几何摘录

需要注意的细节

•常用头文件#include<math.h>

•计算几何中一般来说使用double型比较频繁，请注意数据类型的选择，该用实数的时候就用double，而float容易失去精度。

•判断double型的x是否为0，应当用x<eps && x>-eps（或者fabs(x)<eps），其中eps代表某个精度，常常取eps=0.000001，还有其他类似情况也要注意double类型的精度问题，int(x+eps)，避免x=4.999999999

•圆周率取3.141592654或者更精确，或者用acos(-1)

•角度制和弧度制的转换，C/C++中的三角函数均为弧度制

•尽量少用除法，开方，三角函数，容易失去精度。用除法时注意除数不为0

•输出的时候要小心-0.00000，比如

•a=-0.0000001，printf(“%.5lf”,a);

1.判断点在直线上
•利用三点共线的等价条件α×β==0

•直线上取两不同点P1,P2，若点P在直线上，则fabs((P1- P) × (P2 - P )) < eps

•如果该题目需要编写求三角形面积的函数，那直接判断这三个点形成的三角形面积是否<eps

2.判断点在线段上

•判断点P(x,y)是否在线段P1P2上，其中P1(x1,y1),P2(x2,y2)

•需要验证两条

•（1）点P在P1P2所在直线上，即三点共线

•（2）点P在P1P2为对角线的矩形内

•其中（2）利用
min(x1,x2)<=x<=max(x1,x2)&&
min(y1,y2)<=y<=max(y1,y2)

如何判断是否同线？由叉积的原理知道如果p1，p2，p3共线的话那么(p2-p1)X(p3-p1)=0。因此如果p1，p2，p3共线，p1，p2，p4共线，那么两条直线共线。direction（）求叉积，叉积为0说明共线。

如何判断是否平行？由向量可以判断出两直线是否平行。如果两直线平行，那么向量p1p2、p3p4也是平等的。即((p1.x-p2.x)*(p3.y-p4.y)-(p1.y-p2.y)*(p3.x-p4.x))==0说明向量平等。

如何求出交点？这里也用到叉积的原理。假设交点为p0(x0,y0)。则有：

(p1-p0)X(p2-p0)=0

(p3-p0)X(p2-p0)=0

展开后即是

(y1-y2)x0+(x2-x1)y0+x1y2-x2y1=0

(y3-y4)x0+(x4-x3)y0+x3y4-x4y3=0

将x0,y0作为变量求解二元一次方程组。

假设有二元一次方程组

a1x+b1y+c1=0;

a2x+b2y+c2=0

那么

x=(c1*b2-c2*b1)/(a2*b1-a1*b2);

y=(a2*c1-a1*c2)/(a1*b2-a2*b1);

因为此处两直线不会平行，所以分母不会为0。

•struct point{   //构造点的数据类型，也可作向量使用

•     doublex;

•     double y;

•   point operator+(point p2);

•     double operator*(point p2);

•}v1,v2;

线段判交

•判断P1P2是否和P3P4相交，其中Pi坐标为(xi,yi)，同样需要满足两个条件

•（1）快速排斥试验：
以P1P2为对角线的矩形S1是否和以P3P4为对角线的矩形S2相交，即
min(x1,x2)<=max(x3,x4)&&
min(x3,x4)<=max(x1,x2) &&
min(y1,y2)<=max(y3,y4) &&
min(y3,y4)<=max(y1,y2)

     此代码中有很多模板，特转一下

#include<cmath>#include<cstdio>#include<cstdlib>#include<algorithm>using std::swap;#define eps 1e-8#define zero(x) (((x)>0?(x):-(x))<eps)struct Point{double x,y;};struct Line{Point a,b;};//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,double x2,double y2,double x0,double y0){return (x1-x0)*(y2-y0)-(x2-x0)*(y1-y0);}//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,double x2,double y2,double x0,double y0){return (x1-x0)*(x2-x0)+(y1-y0)*(y2-y0);}//两点距离double distance(Point p1,Point p2){return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));}double distance(double x1,double y1,double x2,double y2){return sqrt((x1-x2)*(x1-x2)+(y1-y2)*(y1-y2));}//判三点共线int dots_inline(Point p1,Point p2,Point p3){return zero(xmult(p1,p2,p3));}int dots_inline(double x1,double y1,double x2,double y2,double x3,double y3){return zero(xmult(x1,y1,x2,y2,x3,y3));}//判点是否在线段上,包括端点int dot_online_in(Point p,Line l){return zero(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,Point l2){return zero(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,double x1,double y1,double x2,double y2){return zero(xmult(x,y,x1,y1,x2,y2))&&(x1-x)*(x2-x)<eps&&(y1-y)*(y2-y)<eps;}//判点是否在线段上,不包括端点int dot_online_ex(Point p,Line l){return dot_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,Point l2){return dot_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,double x1,double y1,double x2,double y2){return dot_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0int same_side(Point p1,Point p2,Line l){return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)>eps;}int same_side(Point p1,Point p2,Point l1,Point l2){return xmult(l1,p1,l2)*xmult(l1,p2,l2)>eps;}//判两点在线段异侧,点在线段上返回0int opposite_side(Point p1,Point p2,Line l){return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)<-eps;}int opposite_side(Point p1,Point p2,Point l1,Point l2){return xmult(l1,p1,l2)*xmult(l1,p2,l2)<-eps;}//判两直线平行int parallel(Line u,Line v){return zero((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, Point v1,Point v2){return zero((u1.x-u2.x)*(v1.y-v2.y)-(v1.x-v2.x)*(u1.y-u2.y));}//判两直线垂直int perpendicular(Line u,Line v){return zero((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,Point v1,Point v2){return zero((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);return dot_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,Point v1,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);return dot_online_in(u1,v1,v2)||dot_online_in(u2,v1,v2)||dot_online_in(v1,u1,u2)||dot_online_in(v2,u1,u2);}//判两线段相交,不包括端点和部分重合int intersect_ex(Line u,Line v){return opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);}int intersect_ex(Point u1,Point u2,Point v1,Point v2){return opposite_side(u1,u2,v1,v2)&&opposite_side(v1,v2,u1,u2);}//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){Point ret=u.a;double t=((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;return ret;}Point intersection(Point u1,Point u2,Point v1,Point v2){Point ret=u1;double t=((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;return ret;}//点到直线上的最近点Point ptoline(Point p,Line l){Point t=p;t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;return intersection(p,t,l.a,l.b);}Point ptoline(Point p,Point l1,Point l2){Point t=p;t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;return intersection(p,t,l1,l2);}//点到直线距离double disptoline(Point p,Line l){return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);}double disptoline(Point p,Point l1,Point l2){return fabs(xmult(p,l1,l2))/distance(l1,l2);}double disptoline(double x,double y,double x1,double y1,double x2,double y2){return fabs(xmult(x,y,x1,y1,x2,y2))/distance(x1,y1,x2,y2);}//点到线段上的最近点Point ptoseg(Point p,Line l){Point t=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)return distance(p,l.a)<distance(p,l.b)?l.a:l.b;return intersection(p,t,l.a,l.b);}Point ptoseg(Point p,Point l1,Point l2){Point t=p;t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;if (xmult(l1,t,p)*xmult(l2,t,p)>eps)return distance(p,l1)<distance(p,l2)?l1:l2;return intersection(p,t,l1,l2);}//点到线段距离double disptoseg(Point p,Line l){Point t=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)return distance(p,l.a)<distance(p,l.b)?distance(p,l.a):distance(p,l.b);return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);}double disptoseg(Point p,Point l1,Point l2){Point t=p;t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;if (xmult(l1,t,p)*xmult(l2,t,p)>eps)return distance(p,l1)<distance(p,l2)?distance(p,l1):distance(p,l2);return fabs(xmult(p,l1,l2))/distance(l1,l2);}//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,double angle,double scale){Point ret=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;return ret;}int main(){    int t, i;    //freopen("a.txt","r",stdin);    double x1, y1, x2, y2;    Line l0, l[5];    scanf("%d",&t);    while ( t-- )    {        scanf("%lf%lf%lf%lf",&l0.a.x, &l0.a.y, &l0.b.x, &l0.b.y);        scanf("%lf%lf%lf%lf",&x1, &y1, &x2, &y2);        if ( x1 > x2 ) swap (x1, x2);        if ( y1 > y2 ) swap (y1, y2);        l[1].a.x = x1; l[1].a.y = y1; l[1].b.x = x2; l[1].b.y = y1;        l[2].a.x = x2; l[2].a.y = y1; l[2].b.x = x2; l[2].b.y = y2;        l[3].a.x = x2; l[3].a.y = y2; l[3].b.x = x1; l[3].b.y = y2;        l[4].a.x = x1; l[4].a.y = y2; l[4].b.x = x1; l[4].b.y = y1;        bool flag = false;        for ( i = 1; i <= 4; i++ )            if ( intersect_in (l[i],l0) ) flag = true;        if ( flag == false )        {            for ( i = 1; i <= 4; i++ )                if ( xmult(l[i].b, l0.a, l[i].a) < -eps || xmult(l[i].b, l0.b, l[i].a) < -eps )                    break;            if ( i > 4 ) flag = true;        }        if ( flag ) printf("T\n");        else printf("F\n");    }    return 0;}

待补充。。。。