)/2
(以下对圆的内接四边形)
3.ac+bd=D1D2
4.S=sqrt((P-a)(P-b)(P-c)(P-d)),P为半周长
正n边形:
R为外接圆半径,r为内切圆半径
1.中心角A=2PI/n
2.内角C=(n-2)PI/n
3.边长a=2sqrt(R^2-r^2)=2Rsin(A/2)=2rtan(A/2)
4.面积S=nar/2=nr^2tan(A/2)=nR^2sin(A)/2=na^2/(4tan(A/2))
圆:
1.弧长l=rA
2.弦长a=2sqrt(2hr-h^2)=2rsin(A/2)
3.弓形高h=r-sqrt(r^2-a^2/4)=r(1-cos(A/2))=atan(A/4)/2
4.扇形面积S1=rl/2=r^2A/2
5.弓形面积S2=(rl-a(r-h))/2=r^2(A-sin(A))/2
棱柱:
1.体积V=Ah,A为底面积,h为高
2.侧面积S=lp,l为棱长,p为直截面周长
3.全面积T=S+2A
棱锥:
1.体积V=Ah/3,A为底面积,h为高
(以下对正棱锥)
2.侧面积S=lp/2,l为斜高,p为底面周长
3.全面积T=S+A
棱台:
1.体积V=(A1+A2+sqrt(A1A2))h/3,A1.A2为上下底面积,h为高
(以下为正棱台)
2.侧面积S=(p1+p2)l/2,p1.p2为上下底面周长,l为斜高
3.全面积T=S+A1+A2
圆柱:
1.侧面积S=2PIrh
2.全面积T=2PIr(h+r)
3.体积V=PIr^2h
圆锥:
1.母线l=sqrt(h^2+r^2)
2.侧面积S=PIrl
3.全面积T=PIr(l+r)
4.体积V=PIr^2h/3
圆台:
1.母线l=sqrt(h^2+(r1-r2)^2)
2.侧面积S=PI(r1+r2)l
3.全面积T=PIr1(l+r1)+PIr2(l+r2)
4.体积V=PI(r1^2+r2^2+r1r2)h/3
球:
1.全面积T=4PIr^2
2.体积V=4PIr^3/3
球台:
1.侧面积S=2PIrh
2.全面积T=PI(2rh+r1^2+r2^2)
3.体积V=PIh(3(r1^2+r2^2)+h^2)/6
球扇形:
1.全面积T=PIr(2h+r0),h为球冠高,r0为球冠底面半径
2.体积V=2PIr^2h/3
1.3多边形
#include
#include
#defineMAXN1000
#defineoffset10000
#defineeps1e-8
#definezero(x)(((x)>0?
(x):
-(x))#define_sign(x)((x)>eps?
1:
((x)<-eps?
2:
0))
structpoint{doublex,y;};
structline{pointa,b;};
doublexmult(pointp1,pointp2,pointp0){
return(p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
//判定凸多边形,顶点按顺时针或逆时针给出,允许相邻边共线
intis_convex(intn,point*p){
inti,s[3]={1,1,1};
for(i=0;is[_sign(xmult(p[(i+1)%n],p[(i+2)%n],p[i]))]=0;
returns[1]|s[2];
}
//判定凸多边形,顶点按顺时针或逆时针给出,不允许相邻边共线
intis_convex_v2(intn,point*p){
inti,s[3]={1,1,1};
for(i=0;is[_sign(xmult(p[(i+1)%n],p[(i+2)%n],p[i]))]=0;
returns[0]&&s[1]|s[2];
}
//判点在凸多边形内或多边形边上,顶点按顺时针或逆时针给出
intinside_convex(pointq,intn,point*p){
inti,s[3]={1,1,1};
for(i=0;is[_sign(xmult(p[(i+1)%n],q,p[i]))]=0;
returns[1]|s[2];
}
//判点在凸多边形内,顶点按顺时针或逆时针给出,在多边形边上返回0
intinside_convex_v2(pointq,intn,point*p){
inti,s[3]={1,1,1};
for(i=0;is[_sign(xmult(p[(i+1)%n],q,p[i]))]=0;
returns[0]&&s[1]|s[2];
}
//判点在任意多边形内,顶点按顺时针或逆时针给出
//on_edge表示点在多边形边上时的返回值,offset为多边形坐标上限
intinside_polygon(pointq,intn,point*p,inton_edge=1){
pointq2;
inti=0,count;
while(ifor(count=i=0,q2.x=rand()+offset,q2.y=rand()+offset;iif(zero(xmult(q,p[i],p[(i+1)%n]))&&(p[i].x-q.x)*(p[(i+1)%n].x-q.x)returnon_edge;
elseif(zero(xmult(q,q2,p[i])))
break;
elseif(xmult(q,p[i],q2)*xmult(q,p[(i+1)%n],q2)<-eps&&xmult(p[i],q,p[(i+1)%n])*xmult(p[i],q2,p[(i+1)%n])<-eps)
count++;
returncount&1;
}
inlineintopposite_side(pointp1,pointp2,pointl1,pointl2){
returnxmult(l1,p1,l2)*xmult(l1,p2,l2)<-eps;
}
inlineintdot_online_in(pointp,pointl1,pointl2){
returnzero(xmult(p,l1,l2))&&(l1.x-p.x)*(l2.x-p.x)}
//判线段在任意多边形内,顶点按顺时针或逆时针给出,与边界相交返回1
intinside_polygon(pointl1,pointl2,intn,point*p){
pointt[MAXN],tt;
inti,j,k=0;
if(!
inside_polygon(l1,n,p)||!
inside_polygon(l2,n,p))
return0;
for(i=0;iif(opposite_side(l1,l2,p[i],p[(i+1)%n])&&opposite_side(p[i],p[(i+1)%n],l1,l2))
return0;
elseif(dot_online_in(l1,p[i],p[(i+1)%n]))
t[k++]=l1;
elseif(dot_online_in(l2,p[i],p[(i+1)%n]))
t[k++]=l2;
elseif(dot_online_in(p[i],l1,l2))
t[k++]=p[i];
for(i=0;ifor(j=i+1;jtt.x=(t[i].x+t[j].x)/2;
tt.y=(t[i].y+t[j].y)/2;
if(!
inside_polygon(tt,n,p))
return0;
}
return1;
}
pointintersection(lineu,linev){
pointret=u.a;
doublet=((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))
/((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
ret.x+=(u.b.x-u.a.x)*t;
ret.y+=(u.b.y-u.a.y)*t;
returnret;
}
pointbarycenter(pointa,pointb,pointc){
lineu,v;
u.a.x=(a.x+b.x)/2;
u.a.y=(a.y+b.y)/2;
u.b=c;
v.a.x=(a.x+c.x)/2;
v.a.y=(a.y+c.y)/2;
v.b=b;
returnintersection(u,v);
}
//多边形重心
pointbarycenter(intn,point*p){
pointret,t;
doublet1=0,t2;
inti;
ret.x=ret.y=0;
for(i=1;iif(fabs(t2=xmult(p[0],p[i],p[i+1]))>eps){
t=barycenter(p[0],p[i],p[i+1]);
ret.x+=t.x*t2;
ret.y+=t.y*t2;
t1+=t2;
}
if(fabs(t1)>eps)
ret.x/=t1,ret.y/=t1;
returnret;
}
1.4多边形切割
//多边形切割
//可用于半平面交
#defineMAXN100
#defineeps1e-8
#definezero(x)(((x)>0?
(x):
-(x))structpoint{doublex,y;};
doublexmult(pointp1,pointp2,pointp0){
return(p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
intsame_side(pointp1,pointp2,pointl1,pointl2){
returnxmult(l1,p1,l2)*xmult(l1,p2,l2)>eps;
}
pointintersection(pointu1,pointu2,pointv1,pointv2){
pointret=u1;
doublet=((u1.x-v1.x)*(v1.y-v2.y)-(u1.y-v1.y)*(v1.x-v2.x))
/((u1.x-u2.x)*(v1.y-v2.y)-(u1.y-u2.y)*(v1.x-v2.x));
ret.x+=(u2.x-u1.x)*t;
ret.y+=(u2.y-u1.y)*t;
returnret;
}
//将多边形沿l1,l2确定的直线切割在side侧切割,保证l1,l2,side不共线
voidpolygon_cut(int&n,point*p,pointl1,pointl2,pointside){
pointpp[100];
intm=0,i;
for(i=0;iif(same_side(p[i],side,l1,l2))
pp[m++]=p[i];
if(!
same_side(p[i],p[(i+1)%n],l1,l2)&&!
(zero(xmult(p[i],l1,l2))&&zero(xmult(p[(i+1)%n],l1,l2))))
pp[m++]=intersection(p[i],p[(i+1)%n],l1,l2);
}
for(n=i=0;iif(!
i||!
zero(pp[i].x-pp[i-1].x)||!
zero(pp[i].y-pp[i-1].y))
p[n++]=pp[i];
if(zero(p[n-1].x-p[0].x)&&zero(p[n-1].y-p[0].y))
n--;
if(n<3)
n=0;
}
1.5浮点函数
//浮点几何函数库
#include
#defineeps1e-8
#definezero(x)(((x)>0?
(x):
-(x))structpoint{doublex,y;};
structline{pointa,b;};
//计算crossproduct(P1-P0)x(P2-P0)
doublexmult(pointp1,pointp2,pointp0){
return(p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
doublexmult(doublex1,doubley1,doublex2,doubley2,doublex0,doubley0){
return(x1-x0)*(y2-y0)-(x2-x0)*(y1-y0);
}
//计算dotproduct(P1-P0).(P2-P0)
doubledmult(pointp1,pointp2,pointp0){
return(p1.x-p0.x)*(p2.x-p0.x)+(p1.y-p0.y)*(p2.y-p0.y);
}
doubledmult(doublex1,doubley1,doublex2,doubley2,doublex0,doubley0){
return(x1-x0)*(x2-x0)+(y1-y0)*(y2-y0);
}
//两点距离
doubledistance(pointp1,pointp2){
returnsqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}
doubledistance(doublex1,doubley1,doublex2,doubley2){
returnsqrt((x1-x2)*(x1-x2)+(y1-y2)*(y1-y2));
}
//判三点共线
intdots_inline(pointp1,pointp2,pointp3){
returnzero(xmult(p1,p2,p3));
}
intdots_inline(doublex1,doubley1,doublex2,doubley2,doublex3,doubley3){
returnzero(xmult(x1,y1,x2,y2,x3,y3));
}
//判点是否在线段上,包括端点
intdot_online_in(pointp,linel){
returnzero(xmult(p,l.a,l.b))&&(l.a.x-p.x)*(l.b.x-p.x)}
intdot_online_in(pointp,pointl1,pointl2){
returnzero(xmult(p,l1,l2))&&(l1.x-p.x)*(l2.x-p.x)}
intdot_online_in(doublex,doubley,doublex1,doubley1,doublex2,doubley2){
returnzero(xmult(x,y,x1,y1,x2,y2))&&(x1-x)*(x2-x)}
//判点是否在线段上,不包括端点
intdot_online_ex(pointp,linel){
returndot_online_in(p,l)&&(!
zero(p.x-l.a.x)||!
zero(p.y-l.a.y))&&(!
zero(p.x-l.b.x)||!
zero(p.y-l.b.y));
}
intdot_online_ex(pointp,pointl1,pointl2){
returndot_online_in(p,l1,l2)&&(!
zero(p.x-l1.x)||!
zero(p.y-l1.y))&&(!
zero(p.x-l2.x)||!
zero(p.y-l2.y));
}
intdot_online_ex(doublex,doubley,doublex1,doubley1,doublex2,doubley2){
returndot_online_in(x,y,x1,y1,x2,y2)&&(!
zero(x-x1)||!
zero(y-y1))&&(!
zero(x-x2)||!
zero(y-y2));
}
//判两点在线段同侧,点在线段上返回0
intsame_side(pointp1,pointp2,linel){
returnxmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)>eps;
}
intsame_side(pointp1,pointp2,pointl1,pointl2){
returnxmult(l1,p1,l2)*xmult(l1,p2,l2)>eps;
}
//判两点在线段异侧,点在线段上返回0
intopposite_side(pointp1,pointp2,linel){
returnxmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)<-eps;
}
intopposite_side(pointp1,pointp2,pointl1,pointl2){
returnxmult(l1,p1,l2)*xmult(l1,p2,l2)<-eps;
}
//判两直线平行
intparallel(lineu,linev){
returnzero((u.a.x-u.b.x)*(v.a.y-v.b.y)-(v.a.x-v.b.x)*(u.a.y-u.b.y));
}
intparallel(pointu1,pointu2,pointv1,pointv2){
returnzero((u1.x-u2.x)*(v1.y-v2.y)-(v1.x-v2.x)*(u1.y-u2.y));
}
//判两直线垂直
intperpendicular(lineu,linev){
returnzero((u.a.x-u.b.x)*(v.a.x-v.b.x)+(u.a.y-u.b.y)*(v.a.y-v.b.y));
}
intperpendicular(pointu1,pointu2,pointv1,pointv2){
returnzero((u1.x-u2.x)*(v1.x-v2.x)+(u1.y-u2.y)*(v1.y-v2.y));
}
//判两线段相交,包括端点和部分重合
intintersect_in(lineu,linev){
if(!
dots_inline(u.a,u.b,v.a)||!
dots_inline(u.a,u.b,v.b))
return!
same_side(u.a,u.b,v)&&!