几何公式
1、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))
2、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
3、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))
4、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
5、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
6、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
7、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
8、圆柱:
1. 侧面积 S=2PIrh
2. 全面积 T=2PIr(h+r)
3. 体积 V=PIr^2h
9、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
10、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
11、球:
1. 全面积 T=4PIr^2
2. 体积 V=4PIr^3/3
12、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
13、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
2、点、线之间的运算
//浮点几何函数库
#include <math.h>
#define eps 1e-8
#define zero(x)(((x)>0?(x):-(x))<eps)
struct point{double x,y;};
struct line{point a,b;};
1、计算cross product (P1-P0)x(P2-P0)
double xmult(point p1,point p2,point p0){
return(p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
double xmult(double x1,double y1,doublex2,double y2,double x0,double y0){
return(x1-x0)*(y2-y0)-(x2-x0)*(y1-y0);
}
2、计算dot product (P1-P0).(P2-P0)
double dmult(point p1,point p2,point p0){
return(p1.x-p0.x)*(p2.x-p0.x)+(p1.y-p0.y)*(p2.y-p0.y);
}
double dmult(double x1,double y1,doublex2,double y2,double x0,double y0){
return(x1-x0)*(x2-x0)+(y1-y0)*(y2-y0);
}
3、两点距离
double distance(point p1,point p2){
returnsqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}
double distance(double x1,double y1,doublex2,double y2){
returnsqrt((x1-x2)*(x1-x2)+(y1-y2)*(y1-y2));
}
4、判三点共线
int dots_inline(point p1,point p2,pointp3){
returnzero(xmult(p1,p2,p3));
}
int dots_inline(double x1,double y1,doublex2,double y2,double x3,double y3){
returnzero(xmult(x1,y1,x2,y2,x3,y3));
}
5、判点是否在线段上,包括端点
int dot_online_in(point p,line l){
returnzero(xmult(p,l.a,l.b))&&(l.a.x-p.x)*(l.b.x-p.x)<eps&&(l.a.y-p.y)*(l.b.y-p.y)<eps;
}
int dot_online_in(point p,point l1,pointl2){
returnzero(xmult(p,l1,l2))&&(l1.x-p.x)*(l2.x-p.x)<eps&&(l1.y-p.y)*(l2.y-p.y)<eps;
}
int dot_online_in(double x,double y,doublex1,double y1,double x2,double y2){
returnzero(xmult(x,y,x1,y1,x2,y2))&&(x1-x)*(x2-x)<eps&&(y1-y)*(y2-y)<eps;
}
6、判点是否在线段上,不包括端点
int dot_online_ex(point p,line l){
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));
}
int dot_online_ex(point p,point l1,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));
}
int dot_online_ex(double x,double y,doublex1,double y1,double x2,double y2){
returndot_online_in(x,y,x1,y1,x2,y2)&&(!zero(x-x1)||!zero(y-y1))&&(!zero(x-x2)||!zero(y-y2));
}
7、判两点在线段同侧,点在线段上返回0
int same_side(point p1,point p2,line l){
returnxmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)>eps;
}
int same_side(point p1,point p2,pointl1,point l2){
returnxmult(l1,p1,l2)*xmult(l1,p2,l2)>eps;
}
8、判两点在线段异侧,点在线段上返回0
int opposite_side(point p1,point p2,linel){
returnxmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)<-eps;
}
int opposite_side(point p1,point p2,pointl1,point l2){
returnxmult(l1,p1,l2)*xmult(l1,p2,l2)<-eps;
}
9、判两直线平行
int parallel(line u,line v){
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));
}
int parallel(point u1,point u2,pointv1,point v2){
returnzero((u1.x-u2.x)*(v1.y-v2.y)-(v1.x-v2.x)*(u1.y-u2.y));
}
10、判两直线垂直
int perpendicular(line u,line v){
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));
}
int perpendicular(point u1,point u2,pointv1,point v2){
returnzero((u1.x-u2.x)*(v1.x-v2.x)+(u1.y-u2.y)*(v1.y-v2.y));
}
11、判两线段相交,包括端点和部分重合
int intersect_in(line u,line v){
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)&&!same_side(v.a,v.b,u);
returndot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
int intersect_in(point u1,point u2,pointv1,point v2){
if(!dots_inline(u1,u2,v1)||!dots_inline(u1,u2,v2))
return!same_side(u1,u2,v1,v2)&&!same_side(v1,v2,u1,u2);
returndot_online_in(u1,v1,v2)||dot_online_in(u2,v1,v2)||dot_online_in(v1,u1,u2)||dot_online_in(v2,u1,u2);
}
12、判两线段相交,不包括端点和部分重合
int intersect_ex(line u,line v){
returnopposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
int intersect_ex(point u1,point u2,pointv1,point v2){
returnopposite_side(u1,u2,v1,v2)&&opposite_side(v1,v2,u1,u2);
}
13、计算两直线交点,注意事先判断直线是否平行!
//线段交点请另外判线段相交(同时还是要判断是否平行!)
point intersection(line u,line v){
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;
}
point intersection(point u1,point u2,pointv1,point v2){
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;
}
14、点到直线上的最近点
point ptoline(point p,line l){
pointt=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
returnintersection(p,t,l.a,l.b);
}
point ptoline(point p,point l1,point l2){
pointt=p;
t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
returnintersection(p,t,l1,l2);
}
15、点到直线距离
double disptoline(point p,line l){
returnfabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
double disptoline(point p,point l1,pointl2){
returnfabs(xmult(p,l1,l2))/distance(l1,l2);
}
double disptoline(double x,double y,doublex1,double y1,double x2,double y2){
returnfabs(xmult(x,y,x1,y1,x2,y2))/distance(x1,y1,x2,y2);
}
16、点到线段上的最近点
point ptoseg(point p,line l){
pointt=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
if(xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
returndistance(p,l.a)<distance(p,l.b)?l.a:l.b;
returnintersection(p,t,l.a,l.b);
}
point ptoseg(point p,point l1,point l2){
pointt=p;
t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
if(xmult(l1,t,p)*xmult(l2,t,p)>eps)
returndistance(p,l1)<distance(p,l2)?l1:l2;
returnintersection(p,t,l1,l2);
}
17、点到线段距离
double disptoseg(point p,line l){
pointt=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
if(xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
returndistance(p,l.a)<distance(p,l.b)?distance(p,l.a):distance(p,l.b);
returnfabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
double disptoseg(point p,point l1,pointl2){
pointt=p;
t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
if(xmult(l1,t,p)*xmult(l2,t,p)>eps)
returndistance(p,l1)<distance(p,l2)?distance(p,l1):distance(p,l2);
returnfabs(xmult(p,l1,l2))/distance(l1,l2);
}
18、矢量V以P为顶点逆时针旋转angle并放大scale倍
point rotate(point v,point p,doubleangle,double scale){
pointret=p;
v.x-=p.x,v.y-=p.y;
p.x=scale*cos(angle);
p.y=scale*sin(angle);
ret.x+=v.x*p.x-v.y*p.y;
ret.y+=v.x*p.y+v.y*p.x;
returnret;
}
3、凸边形面积
#include <math.h>
struct point{double x,y;};
1、计算cross product (P1-P0)x(P2-P0)
double xmult(point p1,point p2,point p0){
return(p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
double xmult(double x1,double y1,doublex2,double y2,double x0,double y0){
return(x1-x0)*(y2-y0)-(x2-x0)*(y1-y0);
}
2、计算三角形面积,输入三顶点
double area_triangle(point p1,pointp2,point p3){
returnfabs(xmult(p1,p2,p3))/2;
}
double area_triangle(double x1,doubley1,double x2,double y2,double x3,double y3){
returnfabs(xmult(x1,y1,x2,y2,x3,y3))/2;
}
3、计算三角形面积,输入三边长
double area_triangle(double a,doubleb,double c){
doubles=(a+b+c)/2;
returnsqrt(s*(s-a)*(s-b)*(s-c));
}
4、计算多边形面积,顶点按顺时针或逆时针给出
double area_polygon(int n,point* p){
doubles1=0,s2=0;
inti;
for(i=0;i<n;i++)
s1+=p[(i+1)%n].y*p[i].x,s2+=p[(i+1)%n].y*p[(i+2)%n].x;
returnfabs(s1-s2)/2;
}
4、球面
#include <math.h>
const double pi=acos(-1);
1、计算圆心角lat表示纬度,-90<=w<=90,lng表示经度
//返回两点所在大圆劣弧对应圆心角,0<=angle<=pi
double angle(double lng1,double lat1,doublelng2,double lat2){
doubledlng=fabs(lng1-lng2)*pi/180;
while(dlng>=pi+pi)
dlng-=pi+pi;
if(dlng>pi)
dlng=pi+pi-dlng;
lat1*=pi/180,lat2*=pi/180;
returnacos(cos(lat1)*cos(lat2)*cos(dlng)+sin(lat1)*sin(lat2));
}
2、计算距离,r为球半径
double line_dist(double r,doublelng1,double lat1,double lng2,double lat2){
doubledlng=fabs(lng1-lng2)*pi/180;
while(dlng>=pi+pi)
dlng-=pi+pi;
if(dlng>pi)
dlng=pi+pi-dlng;
lat1*=pi/180,lat2*=pi/180;
returnr*sqrt(2-2*(cos(lat1)*cos(lat2)*cos(dlng)+sin(lat1)*sin(lat2)));
}
3、计算球面距离,r为球半径
inline double sphere_dist(double r,doublelng1,double lat1,double lng2,double lat2){
returnr*angle(lng1,lat1,lng2,lat2);
}
5、三角形
#include <math.h>
struct point{double x,y;};
struct line{point a,b;};
double distance(point p1,point p2){
returnsqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}
point intersection(line u,line v){
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;
}
1、外心
point circumcenter(point a,point b,pointc){
lineu,v;
u.a.x=(a.x+b.x)/2;
u.a.y=(a.y+b.y)/2;
u.b.x=u.a.x-a.y+b.y;
u.b.y=u.a.y+a.x-b.x;
v.a.x=(a.x+c.x)/2;
v.a.y=(a.y+c.y)/2;
v.b.x=v.a.x-a.y+c.y;
v.b.y=v.a.y+a.x-c.x;
returnintersection(u,v);
}
2、内心
point incenter(point a,point b,point c){
lineu,v;
doublem,n;
u.a=a;
m=atan2(b.y-a.y,b.x-a.x);
n=atan2(c.y-a.y,c.x-a.x);
u.b.x=u.a.x+cos((m+n)/2);
u.b.y=u.a.y+sin((m+n)/2);
v.a=b;
m=atan2(a.y-b.y,a.x-b.x);
n=atan2(c.y-b.y,c.x-b.x);
v.b.x=v.a.x+cos((m+n)/2);
v.b.y=v.a.y+sin((m+n)/2);
returnintersection(u,v);
}
3、垂心
point perpencenter(point a,point b,pointc){
lineu,v;
u.a=c;
u.b.x=u.a.x-a.y+b.y;
u.b.y=u.a.y+a.x-b.x;
v.a=b;
v.b.x=v.a.x-a.y+c.y;
v.b.y=v.a.y+a.x-c.x;
returnintersection(u,v);
}
4、重心
//到三角形三顶点距离的平方和最小的点
//三角形内到三边距离之积最大的点
point barycenter(point a,point b,point c){
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);
}
5、费马点，到三角形三顶点距离之和最小的点
point fermentpoint(point a,point b,pointc){
pointu,v;
doublestep=fabs(a.x)+fabs(a.y)+fabs(b.x)+fabs(b.y)+fabs(c.x)+fabs(c.y);
inti,j,k;
u.x=(a.x+b.x+c.x)/3;
u.y=(a.y+b.y+c.y)/3;
while(step>1e-10)
for (k=0;k<10;step/=2,k++)
for(i=-1;i<=1;i++)
for (j=-1;j<=1;j++){
v.x=u.x+step*i;
v.y=u.y+step*j;
if(distance(u,a)+distance(u,b)+distance(u,c)>distance(v,a)+distance(v,b)+distance(v,c))
u=v;
}
returnu;
}
6、三维几何
//三维几何函数库
#include <math.h>
#define eps 1e-8
#define zero(x)(((x)>0?(x):-(x))<eps)
struct point3{double x,y,z;};
struct line3{point3 a,b;};
struct plane3{point3 a,b,c;};
1、计算cross product U x V
point3 xmult(point3 u,point3 v){
point3ret;
ret.x=u.y*v.z-v.y*u.z;
ret.y=u.z*v.x-u.x*v.z;
ret.z=u.x*v.y-u.y*v.x;
returnret;
}
2、计算dot product U . V
double dmult(point3 u,point3 v){
returnu.x*v.x+u.y*v.y+u.z*v.z;
}
3、矢量差 U - V
point3 subt(point3 u,point3 v){
point3ret;
ret.x=u.x-v.x;
ret.y=u.y-v.y;
ret.z=u.z-v.z;
returnret;
}
4、取平面法向量
point3 pvec(plane3 s){
returnxmult(subt(s.a,s.b),subt(s.b,s.c));
}
point3 pvec(point3 s1,point3 s2,point3 s3){
returnxmult(subt(s1,s2),subt(s2,s3));
}
5、两点距离,单参数取向量大小
double distance(point3 p1,point3 p2){
returnsqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y)+(p1.z-p2.z)*(p1.z-p2.z));
}
6、向量大小
double vlen(point3 p){
returnsqrt(p.x*p.x+p.y*p.y+p.z*p.z);
}
7、判三点共线
int dots_inline(point3 p1,point3 p2,point3p3){
returnvlen(xmult(subt(p1,p2),subt(p2,p3)))<eps;
}
8、判四点共面
int dots_onplane(point3 a,point3 b,point3c,point3 d){
returnzero(dmult(pvec(a,b,c),subt(d,a)));
}
9、判点是否在线段上,包括端点和共线
int dot_online_in(point3 p,line3 l){
returnzero(vlen(xmult(subt(p,l.a),subt(p,l.b))))&&(l.a.x-p.x)*(l.b.x-p.x)<eps&&
(l.a.y-p.y)*(l.b.y-p.y)<eps&&(l.a.z-p.z)*(l.b.z-p.z)<eps;
}
int dot_online_in(point3 p,point3 l1,point3l2){
returnzero(vlen(xmult(subt(p,l1),subt(p,l2))))&&(l1.x-p.x)*(l2.x-p.x)<eps&&
(l1.y-p.y)*(l2.y-p.y)<eps&&(l1.z-p.z)*(l2.z-p.z)<eps;
}
10、判点是否在线段上,不包括端点
int dot_online_ex(point3 p,line3 l){
returndot_online_in(p,l)&&(!zero(p.x-l.a.x)||!zero(p.y-l.a.y)||!zero(p.z-l.a.z))&&
(!zero(p.x-l.b.x)||!zero(p.y-l.b.y)||!zero(p.z-l.b.z));
}
int dot_online_ex(point3 p,point3 l1,point3l2){
returndot_online_in(p,l1,l2)&&(!zero(p.x-l1.x)||!zero(p.y-l1.y)||!zero(p.z-l1.z))&&
(!zero(p.x-l2.x)||!zero(p.y-l2.y)||!zero(p.z-l2.z));
}
11、判点是否在空间三角形上,包括边界,三点共线无意义
int dot_inplane_in(point3 p,plane3 s){
returnzero(vlen(xmult(subt(s.a,s.b),subt(s.a,s.c)))-vlen(xmult(subt(p,s.a),subt(p,s.b)))-
vlen(xmult(subt(p,s.b),subt(p,s.c)))-vlen(xmult(subt(p,s.c),subt(p,s.a))));
}
int dot_inplane_in(point3 p,point3s1,point3 s2,point3 s3){
returnzero(vlen(xmult(subt(s1,s2),subt(s1,s3)))-vlen(xmult(subt(p,s1),subt(p,s2)))-
vlen(xmult(subt(p,s2),subt(p,s3)))-vlen(xmult(subt(p,s3),subt(p,s1))));
}
12、判点是否在空间三角形上,不包括边界,三点共线无意义
int dot_inplane_ex(point3 p,plane3 s){
returndot_inplane_in(p,s)&&vlen(xmult(subt(p,s.a),subt(p,s.b)))>eps&&
vlen(xmult(subt(p,s.b),subt(p,s.c)))>eps&&vlen(xmult(subt(p,s.c),subt(p,s.a)))>eps;
}
int dot_inplane_ex(point3 p,point3s1,point3 s2,point3 s3){
returndot_inplane_in(p,s1,s2,s3)&&vlen(xmult(subt(p,s1),subt(p,s2)))>eps&&
vlen(xmult(subt(p,s2),subt(p,s3)))>eps&&vlen(xmult(subt(p,s3),subt(p,s1)))>eps;
}
13、判两点在线段同侧,点在线段上返回0,不共面无意义
int same_side(point3 p1,point3 p2,line3 l){
returndmult(xmult(subt(l.a,l.b),subt(p1,l.b)),xmult(subt(l.a,l.b),subt(p2,l.b)))>eps;
}
int same_side(point3 p1,point3 p2,point3l1,point3 l2){
returndmult(xmult(subt(l1,l2),subt(p1,l2)),xmult(subt(l1,l2),subt(p2,l2)))>eps;
}
14、判两点在线段异侧,点在线段上返回0,不共面无意义
int opposite_side(point3 p1,point3 p2,line3l){
returndmult(xmult(subt(l.a,l.b),subt(p1,l.b)),xmult(subt(l.a,l.b),subt(p2,l.b)))<-eps;
}
int opposite_side(point3 p1,point3p2,point3 l1,point3 l2){
returndmult(xmult(subt(l1,l2),subt(p1,l2)),xmult(subt(l1,l2),subt(p2,l2)))<-eps;
}
15、判两点在平面同侧,点在平面上返回0
int same_side(point3 p1,point3 p2,plane3s){
returndmult(pvec(s),subt(p1,s.a))*dmult(pvec(s),subt(p2,s.a))>eps;
}
int same_side(point3 p1,point3 p2,point3s1,point3 s2,point3 s3){
returndmult(pvec(s1,s2,s3),subt(p1,s1))*dmult(pvec(s1,s2,s3),subt(p2,s1))>eps;
}
16、判两点在平面异侧,点在平面上返回0
int opposite_side(point3 p1,point3p2,plane3 s){
returndmult(pvec(s),subt(p1,s.a))*dmult(pvec(s),subt(p2,s.a))<-eps;
}
int opposite_side(point3 p1,point3p2,point3 s1,point3 s2,point3 s3){
returndmult(pvec(s1,s2,s3),subt(p1,s1))*dmult(pvec(s1,s2,s3),subt(p2,s1))<-eps;
}
17、判两直线平行
int parallel(line3 u,line3 v){
returnvlen(xmult(subt(u.a,u.b),subt(v.a,v.b)))<eps;
}
int parallel(point3 u1,point3 u2,point3v1,point3 v2){
returnvlen(xmult(subt(u1,u2),subt(v1,v2)))<eps;
}
18、判两平面平行
int parallel(plane3 u,plane3 v){
returnvlen(xmult(pvec(u),pvec(v)))<eps;
}
int parallel(point3 u1,point3 u2,point3u3,point3 v1,point3 v2,point3 v3){
returnvlen(xmult(pvec(u1,u2,u3),pvec(v1,v2,v3)))<eps;
}
19、判直线与平面平行
int parallel(line3 l,plane3 s){
returnzero(dmult(subt(l.a,l.b),pvec(s)));
}
int parallel(point3 l1,point3 l2,point3s1,point3 s2,point3 s3){
returnzero(dmult(subt(l1,l2),pvec(s1,s2,s3)));
}
20、判两直线垂直
int perpendicular(line3 u,line3 v){
returnzero(dmult(subt(u.a,u.b),subt(v.a,v.b)));
}
int perpendicular(point3 u1,point3u2,point3 v1,point3 v2){
returnzero(dmult(subt(u1,u2),subt(v1,v2)));
}
21、判两平面垂直
int perpendicular(plane3 u,plane3 v){
returnzero(dmult(pvec(u),pvec(v)));
}
int perpendicular(point3 u1,point3u2,point3 u3,point3 v1,point3 v2,point3 v3){
returnzero(dmult(pvec(u1,u2,u3),pvec(v1,v2,v3)));
}
22、判直线与平面平行
int perpendicular(line3 l,plane3 s){
returnvlen(xmult(subt(l.a,l.b),pvec(s)))<eps;
}
int perpendicular(point3 l1,point3l2,point3 s1,point3 s2,point3 s3){
returnvlen(xmult(subt(l1,l2),pvec(s1,s2,s3)))<eps;
}
23、判两线段相交,包括端点和部分重合
int intersect_in(line3 u,line3 v){
if(!dots_onplane(u.a,u.b,v.a,v.b))
return 0;
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)&&!same_side(v.a,v.b,u);
returndot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
int intersect_in(point3 u1,point3 u2,point3v1,point3 v2){
if(!dots_onplane(u1,u2,v1,v2))
return 0;
if(!dots_inline(u1,u2,v1)||!dots_inline(u1,u2,v2))
return!same_side(u1,u2,v1,v2)&&!same_side(v1,v2,u1,u2);
returndot_online_in(u1,v1,v2)||dot_online_in(u2,v1,v2)||dot_online_in(v1,u1,u2)||dot_online_in(v2,u1,u2);
}
24、判两线段相交,不包括端点和部分重合
int intersect_ex(line3 u,line3 v){
returndots_onplane(u.a,u.b,v.a,v.b)&&opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
int intersect_ex(point3 u1,point3 u2,point3v1,point3 v2){
returndots_onplane(u1,u2,v1,v2)&&opposite_side(u1,u2,v1,v2)&&opposite_side(v1,v2,u1,u2);
}
25、判线段与空间三角形相交,包括交于边界和(部分)包含
int intersect_in(line3 l,plane3 s){
return!same_side(l.a,l.b,s)&&!same_side(s.a,s.b,l.a,l.b,s.c)&&
!same_side(s.b,s.c,l.a,l.b,s.a)&&!same_side(s.c,s.a,l.a,l.b,s.b);
}
int intersect_in(point3 l1,point3 l2,point3s1,point3 s2,point3 s3){
return!same_side(l1,l2,s1,s2,s3)&&!same_side(s1,s2,l1,l2,s3)&&
!same_side(s2,s3,l1,l2,s1)&&!same_side(s3,s1,l1,l2,s2);
}
26、判线段与空间三角形相交,不包括交于边界和(部分)包含
int intersect_ex(line3 l,plane3 s){
returnopposite_side(l.a,l.b,s)&&opposite_side(s.a,s.b,l.a,l.b,s.c)&&
opposite_side(s.b,s.c,l.a,l.b,s.a)&&opposite_side(s.c,s.a,l.a,l.b,s.b);
}
int intersect_ex(point3 l1,point3 l2,point3s1,point3 s2,point3 s3){
returnopposite_side(l1,l2,s1,s2,s3)&&opposite_side(s1,s2,l1,l2,s3)&&
opposite_side(s2,s3,l1,l2,s1)&&opposite_side(s3,s1,l1,l2,s2);
}
27、计算两直线交点,注意事先判断直线是否共面和平行!
//线段交点请另外判线段相交(同时还是要判断是否平行!)
point3 intersection(line3 u,line3 v){
point3ret=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;
ret.z+=(u.b.z-u.a.z)*t;
returnret;
}
point3 intersection(point3 u1,point3u2,point3 v1,point3 v2){
point3ret=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;
ret.z+=(u2.z-u1.z)*t;
returnret;
}
28、计算直线与平面交点,注意事先判断是否平行,并保证三点不共线!
//线段和空间三角形交点请另外判断
point3 intersection(line3 l,plane3 s){
point3ret=pvec(s);
doublet=(ret.x*(s.a.x-l.a.x)+ret.y*(s.a.y-l.a.y)+ret.z*(s.a.z-l.a.z))/
(ret.x*(l.b.x-l.a.x)+ret.y*(l.b.y-l.a.y)+ret.z*(l.b.z-l.a.z));
ret.x=l.a.x+(l.b.x-l.a.x)*t;
ret.y=l.a.y+(l.b.y-l.a.y)*t;
ret.z=l.a.z+(l.b.z-l.a.z)*t;
returnret;
}
point3 intersection(point3 l1,point3l2,point3 s1,point3 s2,point3 s3){
point3ret=pvec(s1,s2,s3);
doublet=(ret.x*(s1.x-l1.x)+ret.y*(s1.y-l1.y)+ret.z*(s1.z-l1.z))/
(ret.x*(l2.x-l1.x)+ret.y*(l2.y-l1.y)+ret.z*(l2.z-l1.z));
ret.x=l1.x+(l2.x-l1.x)*t;
ret.y=l1.y+(l2.y-l1.y)*t;
ret.z=l1.z+(l2.z-l1.z)*t;
returnret;
}
29、计算两平面交线,注意事先判断是否平行,并保证三点不共线!
line3 intersection(plane3 u,plane3 v){
line3ret;
ret.a=parallel(v.a,v.b,u.a,u.b,u.c)?intersection(v.b,v.c,u.a,u.b,u.c):intersection(v.a,v.b,u.a,u.b,u.c);
ret.b=parallel(v.c,v.a,u.a,u.b,u.c)?intersection(v.b,v.c,u.a,u.b,u.c):intersection(v.c,v.a,u.a,u.b,u.c);
returnret;
}
line3 intersection(point3 u1,point3u2,point3 u3,point3 v1,point3 v2,point3 v3){
line3ret;
ret.a=parallel(v1,v2,u1,u2,u3)?intersection(v2,v3,u1,u2,u3):intersection(v1,v2,u1,u2,u3);
ret.b=parallel(v3,v1,u1,u2,u3)?intersection(v2,v3,u1,u2,u3):intersection(v3,v1,u1,u2,u3);
returnret;
}
30、点到直线距离
double ptoline(point3 p,line3 l){
returnvlen(xmult(subt(p,l.a),subt(l.b,l.a)))/distance(l.a,l.b);
}
double ptoline(point3 p,point3 l1,point3l2){
returnvlen(xmult(subt(p,l1),subt(l2,l1)))/distance(l1,l2);
}
31、点到平面距离
double ptoplane(point3 p,plane3 s){
returnfabs(dmult(pvec(s),subt(p,s.a)))/vlen(pvec(s));
}
double ptoplane(point3 p,point3 s1,point3s2,point3 s3){
returnfabs(dmult(pvec(s1,s2,s3),subt(p,s1)))/vlen(pvec(s1,s2,s3));
}
32、直线到直线距离
double linetoline(line3 u,line3 v){
point3n=xmult(subt(u.a,u.b),subt(v.a,v.b));
returnfabs(dmult(subt(u.a,v.a),n))/vlen(n);
}
double linetoline(point3 u1,point3u2,point3 v1,point3 v2){
point3n=xmult(subt(u1,u2),subt(v1,v2));
returnfabs(dmult(subt(u1,v1),n))/vlen(n);
}
33、两直线夹角cos值
double angle_cos(line3 u,line3 v){
returndmult(subt(u.a,u.b),subt(v.a,v.b))/vlen(subt(u.a,u.b))/vlen(subt(v.a,v.b));
}
double angle_cos(point3 u1,point3 u2,point3v1,point3 v2){
returndmult(subt(u1,u2),subt(v1,v2))/vlen(subt(u1,u2))/vlen(subt(v1,v2));
}
34、两平面夹角cos值
double angle_cos(plane3 u,plane3 v){
returndmult(pvec(u),pvec(v))/vlen(pvec(u))/vlen(pvec(v));
}
double angle_cos(point3 u1,point3 u2,point3u3,point3 v1,point3 v2,point3 v3){
returndmult(pvec(u1,u2,u3),pvec(v1,v2,v3))/vlen(pvec(u1,u2,u3))/vlen(pvec(v1,v2,v3));
}
35、直线平面夹角sin值
double angle_sin(line3 l,plane3 s){
returndmult(subt(l.a,l.b),pvec(s))/vlen(subt(l.a,l.b))/vlen(pvec(s));
}
double angle_sin(point3 l1,point3 l2,point3s1,point3 s2,point3 s3){
returndmult(subt(l1,l2),pvec(s1,s2,s3))/vlen(subt(l1,l2))/vlen(pvec(s1,s2,s3));
}
转自：https://blog.csdn.net/u011988082/article/details/45787209