几何模板.docx
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几何模板
大几何模板
大几何模板1
(1)凸包2
(2)判断两条线段是否相交(平行,不平行)4
(3)三角形的外接圆(已知不在同一直线上的三点求经过三点的圆)4
(4)三角形的垂心内心重心中垂线6
(5)求直线的交点10
(6)根据线段两端点的坐标求垂直平分线上除中点外的另一点11
(7)根据两点坐标求直线方程11
(8)差积的应用12
(9)三角形的面积公式12
(10)三角形的内接圆12
(11)多边形的面积(适合凹多边形)13
(12)判断点是否在线段上14
(13)平面上两个点之间的距离14
(14)p点关于直线L的对称点14
(15)判断一个矩形是否在另一个矩形中15
(16)直线和圆的交点+点关于线的对称点+点到线的距离+直线方程16
(17)判断点是否在多边形内21
(18)N点中三个点组成三角形面积最大28
(19)扇形的重心31
(20)多边形的重心32
(21)判断N点是否共面33
(22)求共线的点最多为多少34
(23)N个矩形的相交的面积36
(24)三角形外接圆+圆的参数方程38
(25)判断线段是否有交点并求交点43
(26)简单多边形的核46
(27)线段重叠+投影50
(28)二分+圆的参数方程53
(29)Pick公式54
(30)根据经度纬度求球面距离56
(31)两圆切线的交点58
(31)两圆切线的交点(32)线段与三角形的交60
(33)最近最远点对68
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(1)凸包
/*凸包cug_1038*/
#include
#include
structpoint
{
intx,y;
}pp;
pointp[100005];
intstack[100005],top;
intdis(pointa,pointb)
{
return((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y));
}
intmulti(pointb,pointc,pointa)
{
return(b.x-a.x)*(c.y-a.y)-(b.y-a.y)*(c.x-a.x);
}
voidswap(pointp[],ints,intt)
{
pointtmp;
tmp=p[s];
p[s]=p[t];
p[t]=tmp;
}
intcmp(constvoid*a,constvoid*b)
{
point*c=(point*)a;
point*d=(point*)b;
doublek=multi(*c,*d,pp);
if(k<0)return1;
elseif(k==0&&dis(*c,pp)>=dis(*d,pp))return1;
elsereturn-1;
}
voidGraham(pointp[],intn,intstack[],int&top)
{
inti,u;
u=0;
for(i=1;iif(p[i].y==p[u].y&&p[i].x
elseif(p[i].y
}
swap(p,0,u);
pp=p[0];
qsort(p+1,n-1,sizeof(p[0]),cmp);
stack[0]=0;
stack[1]=1;
top=1;
for(i=2;iwhile(multi(p[i],p[stack[top]],p[stack[top-1]])>=0){
if(top==0)break;
top--;
}
top++;
stack[top]=i;
}
}
intmain()
{
intca,i,j,n;
intarea;
scanf("%d",&ca);
for(i=1;i<=ca;i++){
scanf("%d",&n);
for(j=0;jscanf("%d%d",&p[j].x,&p[j].y);
}
Graham(p,n,stack,top);
area=0;
for(j=1;j<=top-1;j++){
area+=abs(multi(p[stack[0]],p[stack[j]],p[stack[j+1]]));
}
printf("%.1lf\n",(double)area/2);
}
return0;
}
---------------------------------------------------------------------------------------------------------------------
(2)判断两条线段是否相交(平行,不平行)
boolisIntersected(TPoints1,TPointe1,TPoints2,TPointe2)
{
//判断线段是否相交
//1.快速排斥试验判断以两条线段为对角线的两个矩形是否相交
//2.跨立试验
if(
(max(s1.x,e1.x)>=min(s2.x,e2.x))&&
(max(s2.x,e2.x)>=min(s1.x,e1.x))&&
(max(s1.y,e1.y)>=min(s2.y,e2.y))&&
(max(s2.y,e2.y)>=min(s1.y,e1.y))&&
(multi(s2,e1,s1)*multi(e1,e2,s1)>=0)&&
(multi(s1,e2,s2)*multi(e2,e1,s2)>=0)
)returntrue;
returnfalse;
}
(3)三角形的外接圆(已知不在同一直线上的三点求经过三点的圆)
/*三角形的外接圆pku_1329*/
#include
#include
constdoubleeps=1e-6;
typedefstructTPoint
{
doublex;
doubley;
}TPoint;
typedefstructTTriangle
{
TPointt[3];
}TTriangle;
typedefstructTCircle
{
TPointcentre;
doubler;
}TCircle;
doubledistance(TPointp1,TPointp2)
{
//计算平面上两个点之间的距离
returnsqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}
doubletriangleArea(TTrianglet)
{
//已知三角形三个顶点的坐标,求三角形的面积
returnfabs(t.t[0].x*t.t[1].y+t.t[1].x*t.t[2].y+t.t[2].x*t.t[0].y
-t.t[1].x*t.t[0].y-t.t[2].x*t.t[1].y-t.t[0].x*t.t[2].y)/2;
}
TCirclecircumcircleOfTriangle(TTrianglet)
{
//三角形的外接圆
TCircletmp;
doublea,b,c,c1,c2;
doublexA,yA,xB,yB,xC,yC;
a=distance(t.t[0],t.t[1]);
b=distance(t.t[1],t.t[2]);
c=distance(t.t[2],t.t[0]);
//根据S=a*b*c/R/4;求半径R
tmp.r=a*b*c/triangleArea(t)/4;
xA=t.t[0].x;yA=t.t[0].y;
xB=t.t[1].x;yB=t.t[1].y;
xC=t.t[2].x;yC=t.t[2].y;
c1=(xA*xA+yA*yA-xB*xB-yB*yB)/2;
c2=(xA*xA+yA*yA-xC*xC-yC*yC)/2;
tmp.centre.x=-(c1*(yA-yC)-c2*(yA-yB))/
((xA-xB)*(yA-yC)-(xA-xC)*(yA-yB));
tmp.centre.y=-(c1*(xA-xC)-c2*(xA-xB))/
((yA-yB)*(xA-xC)-(yA-yC)*(xA-xB));
returntmp;
}
intmain()
{
TTrianglet;
TCirclecircle;
doublec,d,e;
while(scanf("%lf%lf%lf%lf%lf%lf",&t.t[0].x,&t.t[0].y,
&t.t[1].x,&t.t[1].y,&t.t[2].x,&t.t[2].y)!
=EOF){
circle=circumcircleOfTriangle(t);
//printf("%lf%lf%lf\n",circle.centre.x,circle.centre.y,circle.r);
if(fabs(circle.centre.x)elseif(circle.centre.x<0)printf("(x-%.3lf)^2+",-circle.centre.x);
elseprintf("(x+%.3lf)^2+",circle.centre.x);
if(fabs(circle.centre.y)elseif(circle.centre.y<0)printf("(y-%.3lf)^2=",-circle.centre.y);
elseprintf("(y+%.3lf)^2=",circle.centre.y);
printf("%.3lf^2\n",circle.r);
c=2*circle.centre.x;
d=2*circle.centre.y;
e=circle.centre.x*circle.centre.x+
circle.centre.y*circle.centre.y-circle.r*circle.r;
printf("x^2+y^2");
//if(fabs(c)if(c<0)printf("-%.3lfx",-c);
elseprintf("+%.3lfx",c);
if(d<0)printf("-%.3lfy",-d);
elseprintf("+%.3lfy",d);
if(e<0)printf("-%.3lf=0\n",-e);
elseprintf("+%.3lf=0\n",e);
printf("\n");
}
return0;
}
(4)三角形的垂心内心重心中垂线
/*cug_1011_垂心内心重心中垂线.cpp*/
#include
#include
usingnamespacestd;
constdoubleeps=1e-6;
structpoint
{
doublex,y;
};
voidK()
{
//到三边距离和最短
}
voidL(doublea,doubleb,doublec,doubleA,doubleB,doubleC)
{//垂线的交点
doublet1,t2,t3;
t1=c*cos(A)/cos(M_PI/2-C);
t2=c*cos(B)/cos(M_PI/2-C);
t3=a*cos(C)/cos(M_PI/2-A);
t1+=t2+t3;
printf("%.3lf\n",t1);
}
structTLine
{
doublea,b,c;
};
TLinelineFromSegment(pointp1,pointp2)
{
//线段所在直线,返回直线方程的三个系统
TLinetmp;
tmp.a=p2.y-p1.y;
tmp.b=p1.x-p2.x;
tmp.c=p2.x*p1.y-p1.x*p2.y;
returntmp;
}
pointLineInter(TLinel1,TLinel2)
{
//求两直线得交点坐标
pointtmp;
if(fabs(l1.b)tmp.x=-l1.c/l1.a;
tmp.y=(-l2.c-l2.a*tmp.x)/l2.b;
}
else{
tmp.x=(l1.c*l2.b-l1.b*l2.c)/(l1.b*l2.a-l2.b*l1.a);
tmp.y=(-l1.c-l1.a*tmp.x)/l1.b;
}
returntmp;
}
doubledis(pointa,pointb)
{
returnsqrt((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y));
}
voidF(doublea,doubleb,doublec,doubleA,doubleB,doubleC)
{
//到三顶点的距离和最短,费马点
/*当三角形最大的顶角小于120度的时候,三角形内一点到
三顶点之间的距离最小是与三顶点夹角都成120度的点P当
最到顶点大于等于120度,该顶点取最小值
补充一下,当三角形的最大角小于120度时,费尔码点在三
角形内,作法有多种,可以从任二办向外作等边三角形,联
接正三角形的顶点和原三角形的对角,两者的联线即所求。
当三角形的最大角大于等于120度时,
费尔码点在三角形的钝角上。
*/
if(A-2*M_PI/3>-eps){
printf("%.3lf",b+c);
return;
}
elseif(B-2*M_PI/3>-eps){
printf("%.3lf",a+c);
return;
}
elseif(C-2*M_PI/3>-eps){
printf("%.3lf",a+b);
return;
}
pointpa,pb,pc,pc1,pa1;
pa.x=0,pa.y=0;
pb.x=c,pb.y=0;
pc.x=b*cos(A);
pc.y=b*sin(A);
pc1.x=c*cos(-M_PI/3);
pc1.y=c*sin(-M_PI/3);
pa1.x=a*cos(2*M_PI/3-B)+c;
pa1.y=a*sin(2*M_PI/3-B);
TLinel1,l2;
l1=lineFromSegment(pa,pa1);
l2=lineFromSegment(pc,pc1);
pointf=LineInter(l1,l2);
printf("%.3lf",dis(pa,f)+dis(pb,f)+dis(pc,f));
}
voidI(doublea,doubleb,doublec,doubleA,doubleB,doubleC)
{
//角平分线的交点到三顶点的距离和
doublet,ans;
t=(a+b-c)/2;
ans=t/cos(C/2)+(b-t)/cos(A/2)+(a-t)/cos(B/2);
printf("%.3lf",ans);
}
voidG(doublea,doubleb,doublec,doubleA,doubleB,doubleC)
{
//中线的交点
doublet1,t2,t3;
t1=sqrt((b/2)*(b/2)+a*a-2*a*b/2*cos(C));
t2=sqrt((a/2)*(a/2)+c*c-2*a*c/2*cos(B));
t3=sqrt((c/2)*(c/2)+b*b-2*b*c/2*cos(A));
t1+=t2+t3;
printf("%.3lf",t1*2/3);
}
voidO(doublea,doubleb,doublec,doubleA,doubleB,doubleC)
{//垂线的交点
doublet=(A+C-B)/2;
printf("%.3lf\n",3*b/2/cos(t));
}
intmain()
{
inti,ca;
doublea,b,c;
doubleA,B,C;
cin>>ca;
for(i=1;i<=ca;i++){
cin>>a>>b>>c;
A=(b*b+c*c-a*a)/2/b/c;
B=(a*a+c*c-b*b)/2/a/c;
C=(a*a+b*b-c*c)/2/a/b;
A=acos(A),B=acos(B),C=acos(C);
F(a,b,c,A,B,C);
I(a,b,c,A,B,C);
G(a,b,c,A,B,C);
O(a,b,c,A,B,C);
}
return0;
}
============================================================================================--------------------------------------------------------------------------------------------
(5)求直线的交点
/*求直线的交点,注意平形的情况无解,避免RE*/
TPointLineInter(TLinel1,TLinel2)
{
//求两直线得交点坐标
TPointtmp;
doublea1=l1.a;
doubleb1=l1.b;
doublec1=l1.c;
doublea2=l2.a;
doubleb2=l2.b;
doublec2=l2.c;
//注意这里b1=0
if(fabs(b1)tmp.x=-c1/a1;
tmp.y=(-c2-a2*tmp.x)/b2;
}
else{
tmp.x=(c1*b2-b1*c2)/(b1*a2-b2*a1);
tmp.y=(-c1-a1*tmp.x)/b1;
}
//cout<<"交点坐标"<//cout<//cout<returntmp;
}
(6)根据线段两端点的坐标求垂直平分线上除中点外的另一点
TPointGetOtherPoint(TPointpre,TPointtmp)
{
/*根据线段两端点的坐标求垂直平分线上除中点外的另一点*/
doublekx,ky;
TPointother,mid;
mid.x=(pre.x+tmp.x)/2;
mid.y=(pre.y+tmp.y)/2;
kx=pre.x-tmp.x;
ky=pre.y-tmp.y;
if(fabs(kx)other.y=mid.y;
other.x=1.0;
if(fabs(other.x-mid.x)}
elseif(fabs(ky)other.x=mid.x;
other.y=1.0;
if(fabs(other.y-mid.y)}
else{
doublek