计算几何整理总结

MFDYCS
2021-03-03 / 0 评论 / 140 阅读 / 正在检测是否收录...

以前学习的做的计算几何部分笔记,现在整理一下发出来

Sorted & Add By: MFDY

From: kuangbin

二维几何


预备部分

const double eps = 1e-8;
const double inf = 1e20;
const double pi = acos(-1.0);
const int maxp = 1010;
//`Compares a double to zero`
int sgn(double x) {
    if(fabs(x) < eps) return 0;
    if(x < 0) return -1;
    else return 1;
}
//square of a double
inline double sqr(double x){return x*x;}

第一部分—点

  1. point() - Empty constructor
  2. point(double _x,double _y) - constructor
  3. input() - double input
  4. output() - %.2f output
  5. operator == - 比较x和y
  6. operator < - 先比x,后比y
  7. operator - - x,y分别减; 从A点指向B点的矢量AB可用B-A来表示
  8. operator ^ - 叉积:求两个矢量形成的面积;求面积,求顺逆时针方向,判断是否在半平面上
    因为a*b为有向面积,可正可负,a逆时针旋转小于180为正
  9. operator * - 矢量的点积,求投影长度
  10. len() - gives length from origin
  11. len2() - gives square of length from origin
  12. distance(point p) - gives distance from p
  13. operator + point b - returns new point after adding curresponging x and y
  14. operator * double k - returns new point after multiplieing x and y by k
  15. operator / double k - returns new point after divideing x and y by k
  16. rad(point a,point b)- 计算pa和pb的夹角
  17. trunc(double r) - 更改为长度为r的向量
  18. rotleft() - 逆时针旋转90°
  19. rotright() - 顺时针旋转90°
  20. rotate(point p,double angle) - 绕p点逆时针旋转angle°
struct Point{
    double x, y;
    Point(){}
    Point(double _x, double _y){
        x = _x;
        y = _y;
    }
    void input(){
        scanf("%lf%lf", &x, &y);
    }
    void output(){
        printf("%.2f %.2f\n", x, y);
    }
    bool operator == (Point b) const {
        return sgn(x - b.x) == 0 && sgn(y - b.y) == 0;
    }
    bool operator < (Point b) const {
        return sgn(x-b.x) == 0 ? sgn(y-b.y) < 0 : x < b.x;
    }
    Point operator -(const Point &b)const{
        return Point(x-b.x,y-b.y);
    }
    //叉积
    double operator ^(const Point &b)const{
        return x*b.y - y*b.x;
    }
    //点积
    double operator *(const Point &b)const{
        return x*b.x + y*b.y;
    }
    //返回长度
    double len(){
        return hypot(x,y);//库函数
    }
    //返回长度的平方
    double len2(){
        return x*x + y*y;
    }
    //返回两点的距离
    double distance(Point p){
        return hypot(x-p.x,y-p.y);
    }
    Point operator +(const Point &b)const{
        return Point(x+b.x,y+b.y);
    }
    Point operator *(const double &k)const{
        return Point(x*k,y*k);
    }
    Point operator /(const double &k)const{
        return Point(x/k,y/k);
    }
    //`计算pa  和  pb 的夹角`
    //`就是求这个点看a,b 所成的夹角`
    //`测试 LightOJ1203`
    double rad(Point a,Point b){
        Point p = *this;
        return fabs(atan2( fabs((a-p)^(b-p)),(a-p)*(b-p) ));
    }
    //`化为长度为r的向量`
    Point trunc(double r){
        double l = len();
        if(!sgn(l))return *this;
        r /= l;
        return Point(x*r,y*r);
    }
    //`逆时针旋转90度`
    Point rotleft(){
        return Point(-y,x);
    }
    //`顺时针旋转90度`
    Point rotright() {
        return Point(y,-x);
    }
    //`绕着p点逆时针旋转angle`
    Point rotate(Point p,double angle){
        Point v = (*this) - p;
        double c = cos(angle), s = sin(angle);
        return Point(p.x + v.x*c - v.y*s,p.y + v.x*s + v.y*c);
    }
};

第二部分—线

  1. Stores two points
  2. line() - Empty constructor
  3. line(point _s,point _e) - line through _s and _e
  4. operator == - checks if two points are same
  5. line(point p,double angle) - 根据一个点和倾斜角angle确定直线,0≤angle<pi
  6. line(double a,double b,double c) - ax + by + c = 0 转化
  7. input() - inputs s and e
  8. adjust() - orders in such a way that s < e
  9. length() - 求线段长度
  10. angle() - 返回直线倾斜角
  11. relation(point p) - 点和直线关系(向量)
    • 1 点在直线左侧
    • 2 点在直线右侧
    • 3 点在直线上
  12. xmult(Point p) - 点和直线叉积, 用于判断点和直线关系(优先选择!!!)
  13. pointonseg(double p) - 判断点是否在线段上
  14. parallel(line v) - 判断两向量是否平行
  15. segcrossseg(line v) - 两线段相交判断
    • returns 0 不相交
    • returns 1 非规范相交
    • returns 2 规范相交
  16. linecrossseg(line v) - 直线和线段相交判断,同上
  17. linecrossline(line v) - 两直线关系
    • 0 平行
    • 1 重合
    • 2 相交
  18. crosspoint(line v) -求两直线的交点,要保证两直线不平行或重合
  19. dispointtoline(point p) - 点到直线的距离
  20. dispointtoseg(point p) - 点到线段的距离
  21. dissegtoseg(line v) - 返回线段到线段的距离,前提是两线段不相交,相交距离就是 0 了
  22. lineprog(point p) - 返回点 p 在直线上的投影
  23. symmetrypoint(point p) - 返回点 p 关于直线的对称点
struct Line{
    Point s,e;
    Line(){}
    Line(Point _s,Point _e){
        s = _s;
        e = _e;
    }
    bool operator ==(Line v){
        return (s == v.s)&&(e == v.e);
    }
    //`根据一个点和倾斜角angle确定直线,0<=angle<pi`
    Line(Point p,double angle){
        s = p;
        if(sgn(angle-pi/2) == 0){
            e = (s + Point(0,1));
        }
        else{
            e = (s + Point(1,tan(angle)));
        }
    }
    //ax+by+c=0
    Line(double a,double b,double c){
        if(sgn(a) == 0){
            s = Point(0,-c/b);
            e = Point(1,-c/b);
        }
        else if(sgn(b) == 0){
            s = Point(-c/a,0);
            e = Point(-c/a,1);
        }
        else{
            s = Point(0,-c/b);
            e = Point(1,(-c-a)/b);
        }
    }
    void input(){
        s.input();
        e.input();
    }
    void adjust(){
        if(e < s)swap(s,e);
    }
    //求线段长度
    double length(){
        return s.distance(e);
    }
    //`返回直线倾斜角 0<=angle<pi`
    double angle(){
        double k = atan2(e.y-s.y,e.x-s.x);
        if(sgn(k) < 0)k += pi;
        if(sgn(k-pi) == 0)k -= pi;
        return k;
    }
    //点和直线关系(向量)
    //1  在左侧
    //2  在右侧
    //3  在直线上
    int relation(Point p){
        int c = sgn((p-s)^(e-s));
        if(c < 0)return 1;
        else if(c > 0)return 2;
        else return 3;
    }
    // 点和直线的叉积,也可用于判断点和直线关系(优选)
    // return < 0  点在直线左侧, 即点在直线的逆时针方向
    // return > 0  点在直线右侧, 即点在直线的顺时针方向
    // return == 0  点在直线上
    int xmult(Point p) {
        return (s - p) ^ (e - p);
    }
    // 点在线段上的判断
    bool pointonseg(Point p){
        return sgn((p-s)^(e-s)) == 0 && sgn((p-s)*(p-e)) <= 0;
    }
    //`两向量平行(对应直线平行或重合)`
    bool parallel(Line v){
        return sgn((e-s)^(v.e-v.s)) == 0;
    }
    //`两线段相交判断`
    //`2 规范相交`
    //`1 非规范相交`
    //`0 不相交`
    int segcrossseg(Line v){
        int d1 = sgn((e-s)^(v.s-s));
        int d2 = sgn((e-s)^(v.e-s));
        int d3 = sgn((v.e-v.s)^(s-v.s));
        int d4 = sgn((v.e-v.s)^(e-v.s));
        if( (d1^d2)==-2 && (d3^d4)==-2 )return 2;
        return (d1==0 && sgn((v.s-s)*(v.s-e))<=0) ||
            (d2==0 && sgn((v.e-s)*(v.e-e))<=0) ||
            (d3==0 && sgn((s-v.s)*(s-v.e))<=0) ||
            (d4==0 && sgn((e-v.s)*(e-v.e))<=0);
    }
    //`直线和线段相交判断`
    //`-*this line   -v seg`
    //`2 规范相交`
    //`1 非规范相交`
    //`0 不相交`
    int linecrossseg(Line v){
        int d1 = sgn((e-s)^(v.s-s));
        int d2 = sgn((e-s)^(v.e-s));
        if((d1^d2)==-2) return 2;
        return (d1==0||d2==0);
    }
    //`两直线关系`
    //`0 平行`
    //`1 重合`
    //`2 相交`
    int linecrossline(Line v){
        if((*this).parallel(v))
            return v.relation(s)==3;
        return 2;
    }
    //`求两直线的交点`
    //`要保证两直线不平行或重合`
    Point crosspoint(Line v){
        double a1 = (v.e-v.s)^(s-v.s);
        double a2 = (v.e-v.s)^(e-v.s);
        return Point((s.x*a2-e.x*a1)/(a2-a1),(s.y*a2-e.y*a1)/(a2-a1));
    }
    //点到直线的距离
    double dispointtoline(Point p){
        return fabs((p-s)^(e-s))/length();
    }
    //点到线段的距离
    double dispointtoseg(Point p){
        if(sgn((p-s)*(e-s))<0 || sgn((p-e)*(s-e))<0)
            return min(p.distance(s),p.distance(e));
        return dispointtoline(p);
    }
    //`返回线段到线段的距离`
    //`前提是两线段不相交,相交距离就是0了`
    double dissegtoseg(Line v){
        return min(min(dispointtoseg(v.s),dispointtoseg(v.e)),min(v.dispointtoseg(s),v.dispointtoseg(e)));
    }
    //`返回点p在直线上的投影`
    Point lineprog(Point p){
        return s + ( ((e-s)*((e-s)*(p-s)))/((e-s).len2()) );
    }
    //`返回点p关于直线的对称点`
    Point symmetrypoint(Point p){
        Point q = lineprog(p);
        return Point(2*q.x-p.x,2*q.y-p.y);
    }
};

第三部分—圆

  1. circle(Point a,Point b,Point c) - 三角形的外接圆
  2. circle(Point a,Point b,Point c,bool t) - 三角形的内切圆
  3. input() - 输入
  4. output() - 输出
  5. area() - 面积
  6. circumference() - 周长
  7. relation() - 点和圆的关系
    • 圆外
    • 圆上
    • 圆内
  8. relationseg(Line v) - 线段和圆的关系
  9. relationline(Line v) - 直线和圆的关系
  10. relationcircle(circle v) - 两圆的关系
  11. pointcrosscircle(circle v,Point &p1,Point &p2) - 两个圆的交点个数
  12. pointcrossline(Line v,Point &p1,Point &p2) - 求直线和圆的交点,返回交点个数
  13. gercircle() - 得到过a,b两点,半径为r1的两个圆
  14. getcircle() - 得到与直线u相切,过点q,半径为r1的圆
  15. getcircle() - 同时与直线u,v相切,半径为r1的圆
  16. getcircle() - 同时与不相交圆cx,cy相切,半径为r1的圆
  17. tangentline(Point q,Line &u,Line &v) - 过一点作圆的切线(先判断点和圆的关系)
  18. areacircle(circle v) - 求两圆相交的面积
  19. areatriangle(Point a,Point b) - 求圆和三角形pab的相交面积
struct circle{
    Point p;//圆心
    double r;//半径
    circle(){}
    circle(Point _p,double _r){
        p = _p;
        r = _r;
    }
    circle(double x,double y,double _r){
        p = Point(x,y);
        r = _r;
    }
    //`三角形的外接圆`
    //`需要Point的+ /  rotate()  以及Line的crosspoint()`
    //`利用两条边的中垂线得到圆心`
    //`测试:UVA12304`
    circle(Point a,Point b,Point c){
        Line u = Line((a+b)/2,((a+b)/2)+((b-a).rotleft()));
        Line v = Line((b+c)/2,((b+c)/2)+((c-b).rotleft()));
        p = u.crosspoint(v);
        r = p.distance(a);
    }
    //`三角形的内切圆`
    //`参数bool t没有作用,只是为了和上面外接圆函数区别`
    //`测试:UVA12304`
    circle(Point a,Point b,Point c,bool t){
        Line u,v;
        double m = atan2(b.y-a.y,b.x-a.x), n = atan2(c.y-a.y,c.x-a.x);
        u.s = a;
        u.e = u.s + Point(cos((n+m)/2),sin((n+m)/2));
        v.s = b;
        m = atan2(a.y-b.y,a.x-b.x) , n = atan2(c.y-b.y,c.x-b.x);
        v.e = v.s + Point(cos((n+m)/2),sin((n+m)/2));
        p = u.crosspoint(v);
        r = Line(a,b).dispointtoseg(p);
    }
    //输入
    void input(){
        p.input();
        scanf("%lf",&r);
    }
    //输出
    void output(){
        printf("%.2lf %.2lf %.2lf\n",p.x,p.y,r);
    }
    bool operator == (circle v){
        return (p==v.p) && sgn(r-v.r)==0;
    }
    bool operator < (circle v)const{
        return ((p<v.p)||((p==v.p)&&sgn(r-v.r)<0));
    }
    //面积
    double area(){
        return pi*r*r;
    }
    //周长
    double circumference(){
        return 2*pi*r;
    }
    //`点和圆的关系`
    //`0 圆外`
    //`1 圆上`
    //`2 圆内`
    int relation(Point b){
        double dst = b.distance(p);
        if(sgn(dst-r) < 0)return 2;
        else if(sgn(dst-r)==0)return 1;
        return 0;
    }
    //`线段和圆的关系`
    //`比较的是圆心到线段的距离和半径的关系`
    int relationseg(Line v){
        double dst = v.dispointtoseg(p);
        if(sgn(dst-r) < 0)return 2;
        else if(sgn(dst-r) == 0)return 1;
        return 0;
    }
    //`直线和圆的关系`
    //`比较的是圆心到直线的距离和半径的关系`
    int relationline(Line v){
        double dst = v.dispointtoline(p);
        if(sgn(dst-r) < 0)return 2;
        else if(sgn(dst-r) == 0)return 1;
        return 0;
    }
    //`两圆的关系`
    //`5 相离`
    //`4 外切`
    //`3 相交`
    //`2 内切`
    //`1 内含`
    //`需要Point的distance`
    //`测试:UVA12304`
    int relationcircle(circle v){
        double d = p.distance(v.p);
        if(sgn(d-r-v.r) > 0) return 5;
        if(sgn(d-r-v.r) == 0) return 4;
        double l = fabs(r - v.r);
        if(sgn(d-r-v.r)<0 && sgn(d-l)>0) return 3;
        if(sgn(d-l)==0) return 2;
        if(sgn(d-l)<0) return 1;
    }
    //`求两个圆的交点,返回0表示没有交点,返回1是一个交点,2是两个交点`
    //`需要relationcircle`
    //`测试:UVA12304`
    int pointcrosscircle(circle v,Point &p1,Point &p2){
        int rel = relationcircle(v);
        if(rel == 1 || rel == 5)return 0;
        double d = p.distance(v.p);
        double l = (d*d+r*r-v.r*v.r)/(2*d);
        double h = sqrt(r*r-l*l);
        Point tmp = p + (v.p-p).trunc(l);
        p1 = tmp + ((v.p-p).rotleft().trunc(h));
        p2 = tmp + ((v.p-p).rotright().trunc(h));
        if(rel == 2 || rel == 4)
            return 1;
        return 2;
    }
    //`求直线和圆的交点,返回交点个数`
    int pointcrossline(Line v,Point &p1,Point &p2){
        if(!(*this).relationline(v))return 0;
        Point a = v.lineprog(p);
        double d = v.dispointtoline(p);
        d = sqrt(r*r-d*d);
        if(sgn(d) == 0){
            p1 = a;
            p2 = a;
            return 1;
        }
        p1 = a + (v.e-v.s).trunc(d);
        p2 = a - (v.e-v.s).trunc(d);
        return 2;
    }
    //`得到过a,b两点,半径为r1的两个圆`
    int gercircle(Point a,Point b,double r1,circle &c1,circle &c2){
        circle x(a,r1),y(b,r1);
        int t = x.pointcrosscircle(y,c1.p,c2.p);
        if(!t)return 0;
        c1.r = c2.r = r;
        return t;
    }
    //`得到与直线u相切,过点q,半径为r1的圆`
    //`测试:UVA12304`
    int getcircle(Line u,Point q,double r1,circle &c1,circle &c2){
        double dis = u.dispointtoline(q);
        if(sgn(dis-r1*2)>0)return 0;
        if(sgn(dis) == 0){
            c1.p = q + ((u.e-u.s).rotleft().trunc(r1));
            c2.p = q + ((u.e-u.s).rotright().trunc(r1));
            c1.r = c2.r = r1;
            return 2;
        }
        Line u1 = Line((u.s + (u.e-u.s).rotleft().trunc(r1)),(u.e + (u.e-u.s).rotleft().trunc(r1)));
        Line u2 = Line((u.s + (u.e-u.s).rotright().trunc(r1)),(u.e + (u.e-u.s).rotright().trunc(r1)));
        circle cc = circle(q,r1);
        Point p1,p2;
        if(!cc.pointcrossline(u1,p1,p2))cc.pointcrossline(u2,p1,p2);
        c1 = circle(p1,r1);
        if(p1 == p2){
            c2 = c1;
            return 1;
        }
        c2 = circle(p2,r1);
        return 2;
    }
    //`同时与直线u,v相切,半径为r1的圆`
    //`测试:UVA12304`
    int getcircle(Line u,Line v,double r1,circle &c1,circle &c2,circle &c3,circle &c4){
        if(u.parallel(v))return 0;//两直线平行
        Line u1 = Line(u.s + (u.e-u.s).rotleft().trunc(r1),u.e + (u.e-u.s).rotleft().trunc(r1));
        Line u2 = Line(u.s + (u.e-u.s).rotright().trunc(r1),u.e + (u.e-u.s).rotright().trunc(r1));
        Line v1 = Line(v.s + (v.e-v.s).rotleft().trunc(r1),v.e + (v.e-v.s).rotleft().trunc(r1));
        Line v2 = Line(v.s + (v.e-v.s).rotright().trunc(r1),v.e + (v.e-v.s).rotright().trunc(r1));
        c1.r = c2.r = c3.r = c4.r = r1;
        c1.p = u1.crosspoint(v1);
        c2.p = u1.crosspoint(v2);
        c3.p = u2.crosspoint(v1);
        c4.p = u2.crosspoint(v2);
        return 4;
    }
    //`同时与不相交圆cx,cy相切,半径为r1的圆`
    //`测试:UVA12304`
    int getcircle(circle cx,circle cy,double r1,circle &c1,circle &c2){
        circle x(cx.p,r1+cx.r),y(cy.p,r1+cy.r);
        int t = x.pointcrosscircle(y,c1.p,c2.p);
        if(!t)return 0;
        c1.r = c2.r = r1;
        return t;
    }

    //`过一点作圆的切线(先判断点和圆的关系)`
    //`测试:UVA12304`
    int tangentline(Point q,Line &u,Line &v){
        int x = relation(q);
        if(x == 2)return 0;
        if(x == 1){
            u = Line(q,q + (q-p).rotleft());
            v = u;
            return 1;
        }
        double d = p.distance(q);
        double l = r*r/d;
        double h = sqrt(r*r-l*l);
        u = Line(q,p + ((q-p).trunc(l) + (q-p).rotleft().trunc(h)));
        v = Line(q,p + ((q-p).trunc(l) + (q-p).rotright().trunc(h)));
        return 2;
    }
    //`求两圆相交的面积`
    double areacircle(circle v){
        int rel = relationcircle(v);
        if(rel >= 4)return 0.0;
        if(rel <= 2)return min(area(),v.area());
        double d = p.distance(v.p);
        double hf = (r+v.r+d)/2.0;
        double ss = 2*sqrt(hf*(hf-r)*(hf-v.r)*(hf-d));
        double a1 = acos((r*r+d*d-v.r*v.r)/(2.0*r*d));
        a1 = a1*r*r;
        double a2 = acos((v.r*v.r+d*d-r*r)/(2.0*v.r*d));
        a2 = a2*v.r*v.r;
        return a1+a2-ss;
    }
    //`求圆和三角形pab的相交面积`
    //`测试:POJ3675 HDU3982 HDU2892`
    double areatriangle(Point a,Point b){
        if(sgn((p-a)^(p-b)) == 0)return 0.0;
        Point q[5];
        int len = 0;
        q[len++] = a;
        Line l(a,b);
        Point p1,p2;
        if(pointcrossline(l,q[1],q[2])==2){
            if(sgn((a-q[1])*(b-q[1]))<0)q[len++] = q[1];
            if(sgn((a-q[2])*(b-q[2]))<0)q[len++] = q[2];
        }
        q[len++] = b;
        if(len == 4 && sgn((q[0]-q[1])*(q[2]-q[1]))>0)swap(q[1],q[2]);
        double res = 0;
        for(int i = 0;i < len-1;i++){
            if(relation(q[i])==0||relation(q[i+1])==0){
                double arg = p.rad(q[i],q[i+1]);
                res += r*r*arg/2.0;
            }
            else{
                res += fabs((q[i]-p)^(q[i+1]-p))/2.0;
            }
        }
        return res;
    }
};

第四部分—多边形

  1. n,p line l for each side
  2. input(int _n) - 输入一个n边形
  3. add(point q) - 在列表末尾添加一个点
  4. getline() - populates line array
  5. cmp - comparision in convex_hull order
  6. norm() - 进行极角排序,首先需要找到最左下角的点,需要重载号好point的 < 操作符
  7. getconvex(polygon &convex) - 得到凸包
    注意如果有影响,要特判下所有点共点,或者共线的特殊情况
  8. Graham(polygon &convex) - 得到凸包的另外一种方法
  9. isconvex() - 判断是不是凸的
  10. relationpoint(point q) - 判断点和任意多边形的关系
    • 3 点上 2 边上
    • 1 内部 0 外部
  11. convexcut(line u,polygon &po) - 直线u切割凸多边形左侧,注意直线方向
  12. gercircumference() - returns 周长 //测试 LightOJ1239
  13. getarea() - returns 面积
  14. getdir() - returns 方向:0表示顺时针, 1表示逆时针
  15. getbarycentre() - returns 重心
  16. areacircle(circle c) - 多边形和圆交的面积
  17. relationcircle(circle c) - 多边形和圆关系
    • 2 圆完全在多边形内
    • 1 圆在多边形里面,碰到了多边形边界
    • 0 其它
struct polygon{
    int n;
    Point p[maxp];
    Line l[maxp];
    void input(int _n){
        n = _n;
        for(int i = 0;i < n;i++)
            p[i].input();
    }
    void add(Point q){
        p[n++] = q;
    }
    void getline(){
        for(int i = 0;i < n;i++){
            l[i] = Line(p[i],p[(i+1)%n]);
        }
    }
    struct cmp{
        Point p;
        cmp(const Point &p0){p = p0;}
        bool operator()(const Point &aa,const Point &bb){
            Point a = aa, b = bb;
            int d = sgn((a-p)^(b-p));
            if(d == 0){
                return sgn(a.distance(p)-b.distance(p)) < 0;
            }
            return d > 0;
        }
    };
    //`进行极角排序`
    //`首先需要找到最左下角的点`
    //`需要重载号好Point的 < 操作符(min函数要用) `
    void norm(){
        Point mi = p[0];
        for(int i = 1;i < n;i++)mi = min(mi,p[i]);
        sort(p,p+n,cmp(mi));
    }
    //`得到凸包`
    //`得到的凸包里面的点编号是0$\sim$n-1的`
    //`两种凸包的方法`
    //`注意如果有影响,要特判下所有点共点,或者共线的特殊情况`
    //`测试 LightOJ1203  LightOJ1239`
    void getconvex(polygon &convex){
        sort(p,p+n);
        convex.n = n;
        for(int i = 0;i < min(n,2);i++){
            convex.p[i] = p[i];
        }
        if(convex.n == 2 && (convex.p[0] == convex.p[1]))convex.n--;//特判
        if(n <= 2)return;
        int &top = convex.n;
        top = 1;
        for(int i = 2;i < n;i++){
            while(top && sgn((convex.p[top]-p[i])^(convex.p[top-1]-p[i])) <= 0)
                top--;
            convex.p[++top] = p[i];
        }
        int temp = top;
        convex.p[++top] = p[n-2];
        for(int i = n-3;i >= 0;i--){
            while(top != temp && sgn((convex.p[top]-p[i])^(convex.p[top-1]-p[i])) <= 0)
                top--;
            convex.p[++top] = p[i];
        }
        if(convex.n == 2 && (convex.p[0] == convex.p[1]))convex.n--;//特判
        convex.norm();//`原来得到的是顺时针的点,排序后逆时针`
    }
    //`得到凸包的另外一种方法`
    //`测试 LightOJ1203  LightOJ1239`
    void Graham(polygon &convex){
        norm();
        int &top = convex.n;
        top = 0;
        if(n == 1){
            top = 1;
            convex.p[0] = p[0];
            return;
        }
        if(n == 2){
            top = 2;
            convex.p[0] = p[0];
            convex.p[1] = p[1];
            if(convex.p[0] == convex.p[1])top--;
            return;
        }
        convex.p[0] = p[0];
        convex.p[1] = p[1];
        top = 2;
        for(int i = 2;i < n;i++){
            while( top > 1 && sgn((convex.p[top-1]-convex.p[top-2])^(p[i]-convex.p[top-2])) <= 0 )
                top--;
            convex.p[top++] = p[i];
        }
        if(convex.n == 2 && (convex.p[0] == convex.p[1]))convex.n--;//特判
    }
    //`判断是不是凸的`
    bool isconvex() {
        bool s[3];
        memset(s, false, sizeof(s));
        for(int i = 0;i < n;i++){
            int j = (i+1)%n;
            int k = (j+1)%n;
            s[sgn((p[j]-p[i])^(p[k]-p[i]))+1] = true;
            if(s[0] && s[2])return false;
        }
        return true;
    }
    //`判断点和任意多边形的关系`
    //` 3 点上`
    //` 2 边上`
    //` 1 内部`
    //` 0 外部`
    int relationpoint(Point q){
        for(int i = 0;i < n;i++){
            if(p[i] == q)return 3;
        }
        getline();
        for(int i = 0;i < n;i++){
            if(l[i].pointonseg(q))return 2;
        }
        int cnt = 0;
        for(int i = 0;i < n;i++){
            int j = (i+1)%n;
            int k = sgn((q-p[j])^(p[i]-p[j]));
            int u = sgn(p[i].y-q.y);
            int v = sgn(p[j].y-q.y);
            if(k > 0 && u < 0 && v >= 0)cnt++;
            if(k < 0 && v < 0 && u >= 0)cnt--;
        }
        return cnt != 0;
    }
    //`直线u切割凸多边形左侧`
    //`注意直线方向`
    //`测试:HDU3982`
    void convexcut(Line u,polygon &po){
        int &top = po.n;//注意引用
        top = 0;
        for(int i = 0;i < n;i++){
            int d1 = sgn((u.e-u.s)^(p[i]-u.s));
            int d2 = sgn((u.e-u.s)^(p[(i+1)%n]-u.s));
            if(d1 >= 0)po.p[top++] = p[i];
            if(d1*d2 < 0)po.p[top++] = u.crosspoint(Line(p[i],p[(i+1)%n]));
        }
    }
    //`得到周长`
    //`测试 LightOJ1239`
    double getcircumference(){
        double sum = 0;
        for(int i = 0;i < n;i++){
            sum += p[i].distance(p[(i+1)%n]);
        }
        return sum;
    }
    //`得到面积`
    double getarea(){
        double sum = 0;
        for(int i = 0;i < n;i++){
            sum += (p[i]^p[(i+1)%n]);
        }
        return fabs(sum)/2;
    }
    //`得到方向`
    //` 1 表示逆时针,0表示顺时针`
    bool getdir(){
        double sum = 0;
        for(int i = 0;i < n;i++)
            sum += (p[i]^p[(i+1)%n]);
        if(sgn(sum) > 0)return 1;
        return 0;
    }
    //`得到重心`
    Point getbarycentre(){
        Point ret(0,0);
        double area = 0;
        for(int i = 1;i < n-1;i++){
            double tmp = (p[i]-p[0])^(p[i+1]-p[0]);
            if(sgn(tmp) == 0)continue;
            area += tmp;
            ret.x += (p[0].x+p[i].x+p[i+1].x)/3*tmp;
            ret.y += (p[0].y+p[i].y+p[i+1].y)/3*tmp;
        }
        if(sgn(area)) ret = ret/area;
        return ret;
    }
    //`多边形和圆交的面积`
    //`测试:POJ3675 HDU3982 HDU2892`
    double areacircle(circle c){
        double ans = 0;
        for(int i = 0;i < n;i++){
            int j = (i+1)%n;
            if(sgn( (p[j]-c.p)^(p[i]-c.p) ) >= 0)
                ans += c.areatriangle(p[i],p[j]);
            else ans -= c.areatriangle(p[i],p[j]);
        }
        return fabs(ans);
    }
    //`多边形和圆关系`
    //` 2 圆完全在多边形内`
    //` 1 圆在多边形里面,碰到了多边形边界`
    //` 0 其它`
    int relationcircle(circle c){
        getline();
        int x = 2;
        if(relationpoint(c.p) != 1)return 0;//圆心不在内部
        for(int i = 0;i < n;i++){
            if(c.relationseg(l[i])==2)return 0;
            if(c.relationseg(l[i])==1)x = 1;
        }
        return x;
    }
};

其他

//`AB X AC`
double cross(Point A,Point B,Point C){
    return (B-A)^(C-A);
}
//`AB*AC`
double dot(Point A,Point B,Point C){
    return (B-A)*(C-A);
}

最小矩形面积覆盖

//` A 必须是凸包(而且是逆时针顺序)`
//` 测试 UVA 10173`
double minRectangleCover(polygon A){
    //`要特判A.n < 3的情况`
    if(A.n < 3)return 0.0;
    A.p[A.n] = A.p[0];
    double ans = -1;
    int r = 1, p = 1, q;
    for(int i = 0;i < A.n;i++){
        //`卡出离边A.p[i] - A.p[i+1]最远的点`
        while( sgn( cross(A.p[i],A.p[i+1],A.p[r+1]) - cross(A.p[i],A.p[i+1],A.p[r]) ) >= 0 )
            r = (r+1)%A.n;
        //`卡出A.p[i] - A.p[i+1]方向上正向n最远的点`
        while(sgn( dot(A.p[i],A.p[i+1],A.p[p+1]) - dot(A.p[i],A.p[i+1],A.p[p]) ) >= 0 )
            p = (p+1)%A.n;
        if(i == 0)q = p;
        //`卡出A.p[i] - A.p[i+1]方向上负向最远的点`
        while(sgn(dot(A.p[i],A.p[i+1],A.p[q+1]) - dot(A.p[i],A.p[i+1],A.p[q])) <= 0)
            q = (q+1)%A.n;
        double d = (A.p[i] - A.p[i+1]).len2();
        double tmp = cross(A.p[i],A.p[i+1],A.p[r]) *
            (dot(A.p[i],A.p[i+1],A.p[p]) - dot(A.p[i],A.p[i+1],A.p[q]))/d;
        if(ans < 0 || ans > tmp)ans = tmp;
    }
    return ans;
}

直线切凸多边形

//`多边形是逆时针的,在q1q2的左侧`
//`测试:HDU3982`
vector<Point> convexCut(const vector<Point> &ps,Point q1,Point q2){
    vector<Point>qs;
    int n = ps.size();
    for(int i = 0;i < n;i++){
        Point p1 = ps[i], p2 = ps[(i+1)%n];
        int d1 = sgn((q2-q1)^(p1-q1)), d2 = sgn((q2-q1)^(p2-q1));
        if(d1 >= 0)
            qs.push_back(p1);
        if(d1 * d2 < 0)
            qs.push_back(Line(p1,p2).crosspoint(Line(q1,q2)));
    }
    return qs;
}

半平面交

//`测试 POJ3335 POJ1474 POJ1279`
//***************************
struct halfplane:public Line{
    double angle;
    halfplane(){}
    //`表示向量s->e逆时针(左侧)的半平面`
    halfplane(Point _s,Point _e){
        s = _s;
        e = _e;
    }
    halfplane(Line v){
        s = v.s;
        e = v.e;
    }
    void calcangle(){
        angle = atan2(e.y-s.y,e.x-s.x);
    }
    bool operator <(const halfplane &b)const{
        return angle < b.angle;
    }
};
struct halfplanes{
    int n;
    halfplane hp[maxp];
    Point p[maxp];
    int que[maxp];
    int st,ed;
    void push(halfplane tmp){
        hp[n++] = tmp;
    }
    //去重
    void unique(){
        int m = 1;
        for(int i = 1;i < n;i++){
            if(sgn(hp[i].angle-hp[i-1].angle) != 0)
                hp[m++] = hp[i];
            else if(sgn( (hp[m-1].e-hp[m-1].s)^(hp[i].s-hp[m-1].s) ) > 0)
                hp[m-1] = hp[i];
        }
        n = m;
    }
    bool halfplaneinsert(){
        for(int i = 0;i < n;i++)hp[i].calcangle();
        sort(hp,hp+n);
        unique();
        que[st=0] = 0;
        que[ed=1] = 1;
        p[1] = hp[0].crosspoint(hp[1]);
        for(int i = 2;i < n;i++){
            while(st<ed && sgn((hp[i].e-hp[i].s)^(p[ed]-hp[i].s))<0)ed--;
            while(st<ed && sgn((hp[i].e-hp[i].s)^(p[st+1]-hp[i].s))<0)st++;
            que[++ed] = i;
            if(hp[i].parallel(hp[que[ed-1]]))return false;
            p[ed]=hp[i].crosspoint(hp[que[ed-1]]);
        }
        while(st<ed && sgn((hp[que[st]].e-hp[que[st]].s)^(p[ed]-hp[que[st]].s))<0)ed--;
        while(st<ed && sgn((hp[que[ed]].e-hp[que[ed]].s)^(p[st+1]-hp[que[ed]].s))<0)st++;
        if(st+1>=ed)return false;
        return true;
    }
    //`得到最后半平面交得到的凸多边形`
    //`需要先调用halfplaneinsert() 且返回true`
    void getconvex(polygon &con){
        p[st] = hp[que[st]].crosspoint(hp[que[ed]]);
        con.n = ed-st+1;
        for(int j = st,i = 0;j <= ed;i++,j++)
            con.p[i] = p[j];
    }
};
const int maxn = 1010;
struct circles{
    circle c[maxn];
    double ans[maxn];//`ans[i]表示被覆盖了i次的面积`
    double pre[maxn];
    int n;
    circles(){}
    void add(circle cc){
        c[n++] = cc;
    }
    //`x包含在y中`
    bool inner(circle x,circle y){
        if(x.relationcircle(y) != 1)return 0;
        return sgn(x.r-y.r)<=0?1:0;
    }
    //圆的面积并去掉内含的圆
    void init_or(){
        bool mark[maxn] = {0};
        int i,j,k=0;
        for(i = 0;i < n;i++){
            for(j = 0;j < n;j++)
                if(i != j && !mark[j]){
                    if( (c[i]==c[j])||inner(c[i],c[j]) )break;
                }
            if(j < n)mark[i] = 1;
        }
        for(i = 0;i < n;i++)
            if(!mark[i])
                c[k++] = c[i];
        n = k;
    }
    //`圆的面积交去掉内含的圆`
    void init_add(){
        int i,j,k;
        bool mark[maxn] = {0};
        for(i = 0;i < n;i++){
            for(j = 0;j < n;j++)
                if(i != j && !mark[j]){
                    if( (c[i]==c[j])||inner(c[j],c[i]) )break;
                }
            if(j < n)mark[i] = 1;
        }
        for(i = 0;i < n;i++)
            if(!mark[i])
                c[k++] = c[i];
        n = k;
    }
    //`半径为r的圆,弧度为th对应的弓形的面积`
    double areaarc(double th,double r){
        return 0.5*r*r*(th-sin(th));
    }
    //`测试SPOJVCIRCLES SPOJCIRUT`
    //`SPOJVCIRCLES求n个圆并的面积,需要加上init\_or()去掉重复圆(否则WA)`
    //`SPOJCIRUT 是求被覆盖k次的面积,不能加init\_or()`
    //`对于求覆盖多少次面积的问题,不能解决相同圆,而且不能init\_or()`
    //`求多圆面积并,需要init\_or,其中一个目的就是去掉相同圆`
    void getarea(){
        memset(ans,0,sizeof(ans));
        vector<pair<double,int> >v;
        for(int i = 0;i < n;i++){
            v.clear();
            v.push_back(make_pair(-pi,1));
            v.push_back(make_pair(pi,-1));
            for(int j = 0;j < n;j++)
                if(i != j){
                    Point q = (c[j].p - c[i].p);
                    double ab = q.len(),ac = c[i].r, bc = c[j].r;
                    if(sgn(ab+ac-bc)<=0){
                        v.push_back(make_pair(-pi,1));
                        v.push_back(make_pair(pi,-1));
                        continue;
                    }
                    if(sgn(ab+bc-ac)<=0)continue;
                    if(sgn(ab-ac-bc)>0)continue;
                    double th = atan2(q.y,q.x), fai = acos((ac*ac+ab*ab-bc*bc)/(2.0*ac*ab));
                    double a0 = th-fai;
                    if(sgn(a0+pi)<0)a0+=2*pi;
                    double a1 = th+fai;
                    if(sgn(a1-pi)>0)a1-=2*pi;
                    if(sgn(a0-a1)>0){
                        v.push_back(make_pair(a0,1));
                        v.push_back(make_pair(pi,-1));
                        v.push_back(make_pair(-pi,1));
                        v.push_back(make_pair(a1,-1));
                    }
                    else{
                        v.push_back(make_pair(a0,1));
                        v.push_back(make_pair(a1,-1));
                    }
                }
            sort(v.begin(),v.end());
            int cur = 0;
            for(int j = 0;j < v.size();j++){
                if(cur && sgn(v[j].first-pre[cur])){
                    ans[cur] += areaarc(v[j].first-pre[cur],c[i].r);
                    ans[cur] += 0.5*(Point(c[i].p.x+c[i].r*cos(pre[cur]),c[i].p.y+c[i].r*sin(pre[cur]))^Point(c[i].p.x+c[i].r*cos(v[j].first),c[i].p.y+c[i].r*sin(v[j].first)));
                }
                cur += v[j].second;
                pre[cur] = v[j].first;
            }
        }
        for(int i = 1;i < n;i++)
            ans[i] -= ans[i+1];
    }
};

三维几何

const double eps = 1e-8;
int sgn(double x){
    if(fabs(x) < eps)return 0;
    if(x < 0)return -1;
    else return 1;
}
struct Point3{
    double x,y,z;
    Point3(double _x = 0,double _y = 0,double _z = 0){
        x = _x;
        y = _y;
        z = _z;
    }
    void input(){
        scanf("%lf%lf%lf",&x,&y,&z);
    }
    void output(){
        scanf("%.2lf %.2lf %.2lf\n",x,y,z);
    }
    bool operator ==(const Point3 &b)const{
        return sgn(x-b.x) == 0 && sgn(y-b.y) == 0 && sgn(z-b.z) == 0;
    }
    bool operator <(const Point3 &b)const{
        return sgn(x-b.x)==0?(sgn(y-b.y)==0?sgn(z-b.z)<0:y<b.y):x<b.x;
    }
    double len(){
        return sqrt(x*x+y*y+z*z);
    }
    double len2(){
        return x*x+y*y+z*z;
    }
    double distance(const Point3 &b)const{
        return sqrt((x-b.x)*(x-b.x)+(y-b.y)*(y-b.y)+(z-b.z)*(z-b.z));
    }
    Point3 operator -(const Point3 &b)const{
        return Point3(x-b.x,y-b.y,z-b.z);
    }
    Point3 operator +(const Point3 &b)const{
        return Point3(x+b.x,y+b.y,z+b.z);
    }
    Point3 operator *(const double &k)const{
        return Point3(x*k,y*k,z*k);
    }
    Point3 operator /(const double &k)const{
        return Point3(x/k,y/k,z/k);
    }
    //点乘
    double operator *(const Point3 &b)const{
        return x*b.x+y*b.y+z*b.z;
    }
    //叉乘
    Point3 operator ^(const Point3 &b)const{
        return Point3(y*b.z-z*b.y,z*b.x-x*b.z,x*b.y-y*b.x);
    }
    double rad(Point3 a,Point3 b){
        Point3 p = (*this);
        return acos( ( (a-p)*(b-p) )/ (a.distance(p)*b.distance(p)) );
    }
    //变换长度
    Point3 trunc(double r){
        double l = len();
        if(!sgn(l))return *this;
        r /= l;
        return Point3(x*r,y*r,z*r);
    }
};

struct Line3
{
    Point3 s,e;
    Line3(){}
    Line3(Point3 _s,Point3 _e) {
        s = _s;
        e = _e;
    }
    bool operator ==(const Line3 v) {
        return (s==v.s)&&(e==v.e);
    }
    void input() {
        s.input();
        e.input();
    }
    double length() {
        return s.distance(e);
    }
    //点到直线距离
    double dispointtoline(Point3 p) {
        return ((e-s)^(p-s)).len()/s.distance(e);
    }
    //点到线段距离
    double dispointtoseg(Point3 p) {
        if(sgn((p-s)*(e-s)) < 0 || sgn((p-e)*(s-e)) < 0)
            return min(p.distance(s),e.distance(p));
        return dispointtoline(p);
    }
    //`返回点p在直线上的投影`
    Point3 lineprog(Point3 p) {
        return s + ( ((e-s)*((e-s)*(p-s)))/((e-s).len2()) );
    }
    //`p绕此向量逆时针arg角度`
    Point3 rotate(Point3 p,double ang) {
        if(sgn(((s-p)^(e-p)).len()) == 0)return p;
        Point3 f1 = (e-s)^(p-s);
        Point3 f2 = (e-s)^(f1);
        double len = ((s-p)^(e-p)).len()/s.distance(e);
        f1 = f1.trunc(len); f2 = f2.trunc(len);
        Point3 h = p+f2;
        Point3 pp = h+f1;
        return h + ((p-h)*cos(ang)) + ((pp-h)*sin(ang));
    }
    //`点在直线上`
    bool pointonseg(Point3 p) {
        return sgn( ((s-p)^(e-p)).len() ) == 0 && sgn((s-p)*(e-p)) == 0;
    }
};

struct Plane
{
    Point3 a,b,c,o;//`平面上的三个点,以及法向量`
    Plane(){}
    Plane(Point3 _a,Point3 _b,Point3 _c) {
        a = _a;
        b = _b;
        c = _c;
        o = pvec();
    }
    Point3 pvec() {
        return (b-a)^(c-a);
    }
    //`ax+by+cz+d = 0`
    Plane(double _a,double _b,double _c,double _d) {
        o = Point3(_a,_b,_c);
        if(sgn(_a) != 0)
            a = Point3((-_d-_c-_b)/_a,1,1);
        else if(sgn(_b) != 0)
            a = Point3(1,(-_d-_c-_a)/_b,1);
        else if(sgn(_c) != 0)
            a = Point3(1,1,(-_d-_a-_b)/_c);
    }
    //`点在平面上的判断`
    bool pointonplane(Point3 p) {
        return sgn((p-a)*o) == 0;
    }
    //`两平面夹角`
    double angleplane(Plane f) {
        return acos(o*f.o)/(o.len()*f.o.len());
    }
    //`平面和直线的交点,返回值是交点个数`
    int crossline(Line3 u,Point3 &p) {
        double x = o*(u.e-a);
        double y = o*(u.s-a);
        double d = x-y;
        if(sgn(d) == 0)return 0;
        p = ((u.s*x)-(u.e*y))/d;
        return 1;
    }
    //`点到平面最近点(也就是投影)`
    Point3 pointtoplane(Point3 p) {
        Line3 u = Line3(p,p+o);
        crossline(u,p);
        return p;
    }
    //`平面和平面的交线`
    int crossplane(Plane f,Line3 &u) {
        Point3 oo = o^f.o;
        Point3 v = o^oo;
        double d = fabs(f.o*v);
        if(sgn(d) == 0)return 0;
        Point3 q = a + (v*(f.o*(f.a-a))/d);
        u = Line3(q,q+oo);
        return 1;
    }
};

平面最近点对

const int MAXN = 100010;
const double eps = 1e-8;
const double INF = 1e20;
struct Point{
    double x,y;
    void input(){
        scanf("%lf%lf",&x,&y);
    }
};
double dist(Point a,Point b){
    return sqrt((a.x-b.x)*(a.x-b.x) + (a.y-b.y)*(a.y-b.y));
}
Point p[MAXN];
Point tmpt[MAXN];
bool cmpx(Point a,Point b){
    return a.x < b.x || (a.x == b.x && a.y < b.y);
}
bool cmpy(Point a,Point b){
    return a.y < b.y || (a.y == b.y && a.x < b.x);
}
double Closest_Pair(int left,int right){
    double d = INF;
    if(left == right)return d;
    if(left+1 == right)return dist(p[left],p[right]);
    int mid = (left+right)/2;
    double d1 = Closest_Pair(left,mid);
    double d2 = Closest_Pair(mid+1,right);
    d = min(d1,d2);
    int cnt = 0;
    for(int i = left;i <= right;i++){
        if(fabs(p[mid].x - p[i].x) <= d)
            tmpt[cnt++] = p[i];
    }
    sort(tmpt,tmpt+cnt,cmpy);
    for(int i = 0;i < cnt;i++){
        for(int j = i+1;j < cnt && tmpt[j].y - tmpt[i].y < d;j++)
            d = min(d,dist(tmpt[i],tmpt[j]));
    }
    return d;
}
int main(){
    int n;
    while(scanf("%d",&n) == 1 && n){
        for(int i = 0;i < n;i++)p[i].input();
        sort(p,p+n,cmpx);
        printf("%.2lf\n",Closest_Pair(0,n-1));
    }
    return 0;
}

三维凸包

  例题:HDU - 4273

const double eps = 1e-8;
const int MAXN = 550;
int sgn(double x){
    if(fabs(x) < eps)return 0;
    if(x < 0)return -1;
    else return 1;
}
struct Point3{
    double x,y,z;
    Point3(double _x = 0, double _y = 0, double _z = 0){
        x = _x;
        y = _y;
        z = _z;
    }
    void input(){
        scanf("%lf%lf%lf",&x,&y,&z);
    }
    bool operator ==(const Point3 &b)const{
        return sgn(x-b.x) == 0 && sgn(y-b.y) == 0 && sgn(z-b.z) == 0;
    }
    double len(){
        return sqrt(x*x+y*y+z*z);
    }
    double len2(){
        return x*x+y*y+z*z;
    }
    double distance(const Point3 &b)const{
        return sqrt((x-b.x)*(x-b.x)+(y-b.y)*(y-b.y)+(z-b.z)*(z-b.z));
    }
    Point3 operator -(const Point3 &b)const{
        return Point3(x-b.x,y-b.y,z-b.z);
    }
    Point3 operator +(const Point3 &b)const{
        return Point3(x+b.x,y+b.y,z+b.z);
    }
    Point3 operator *(const double &k)const{
        return Point3(x*k,y*k,z*k);
    }
    Point3 operator /(const double &k)const{
        return Point3(x/k,y/k,z/k);
    }
    //点乘
    double operator *(const Point3 &b)const{
        return x*b.x + y*b.y + z*b.z;
    }
    //叉乘
    Point3 operator ^(const Point3 &b)const{
        return Point3(y*b.z-z*b.y,z*b.x-x*b.z,x*b.y-y*b.x);
    }
};
struct CH3D{
    struct face{
        //表示凸包一个面上的三个点的编号
        int a,b,c;
        //表示该面是否属于最终的凸包上的面
        bool ok;
    };
    //初始顶点数
    int n;
    Point3 P[MAXN];
    //凸包表面的三角形数
    int num;
    //凸包表面的三角形
    face F[8*MAXN];
    int g[MAXN][MAXN];
    //叉乘
    Point3 cross(const Point3 &a,const Point3 &b,const Point3 &c){
        return (b-a)^(c-a);
    }
    //`三角形面积*2`
    double area(Point3 a,Point3 b,Point3 c){
        return ((b-a)^(c-a)).len();
    }
    //`四面体有向面积*6`
    double volume(Point3 a,Point3 b,Point3 c,Point3 d){
        return ((b-a)^(c-a))*(d-a);
    }
    //`正:点在面同向`
    double dblcmp(Point3 &p,face &f){
        Point3 p1 = P[f.b] - P[f.a];
        Point3 p2 = P[f.c] - P[f.a];
        Point3 p3 = p - P[f.a];
        return (p1^p2)*p3;
    }
    void deal(int p,int a,int b){
        int f = g[a][b];
        face add;
        if(F[f].ok){
            if(dblcmp(P[p],F[f]) > eps)
                dfs(p,f);
            else {
                add.a = b;
                add.b = a;
                add.c = p;
                add.ok = true;
                g[p][b] = g[a][p] = g[b][a] = num;
                F[num++] = add;
            }
        }
    }
    //递归搜索所有应该从凸包内删除的面
    void dfs(int p,int now){
        F[now].ok = false;
        deal(p,F[now].b,F[now].a);
        deal(p,F[now].c,F[now].b);
        deal(p,F[now].a,F[now].c);
    }
    bool same(int s,int t){
        Point3 &a = P[F[s].a];
        Point3 &b = P[F[s].b];
        Point3 &c = P[F[s].c];
        return fabs(volume(a,b,c,P[F[t].a])) < eps &&
            fabs(volume(a,b,c,P[F[t].b])) < eps &&
            fabs(volume(a,b,c,P[F[t].c])) < eps;
    }
    //构建三维凸包
    void create(){
        num = 0;
        face add;

        //***********************************
        //此段是为了保证前四个点不共面
        bool flag = true;
        for(int i = 1;i < n;i++){
            if(!(P[0] == P[i])){
                swap(P[1],P[i]);
                flag = false;
                break;
            }
        }
        if(flag)return;
        flag = true;
        for(int i = 2;i < n;i++){
            if( ((P[1]-P[0])^(P[i]-P[0])).len() > eps ){
                swap(P[2],P[i]);
                flag = false;
                break;
            }
        }
        if(flag)return;
        flag = true;
        for(int i = 3;i < n;i++){
            if(fabs( ((P[1]-P[0])^(P[2]-P[0]))*(P[i]-P[0]) ) > eps){
                swap(P[3],P[i]);
                flag = false;
                break;
            }
        }
        if(flag)return;
        //**********************************

        for(int i = 0;i < 4;i++){
            add.a = (i+1)%4;
            add.b = (i+2)%4;
            add.c = (i+3)%4;
            add.ok = true;
            if(dblcmp(P[i],add) > 0)swap(add.b,add.c);
            g[add.a][add.b] = g[add.b][add.c] = g[add.c][add.a] = num;
            F[num++] = add;
        }
        for(int i = 4;i < n;i++)
            for(int j = 0;j < num;j++)
                if(F[j].ok && dblcmp(P[i],F[j]) > eps){
                    dfs(i,j);
                    break;
                }
        int tmp = num;
        num = 0;
        for(int i = 0;i < tmp;i++)
            if(F[i].ok)
                F[num++] = F[i];
    }
    //表面积
    //`测试:HDU3528`
    double area(){
        double res = 0;
        if(n == 3){
            Point3 p = cross(P[0],P[1],P[2]);
            return p.len()/2;
        }
        for(int i = 0;i < num;i++)
            res += area(P[F[i].a],P[F[i].b],P[F[i].c]);
        return res/2.0;
    }
    double volume(){
        double res = 0;
        Point3 tmp = Point3(0,0,0);
        for(int i = 0;i < num;i++)
            res += volume(tmp,P[F[i].a],P[F[i].b],P[F[i].c]);
        return fabs(res/6);
    }
    //表面三角形个数
    int triangle(){
        return num;
    }
    //表面多边形个数
    //`测试:HDU3662`
    int polygon(){
        int res = 0;
        for(int i = 0;i < num;i++){
            bool flag = true;
            for(int j = 0;j < i;j++)
                if(same(i,j)){
                    flag = 0;
                    break;
                }
            res += flag;
        }
        return res;
    }
    //重心
    //`测试:HDU4273`
    Point3 barycenter(){
        Point3 ans = Point3(0,0,0);
        Point3 o = Point3(0,0,0);
        double all = 0;
        for(int i = 0;i < num;i++){
            double vol = volume(o,P[F[i].a],P[F[i].b],P[F[i].c]);
            ans = ans + (((o+P[F[i].a]+P[F[i].b]+P[F[i].c])/4.0)*vol);
            all += vol;
        }
        ans = ans/all;
        return ans;
    }
    //点到面的距离
    //`测试:HDU4273`
    double ptoface(Point3 p,int i){
        double tmp1 = fabs(volume(P[F[i].a],P[F[i].b],P[F[i].c],p));
        double tmp2 = ((P[F[i].b]-P[F[i].a])^(P[F[i].c]-P[F[i].a])).len();
        return tmp1/tmp2;
    }
};
CH3D hull;
int main()
{
    while(scanf("%d",&hull.n) == 1){
        for(int i = 0;i < hull.n;i++)hull.P[i].input();
        hull.create();
        Point3 p = hull.barycenter();
        double ans = 1e20;
        for(int i = 0;i < hull.num;i++)
            ans = min(ans,hull.ptoface(p,i));
        printf("%.3lf\n",ans);
    }
    return 0;
}
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