简单几何(凸包) POJ 1113 Wall

 

题目传送门

题意：求最短路线，使得线上任意一点离城堡至少L距离

分析：先求凸包，答案 = 凸包的长度 + 以L为半径的圆的周长

 

/************************************************
* Author        :Running_Time
* Created Time  :2015/10/25 11:00:48
* File Name     :POJ_1113.cpp
 ************************************************/

#include <cstdio>
#include <algorithm>
#include <iostream>
#include <sstream>
#include <cstring>
#include <cmath>
#include <string>
#include <vector>
#include <queue>
#include <deque>
#include <stack>
#include <list>
#include <map>
#include <set>
#include <bitset>
#include <cstdlib>
#include <ctime>
using namespace std;

#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
typedef long long ll;
const int N = 1e3 + 10;
const int INF = 0x3f3f3f3f;
const int MOD = 1e9 + 7;
const double EPS = 1e-10; 
int dcmp(double x)  {       //三态函数，减少精度问题 
    if (fabs (x) < EPS) return 0; 
    else    return x < 0 ? -1 : 1; 
} 
struct Point    {       //点的定义 
    double x, y; 
    Point (double x=0, double y=0) : x (x), y (y) {} 
    Point operator + (const Point &r) const {       //向量加法 
        return Point (x + r.x, y + r.y); 
    } 
    Point operator - (const Point &r) const {       //向量减法 
        return Point (x - r.x, y - r.y); 
    } 
    Point operator * (double p)  {       //向量乘以标量 
        return Point (x * p, y * p); 
    } 
    Point operator / (double p)  {       //向量除以标量 
        return Point (x / p, y / p); 
    } 
    bool operator < (const Point &r) const {       //点的坐标排序 
        return x < r.x || (x == r.x && y < r.y); 
    } 
    bool operator == (const Point &r) const {       //判断同一个点 
        return dcmp (x - r.x) == 0 && dcmp (y - r.y) == 0; 
    } 
}; 
typedef Point Vector;       //向量的定义 
Point read_point(void)   {      //点的读入 
    double x, y; 
    scanf ("%lf%lf", &x, &y); 
    return Point (x, y); 
} 
double polar_angle(Vector A)  {     //向量极角 
    return atan2 (A.y, A.x); 
} 
double dot(Vector A, Vector B)  {       //向量点积 
    return A.x * B.x + A.y * B.y; 
} 
double cross(Vector A, Vector B)    {       //向量叉积 
    return A.x * B.y - A.y * B.x; 
} 
double length(Vector A) {       //向量长度，点积 
    return sqrt (dot (A, A)); 
} 
double angle(Vector A, Vector B)    {       //向量转角，逆时针，点积 
    return acos (dot (A, B) / length (A) / length (B)); 
} 
double area_triangle(Point a, Point b, Point c) {       //三角形面积，叉积 
    return fabs (cross (b - a, c - a)) / 2.0; 
} 
Vector rotate(Vector A, double rad) {       //向量旋转，逆时针 
    return Vector (A.x * cos (rad) - A.y * sin (rad), A.x * sin (rad) + A.y * cos (rad)); 
} 
Vector nomal(Vector A)  {       //向量的单位法向量 
    double len = length (A); 
    return Vector (-A.y / len, A.x / len); 
} 
Point point_inter(Point p, Vector V, Point q, Vector W)    {        //两直线交点，参数方程 
    Vector U = p - q; 
    double t = cross (W, U) / cross (V, W); 
    return p + V * t; 
} 
double dis_to_line(Point p, Point a, Point b)   {       //点到直线的距离，两点式 
    Vector V1 = b - a, V2 = p - a; 
    return fabs (cross (V1, V2)) / length (V1); 
} 
double dis_to_seg(Point p, Point a, Point b)    {       //点到线段的距离，两点式 
   
    if (a == b) return length (p - a); 
    Vector V1 = b - a, V2 = p - a, V3 = p - b; 
    if (dcmp (dot (V1, V2)) < 0)    return length (V2); 
    else if (dcmp (dot (V1, V3)) > 0)   return length (V3); 
    else    return fabs (cross (V1, V2)) / length (V1); 
} 
Point point_proj(Point p, Point a, Point b)   {     //点在直线上的投影，两点式 
    Vector V = b - a; 
    return a + V * (dot (V, p - a) / dot (V, V)); 
} 
bool inter(Point a1, Point a2, Point b1, Point b2)  {       //判断线段相交，两点式 
    double c1 = cross (a2 - a1, b1 - a1), c2 = cross (a2 - a1, b2 - a1), 
           c3 = cross (b2 - b1, a1 - b1), c4 = cross (b2 - b1, a2 - b1); 
    return dcmp (c1) * dcmp (c2) < 0 && dcmp (c3) * dcmp (c4) < 0; 
} 
bool on_seg(Point p, Point a1, Point a2)    {       //判断点在线段上，两点式 
    return dcmp (cross (a1 - p, a2 - p)) == 0 && dcmp (dot (a1 - p, a2 - p)) < 0; 
} 
double area_poly(Point *p, int n)   {       //多边形面积 
    double ret = 0; 
    for (int i=1; i<n-1; ++i)   { 
        ret += fabs (cross (p[i] - p[0], p[i+1] - p[0])); 
    } 
    return ret / 2; 
} 
vector<Point> convex_hull(vector<Point> &P) {
    sort (P.begin (), P.end ());
    int n = P.size (), k = 0;
    vector<Point> ret (n * 2);
    for (int i=0; i<n; ++i) {
        while (k > 1 && cross (ret[k-1] - ret[k-2], P[i] - ret[k-1]) <= 0)  k--;
        ret[k++] = P[i];
    }
    for (int i=n-2, t=k; i>=0; --i)  {
        while (k > t && cross (ret[k-1] - ret[k-2], P[i] - ret[k-1]) <= 0)  k--;
        ret[k++] = P[i];
    }
    ret.resize (k-1);
    return ret;
}
  
struct Circle   { 
    Point c; 
    double r; 
    Circle () {} 
    Circle (Point c, double r) : c (c), r (r) {} 
    Point point(double a)   { 
        return Point (c.x + cos (a) * r, c.y + sin (a) * r); 
    } 
}; 
struct Line { 
    Point p; 
    Vector v; 
    double r; 
    Line () {} 
    Line (const Point &p, const Vector &v) : p (p), v (v) { 
        r = polar_angle (v); 
    } 
    Point point(double a)   { 
        return p + v * a; 
    } 
}; 
  
  
/* 
    直线相交求交点，返回交点个数，交点保存在P中 
*/
int line_cir_inter(Line L, Circle C, double &t1, double &t2, vector<Point> &P)    { 
    double a = L.v.x, b = L.p.x - C.c.x, c = L.v.y, d = L.p.y - C.c.y; 
    double e = a * a + c * c, f = 2 * (a * b + c * d), g = b * b + d * d - C.r * C.r; 
    double delta = f * f - 4 * e * g; 
    if (dcmp (delta) < 0)   return 0; 
    if (dcmp (delta) == 0)  { 
        t1 = t2 = -f / (2 * e); P.push_back (L.point (t1)); 
        return -1; 
    } 
    t1 = (-f - sqrt (delta)) / (2 * e); P.push_back (L.point (t1)); 
    t2 = (-f + sqrt (delta)) / (2 * e); P.push_back (L.point (t2)); 
    if (dcmp (t1) < 0 || dcmp (t2) < 0) return 0; 
    return 2; 
} 
   
/* 
    两圆相交求交点，返回交点个数。交点保存在P中 
*/
int cir_cir_inter(Circle C1, Circle C2, vector<Point> &P)    { 
    double d = length (C1.c - C2.c); 
    if (dcmp (d) == 0)  { 
        if (dcmp (C1.r - C2.r) == 0)    return -1;      //两圆重叠 
        else    return 0; 
    } 
    if (dcmp (C1.r + C2.r - d) < 0) return 0; 
    if (dcmp (fabs (C1.r - C2.r) - d) < 0)  return 0; 
    double a = polar_angle (C2.c - C1.c); 
    double da = acos ((C1.r * C1.r + d * d - C2.r * C2.r) / (2 * C1.r * d));        //C1C2到C1P1的角？ 
    Point p1 = C1.point (a - da), p2 = C2.point (a + da); 
    P.push_back (p1); 
    if (p1 == p2)   return 1; 
    else    P.push_back (p2); 
    return 2; 
}

const double PI = acos (-1.0);
vector<Point> p;

int main(void)    {
    int n;
    double L;
    while (scanf ("%d%lf", &n, &L) == 2)   {
        p.clear ();
        for (int i=0; i<n; ++i) {
            p.push_back (read_point ());
        }
        vector<Point> q = convex_hull (p);
        double ans = 0;
        q.push_back (q[0]);
        for (int i=0; i<q.size ()-1; ++i)   {
            ans += length (q[i+1] - q[i]);
        }
        ans += 2 * PI * L;
        printf ("%.0f\n", ans);
    }

    return 0;
}