poj 3525 Most Distant Point from the Sea (DC2 + Half Plane)

http://poj.org/problem?id=3525

https://icpcarchive.ecs.baylor.edu/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&category=&problem=1891&mosmsg=Submission+received+with+ID+1236113

　　一直不会半平面交，今天看了半平面交的做法，发现半平面交需要注意的细节还真是太多了，要仔细分析才能写对代码。半平面交较快的做法是用双向队列来维护半平面区域中的点和有向直线。

　　这道题的意思是，给出一个凸包，要求出凸包中的离边界最远的点与边界的距离。十分经典的一道半平面交的题目。

　　这题用的方法是移动有向线段，逐渐逼近到所有直线都交出来的平面为空。这个临界值就是所求的最大距离了。1y～

代码如下：

#include <cstdio>
#include <cstring>
#include <cmath>
#include <set>
#include <vector>
#include <iostream>
#include <algorithm>

using namespace std;

// Point class
struct Point {
    double x, y;
    Point() {}
    Point(double x, double y) : x(x), y(y) {}
} ;
template<class T> T sqr(T x) { return x * x;}
inline double ptDis(Point a, Point b) { return sqrt(sqr(a.x - b.x) + sqr(a.y - b.y));}

// basic calculations
typedef Point Vec;
Vec operator + (Vec a, Vec b) { return Vec(a.x + b.x, a.y + b.y);}
Vec operator - (Vec a, Vec b) { return Vec(a.x - b.x, a.y - b.y);}
Vec operator * (Vec a, double p) { return Vec(a.x * p, a.y * p);}
Vec operator / (Vec a, double p) { return Vec(a.x / p, a.y / p);}

const double EPS = 5e-13;
const double PI = acos(-1.0);
inline int sgn(double x) { return fabs(x) < EPS ? 0 : (x < 0 ? -1 : 1);}
bool operator < (Point a, Point b) { return a.x < b.x || (a.x == b.x && a.y < b.y);}
bool operator == (Point a, Point b) { return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0;}

inline double dotDet(Vec a, Vec b) { return a.x * b.x + a.y * b.y;}
inline double crossDet(Vec a, Vec b) { return a.x * b.y - a.y * b.x;}
inline double crossDet(Point o, Point a, Point b) { return crossDet(a - o, b - o);}
inline double vecLen(Vec x) { return sqrt(sqr(x.x) + sqr(x.y));}
inline double toRad(double deg) { return deg / 180.0 * PI;}
inline double angle(Vec v) { return atan2(v.y, v.x);}
inline double angle(Vec a, Vec b) { return acos(dotDet(a, b) / vecLen(a) / vecLen(b));}
inline double triArea(Point a, Point b, Point c) { return fabs(crossDet(b - a, c - a));}
inline Vec vecUnit(Vec x) { return x / vecLen(x);}
inline Vec rotate(Vec x, double rad) { return Vec(x.x * cos(rad) - x.y * sin(rad), x.x * sin(rad) + x.y * cos(rad));}
Vec normal(Vec x) {
    double len = vecLen(x);
    return Vec(- x.y / len, x.x / len);
}

// Line class
struct Line {
    Point s, t;
    Line() {}
    Line(Point s, Point t) : s(s), t(t) {}
    Point point(double x) {
        return s + (t - s) * x;
    }
    Line move(double x) { // while x > 0 move to (s->t)'s left
        Vec nor = normal(t - s);
        nor = nor * x;
        return Line(s + nor, t + nor);
    }
    Vec vec() { return t - s;}
} ;
typedef Line Seg;

inline bool onSeg(Point x, Point a, Point b) { return sgn(crossDet(a - x, b - x)) == 0 && sgn(dotDet(a - x, b - x)) < 0;}
inline bool onSeg(Point x, Seg s) { return onSeg(x, s.s, s.t);}

// 0 : not intersect
// 1 : proper intersect
// 2 : improper intersect
int segIntersect(Point a, Point c, Point b, Point d) {
    Vec v1 = b - a, v2 = c - b, v3 = d - c, v4 = a - d;
    int a_bc = sgn(crossDet(v1, v2));
    int b_cd = sgn(crossDet(v2, v3));
    int c_da = sgn(crossDet(v3, v4));
    int d_ab = sgn(crossDet(v4, v1));
    if (a_bc * c_da > 0 && b_cd * d_ab > 0) return 1;
    if (onSeg(b, a, c) && c_da) return 2;
    if (onSeg(c, b, d) && d_ab) return 2;
    if (onSeg(d, c, a) && a_bc) return 2;
    if (onSeg(a, d, b) && b_cd) return 2;
    return 0;
}
inline int segIntersect(Seg a, Seg b) { return segIntersect(a.s, a.t, b.s, b.t);}

// point of the intersection of 2 lines
Point lineIntersect(Point P, Vec v, Point Q, Vec w) {
    Vec u = P - Q;
    double t = crossDet(w, u) / crossDet(v, w);
    return P + v * t;
}
inline Point lineIntersect(Line a, Line b) { return lineIntersect(a.s, a.t - a.s, b.s, b.t - b.s);}

// Warning: This is a DIRECTED Distance!!!
double pt2Line(Point x, Point a, Point b) {
    Vec v1 = b - a, v2 = x - a;
    return crossDet(v1, v2) / vecLen(v1);
}
inline double pt2Line(Point x, Line L) { return pt2Line(x, L.s, L.t);}

double pt2Seg(Point x, Point a, Point b) {
    if (a == b) return vecLen(x - a);
    Vec v1 = b - a, v2 = x - a, v3 = x - b;
    if (sgn(dotDet(v1, v2)) < 0) return vecLen(v2);
    if (sgn(dotDet(v1, v3)) > 0) return vecLen(v3);
    return fabs(crossDet(v1, v2)) / vecLen(v1);
}
inline double pt2Seg(Point x, Seg s) { return pt2Seg(x, s.s, s.t);}

// Polygon class
struct Poly {
    vector<Point> pt;
    Poly() { pt.clear();}
    Poly(vector<Point> pt) : pt(pt) {}
    Point operator [] (int x) const { return pt[x];}
    int size() { return pt.size();}
    double area() {
        double ret = 0.0;
        int sz = pt.size();
        for (int i = 1; i < sz; i++) {
            ret += crossDet(pt[i], pt[i - 1]);
        }
        return fabs(ret / 2.0);
    }
} ;

// Circle class
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);
    }
} ;

// Cirlce operations
int lineCircleIntersect(Line L, Circle C, double &t1, double &t2, vector<Point> &sol) {
    double a = L.s.x, b = L.t.x - C.c.x, c = L.s.y, d = L.t.y - C.c.y;
    double e = sqr(a) + sqr(c), f = 2 * (a * b + c * d), g = sqr(b) + sqr(d) - sqr(C.r);
    double delta = sqr(f) - 4.0 * e * g;
    if (sgn(delta) < 0) return 0;
    if (sgn(delta) == 0) {
        t1 = t2 = -f / (2.0 * e);
        sol.push_back(L.point(t1));
        return 1;
    }
    t1 = (-f - sqrt(delta)) / (2.0 * e);
    sol.push_back(L.point(t1));
    t2 = (-f + sqrt(delta)) / (2.0 * e);
    sol.push_back(L.point(t2));
    return 2;
}

int lineCircleIntersect(Line L, Circle C, vector<Point> &sol) {
    Vec dir = L.t - L.s, nor = normal(dir);
    Point mid = lineIntersect(C.c, nor, L.s, dir);
    double len = sqr(C.r) - sqr(ptDis(C.c, mid));
    if (sgn(len) < 0) return 0;
    if (sgn(len) == 0) {
        sol.push_back(mid);
        return 1;
    }
    Vec dis = vecUnit(dir);
    len = sqrt(len);
    sol.push_back(mid + dis * len);
    sol.push_back(mid - dis * len);
    return 2;
}

// -1 : coincide
int circleCircleIntersect(Circle C1, Circle C2, vector<Point> &sol) {
    double d = vecLen(C1.c - C2.c);
    if (sgn(d) == 0) {
        if (sgn(C1.r - C2.r) == 0) {
            return -1;
        }
        return 0;
    }
    if (sgn(C1.r + C2.r - d) < 0) return 0;
    if (sgn(fabs(C1.r - C2.r) - d) > 0) return 0;
    double a = angle(C2.c - C1.c);
    double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2.0 * C1.r * d));
    Point p1 = C1.point(a - da), p2 = C1.point(a + da);
    sol.push_back(p1);
    if (p1 == p2) return 1;
    sol.push_back(p2);
    return 2;
}

void circleCircleIntersect(Circle C1, Circle C2, vector<double> &sol) {
    double d = vecLen(C1.c - C2.c);
    if (sgn(d) == 0) return ;
    if (sgn(C1.r + C2.r - d) < 0) return ;
    if (sgn(fabs(C1.r - C2.r) - d) > 0) return ;
    double a = angle(C2.c - C1.c);
    double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2.0 * C1.r * d));
    sol.push_back(a - da);
    sol.push_back(a + da);
}

int tangent(Point p, Circle C, vector<Vec> &sol) {
    Vec u = C.c - p;
    double dist = vecLen(u);
    if (dist < C.r) return 0;
    if (sgn(dist - C.r) == 0) {
        sol.push_back(rotate(u, PI / 2.0));
        return 1;
    }
    double ang = asin(C.r / dist);
    sol.push_back(rotate(u, -ang));
    sol.push_back(rotate(u, ang));
    return 2;
}

// ptA : points of tangency on circle A
// ptB : points of tangency on circle B
int tangent(Circle A, Circle B, vector<Point> &ptA, vector<Point> &ptB) {
    if (A.r < B.r) {
        swap(A, B);
        swap(ptA, ptB);
    }
    int d2 = sqr(A.c.x - B.c.x) + sqr(A.c.y - B.c.y);
    int rdiff = A.r - B.r, rsum = A.r + B.r;
    if (d2 < sqr(rdiff)) return 0;
    double base = atan2(B.c.y - A.c.y, B.c.x - A.c.x);
    if (d2 == 0 && A.r == B.r) return -1;
    if (d2 == sqr(rdiff)) {
        ptA.push_back(A.point(base));
        ptB.push_back(B.point(base));
        return 1;
    }
    double ang = acos((A.r - B.r) / sqrt(d2));
    ptA.push_back(A.point(base + ang));
    ptB.push_back(B.point(base + ang));
    ptA.push_back(A.point(base - ang));
    ptB.push_back(B.point(base - ang));
    if (d2 == sqr(rsum)) {
        ptA.push_back(A.point(base));
        ptB.push_back(B.point(PI + base));
    } else if (d2 > sqr(rsum)) {
        ang = acos((A.r + B.r) / sqrt(d2));
        ptA.push_back(A.point(base + ang));
        ptB.push_back(B.point(PI + base + ang));
        ptA.push_back(A.point(base - ang));
        ptB.push_back(B.point(PI + base - ang));
    }
    return (int) ptA.size();
}

// -1 : onside
// 0 : outside
// 1 : inside
int ptInPoly(Point p, Poly &poly) {
    int wn = 0, sz = poly.size();
    for (int i = 0; i < sz; i++) {
        if (onSeg(p, poly[i], poly[(i + 1) % sz])) return -1;
        int k = sgn(crossDet(poly[(i + 1) % sz] - poly[i], p - poly[i]));
        int d1 = sgn(poly[i].y - p.y);
        int d2 = sgn(poly[(i + 1) % sz].y - p.y);
        if (k > 0 && d1 <= 0 && d2 > 0) wn++;
        if (k < 0 && d2 <= 0 && d1 > 0) wn--;
    }
    if (wn != 0) return 1;
    return 0;
}

// if DO NOT need a high precision
/*
int ptInPoly(Point p, Poly poly) {
    int sz = poly.size();
    double ang = 0.0, tmp;
    for (int i = 0; i < sz; i++) {
        if (onSeg(p, poly[i], poly[(i + 1) % sz])) return -1;
        tmp = angle(poly[i] - p) - angle(poly[(i + 1) % sz] - p) + PI;
        ang += tmp - floor(tmp / (2.0 * PI)) * 2.0 * PI - PI;
    }
    if (sgn(ang - PI) == 0) return -1;
    if (sgn(ang) == 0) return 0;
    return 1;
}
*/


// Convex Hull algorithms
// return the number of points in convex hull

// andwer's algorithm
// if DO NOT want the points on the side of convex hull, change all "<" into "<="
int andrew(Point *pt, int n, Point *ch) {
    sort(pt, pt + n);
    int m = 0;
    for (int i = 0; i < n; i++) {
        while (m > 1 && crossDet(ch[m - 1] - ch[m - 2], pt[i] - ch[m - 2]) <= 0) m--;
        ch[m++] = pt[i];
    }
    int k = m;
    for (int i = n - 2; i >= 0; i--) {
        while (m > k && crossDet(ch[m - 1] - ch[m - 2], pt[i] - ch[m - 2]) <= 0) m--;
        ch[m++] = pt[i];
    }
    if (n > 1) m--;
    return m;
}

// graham's algorithm
// if DO NOT want the points on the side of convex hull, change all "<=" into "<"
Point origin;
inline bool cmpAng(Point p1, Point p2) { return crossDet(origin, p1, p2) > 0;}
inline bool cmpDis(Point p1, Point p2) { return ptDis(p1, origin) > ptDis(p2, origin);}

void removePt(Point *pt, int &n) {
    int idx = 1;
    for (int i = 2; i < n; i++) {
        if (sgn(crossDet(origin, pt[i], pt[idx]))) pt[++idx] = pt[i];
        else if (cmpDis(pt[i], pt[idx])) pt[idx] = pt[i];
    }
    n = idx + 1;
}

int graham(Point *pt, int n, Point *ch) {
    int top = -1;
    for (int i = 1; i < n; i++) {
        if (pt[i].y < pt[0].y || (pt[i].y == pt[0].y && pt[i].x < pt[0].x)) swap(pt[i], pt[0]);
    }
    origin = pt[0];
    sort(pt + 1, pt + n, cmpAng);
    removePt(pt, n);
    for (int i = 0; i < n; i++) {
        if (i >= 2) {
            while (!(crossDet(ch[top - 1], pt[i], ch[top]) <= 0)) top--;
        }
        ch[++top] = pt[i];
    }
    return top + 1;
}


// Half Plane
// The intersected area is always a convex polygon.
// Directed Line class
struct DLine {
    Point p;
    Vec v;
    double ang;
    DLine() {}
    DLine(Point p, Vec v) : p(p), v(v) { ang = atan2(v.y, v.x);}
    bool operator < (const DLine &L) const { return ang < L.ang;}
    DLine move(double x) { // while x > 0 move to v's left
        Vec nor = normal(v);
        nor = nor * x;
        return DLine(p + nor, v);
    }

} ;

Poly cutPoly(Poly &poly, Point a, Point b) {
    Poly ret = Poly();
    int n = poly.size();
    for (int i = 0; i < n; i++) {
        Point c = poly[i], d = poly[(i + 1) % n];
        if (sgn(crossDet(b - a, c - a)) >= 0) ret.pt.push_back(c);
        if (sgn(crossDet(b - a, c - d)) != 0) {
            Point ip = lineIntersect(a, b - a, c, d - c);
            if (onSeg(ip, c, d)) ret.pt.push_back(ip);
        }
    }
    return ret;
}
inline Poly cutPoly(Poly &poly, DLine L) { return cutPoly(poly, L.p, L.p + L.v);}

inline bool onLeft(DLine L, Point p) { return crossDet(L.v, p - L.p) > 0;}
Point dLineIntersect(DLine a, DLine b) {
    Vec u = a.p - b.p;
    double t = crossDet(b.v, u) / crossDet(a.v, b.v);
    return a.p + a.v * t;
}

Poly halfPlane(DLine *L, int n) {
    Poly ret = Poly();
    sort(L, L + n);
    int fi, la;
    Point *p = new Point[n];
    DLine *q = new DLine[n];
    q[fi = la = 0] = L[0];
    for (int i = 1; i < n; i++) {
        while (fi < la && !onLeft(L[i], p[la - 1])) la--;
        while (fi < la && !onLeft(L[i], p[fi])) fi++;
        q[++la] = L[i];
        if (fabs(crossDet(q[la].v, q[la - 1].v)) < EPS) {
            la--;
            if (onLeft(q[la], L[i].p)) q[la] = L[i];
        }
        if (fi < la) p[la - 1] = dLineIntersect(q[la - 1], q[la]);
    }
    while (fi < la && !onLeft(q[fi], p[la - 1])) la--;
    if (la - fi <= 1) return ret;
    p[la] = dLineIntersect(q[la], q[fi]);
    for (int i = fi; i <= la; i++) ret.pt.push_back(p[i]);
    return ret;
}


// 3D Geometry
void getCoor(double R, double lat, double lng, double &x, double &y, double &z) {
    lat = toRad(lat);
    lng = toRad(lng);
    x = R * cos(lat) * cos(lng);
    y = R * cos(lat) * sin(lng);
    z = R * sin(lat);
}


/****************** template above *******************/

const int N = 111;

bool test(double x, int n, DLine *line) {
    DLine tmp[N];
    for (int i = 0; i < n; i++) {
        tmp[i] = line[i].move(x);
    }
    return halfPlane(tmp, n).size() > 0;
}

double DC2(int n, DLine *line) {
    double h = 0.0, t = 1e5;
    while (t - h > 1e-6) {
        double m = (h + t) / 2.0;
        if (test(m, n, line)) h = m;
        else t = m;
    }
    return h;
}

int main() {
//    freopen("in", "r", stdin);
    DLine line[N];
    Point pt[N];
    int n;
    while (cin >> n && n) {
        for (int i = 0; i < n; i++) {
            cin >> pt[i].x >> pt[i].y;
        }
        pt[n] = pt[0];
        for (int i = 0; i < n; i++) {
            line[i] = DLine(pt[i], pt[i + 1] - pt[i]);
        }
        printf("%.10f\n", DC2(n, line));
    }
    return 0;
}

——written by Lyon