LA 2218 Triathlon (Geometry, Half Plane Intersection)

https://icpcarchive.ecs.baylor.edu/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&problem=219

　　题意是，给出铁人三项运动员在每一项中的速度，问他是否有可能获得第一。

　　如果还没学过半平面交，估计做这题的时候应该是不停yy如何比较每两个运动员之间谁赢谁输，然后忽略了整体情况而导致狂wa不止。

　　如果是利用半平面交，这时我们要假设前两项项目的路程占全程分别是x和y，那么总时间T(x,y)=x/v+y/u+(1-x-y)/w。

　　对于判断当前运动员i是否能获胜，将它与其他运动员j比较，满足Ti<Tj(对于所有i!=j)，那么运动员i就能获得第一。

　　于是有(1/uj-1/ui-1/wj+1/wi)x+(1/vj-1/vi-1/wj+1/wi)y+(1/wj-1/wi)>0，然后，我们就可以划分出多个半平面了。

　　半平面的交都是可行解，如果半平面交集为空，那么运动员i就无法获得第一。

　　在构建半平面的时候，需要对两种必胜/必输状态进行特判，否则在构建半平面交的时候会产生错误。

代码如下：

View Code

#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
;
DLine *q = new DLine
;
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;
const double ep = 1e4;
double v
, u
, w
;

bool halfPlaneTest(DLine *L, int n) {
Poly ret = Poly();
sort(L, L + n);
int fi, la;
Point *p = new Point
;
DLine *q = new DLine
;
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--;
//    cout << la - fi << " ~~" << endl;
if (la - fi <= 1) return false;
return true;
}

bool test(int id, int n) {
DLine tmp
;
int top = 0;
tmp[top++] = DLine(Point(0.0, 0.0), Vec(0.0, -1.0));
tmp[top++] = DLine(Point(0.0, 0.0), Vec(1.0, 0.0));
tmp[top++] = DLine(Point(0.0, 1.0), Vec(-1.0, 1.0));
for (int i = 0; i < n; i++) {
if (i == id) continue;
// 下面的特判必不可少
if (v[i] >= v[id] && u[i] >= u[id] && w[i] >= w[id]) return false;
if (v[i] <= v[id] && u[i] <= u[id] && w[i] <= w[id]) continue;
double A = ep / v[i] - ep / v[id] - ep / w[i] + ep / w[id];
double B = ep / u[i] - ep / u[id] - ep / w[i] + ep / w[id];
Point pt;
if (fabs(A) < fabs(B)) pt = Point(0.0, - (ep / w[i] - ep / w[id]) / B);
else pt = Point(- (ep / w[i] - ep / w[id]) / A, 0.0);
tmp[top++] = DLine(pt, Vec(B, - A));
}
return halfPlaneTest(tmp, top);
}

int main() {
//    freopen("in", "r", stdin);
int n;
while (cin >> n) {
for (int i = 0; i < n; i++) cin >> v[i] >> u[i] >> w[i];
for (int i = 0; i < n; i++) {
if (test(i, n)) puts("Yes");
else puts("No");
}
//        puts("~~~");
}
return 0;
}

——written by Lyon