简单几何(点的位置) POJ 1584 A Round Peg in a Ground Hole

 

题目传送门

题意：判断给定的多边形是否为凸的，peg(pig?)是否在多边形内，且以其为圆心的圆不超出多边形(擦着边也不行)。

分析：判断凸多边形就用凸包，看看点集的个数是否为n。在多边形内用叉积方向来判断，最后再用点到直线的距离和半径比大小(不是线段)

 

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/************************************************ * Author        :Running_Time * Created Time  :2015/11/2 星期一 19:49:13 * File Name     :POJ_1584.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 = 1e5 + 10; const int INF = 0x3f3f3f3f; const int MOD = 1e9 + 7; const double EPS = 1e-10; const double PI = acos (-1.0); 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 () {}     Point (double x, double y) : 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) const {       //向量乘以标量         return Point (x * p, y * p);     }     Point operator / (double p) const {       //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 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 polar_angle(Vector A)  {     //向量极角     return atan2 (A.y, A.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)); } 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 line_line_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 point_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 point_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_line_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 can_seg_seg_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 can_line_seg_inter(Point a1, Point a2, Point b1, Point b2)    {        //判断直线与线段相交，两点式     double c1 = cross (a2 - a1, b1 - a1), c2 = cross (a2 - a1, b2 - a1);     return dcmp (c1 * c2) <= 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_triangle(Point a, Point b, Point c) {       //三角形面积，叉积     return fabs (cross (b - a, c - a)) / 2.0; } double area_poly(Point *p, int n)   {       //多边形面积，叉积     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; } /*     点集凸包，输入点的集合，返回凸包点的集合。     如果不希望在凸包的边上有输入点，把两个 <= 改成 < */ vector<Point> convex_hull(vector<Point> ps) {     sort (ps.begin (), ps.end ());      //x - y排序     ps.erase (unique (ps.begin (), ps.end ()), ps.end ());  //删除重复点     int n = ps.size (), k = 0;     vector<Point> qs (n * 2);     for (int i=0; i<n; ++i) {         while (k > 1 && cross (qs[k-1] - qs[k-2], ps[i] - qs[k-1]) < 0)  k--;         qs[k++] = ps[i];     }     for (int i=n-2, t=k; i>=0; --i)  {         while (k > t && cross (qs[k-1] - qs[k-2], ps[i] - qs[k-1]) < 0)  k--;         qs[k++] = ps[i];     }     qs.resize (k-1);     return qs; }   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; } /*     过点到圆的切线，返回切线条数，切线保存在V中 */ int point_cir_tan(Point p, Circle C, Vector *V) {     Vector u = C.c - p;     double dis = length (u);     if (dis < C.r)  return 0;     else if (dcmp (dis - C.r) == 0) {         V[0] = rotate (u, PI / 2);  return 1;     }     else    {         double ang = asin (C.r / dis);         V[0] = rotate (u, -ang);         V[1] = rotate (u, +ang);         return 0;     } } /*     两圆的公切线，返回公切线条数，切线短点保存在a和b中 */ int cir_cir_tan(Circle A, Circle B, Point *a, Point *b) {     int cnt = 0;     if (A.r < B.r)  {         swap (A, B);    swap (a, b);     }     double d = dot (A.c - B.c, A.c - B.c);     double rsub = A.r - B.r, rsum = A.r + B.r;     if (dcmp (d - rsub) < 0)   return 0;   //内含     double base = polar_angle (B.c - A.c);     if (dcmp (d) == 0 && dcmp (A.r - B.r) == 0) return -1;  //两圆重叠     if (dcmp (d - rsub) == 0)   {       //内切，一条切线         a[cnt] = A.point (base);    b[cnt] = B.point (base);    cnt++;         return 1;     }     //有外公切线     double ang = acos (rsub / d);     a[cnt] = A.point (base + ang);  b[cnt] = B.point (base + ang);  cnt++;     a[cnt] = A.point (base - ang);  b[cnt] = B.point (base - ang);  cnt++;     if (d == rsum)  {         a[cnt] = A.point (base);    b[cnt] = B.point (base + PI);   cnt++;     }     else if (dcmp (d - rsum) > 0)   {       //两条内公切线         double ang2 = acos (rsum / d);         a[cnt] = A.point (base + ang2); b[cnt] = B.point (base + ang2 + PI);    cnt++;         a[cnt] = A.point (base - ang2); b[cnt] = B.point (base - ang2 + PI);    cnt++;     }     return cnt; }   int main(void)    {     int n;     vector<Point> ps;     double r;     Point peg;     while (scanf ("%d", &n) == 1)   {         if (n < 3)  break;         scanf ("%lf", &r);         peg = read_point ();         ps.clear ();         for (int i=0; i<n; ++i) {             ps.push_back (read_point ());         }         vector<Point> qs = convex_hull (ps);         if (qs.size () < n) {             puts ("HOLE IS ILL-FORMED");    continue;         }         qs.push_back (qs[0]);         bool flag = true;         for (int i=0; i<n; ++i) {             if (dcmp (point_to_line (peg, qs[i], qs[i+1]) - r) < 0)  {                 flag = false;   break;             }             if (cross (qs[i+1] - qs[i], peg - qs[i]) < 0) {                 flag = false;   break;             }         }         if (flag)   puts ("PEG WILL FIT");         else    puts ("PEG WILL NOT FIT");     }      //cout << "Time elapsed: " << 1.0 * clock() / CLOCKS_PER_SEC << " s.\n";       return 0; }