大意:甲和乙分别沿着一条直线奔跑。两只狗的速度未知,但已知他们同时出发,同时到达,求它们在奔跑过程中的最远距离与最近距离之差。
思路:由于运动是相对的,因此可以求出甲和乙的相对位移量,认为甲不动,乙静止沿着直线走,因此问题转换为求点到线段的最小或者最大距离。然后模拟整个过程即可。
#include <iostream> #include <cstdlib> #include <cstdio> #include <string> #include <cstring> #include <cmath> #include <vector> #include <queue> #include <stack> #include <algorithm> using namespace std; const double eps = 1e-10; const double PI = acos(-1.0); struct Point { double x, y; Point(double x = 0, double y = 0) : x(x), y(y) { } bool operator < (const Point& a) const { if(a.x != x) return x < a.x; return y < a.y; } }; typedef Point Vector; struct Line { Point P; Vector v; double ang; Line() {} Line(Point P, Vector v) : P(P), v(v) { ang = atan2(v.y, v.x); } bool operator < (const Line& L) const { return ang < L.ang; } }; Vector operator + (Vector A, Vector B) { return Vector(A.x+B.x, A.y+B.y); } Vector operator - (Point A, Point B) { return Vector(A.x-B.x, A.y-B.y); } Vector operator * (Vector A, double p) { return Vector(A.x*p, A.y*p); } Vector operator / (Vector A, double p) { return Vector(A.x/p, A.y/p); } int dcmp(double x) { if(fabs(x) < eps) return 0; else return x < 0 ? -1 : 1; } bool operator == (const Point& a, const Point &b) { return dcmp(a.x-b.x) == 0 && dcmp(a.y-b.y) == 0; } double Dot(Vector A, Vector B) { return A.x*B.x + A.y*B.y; } double Length(Vector A) { return sqrt(Dot(A, A)); } double Angle(Vector A, Vector B) { return acos(Dot(A, B) / Length(A) / Length(B)); } double Cross(Vector A, Vector B) { return A.x*B.y - A.y*B.x; } double Area2(Point A, Point B, Point C) { return fabs(Cross(B-A, C-A)) / 2; } Vector Rotate(Vector A, double rad) { return Vector(A.x*cos(rad)-A.y*sin(rad), A.x*sin(rad) + A.y*cos(rad)); } Point GetLineIntersection(Point P, Vector v, Point Q, Vector w) { Vector u = P-Q; double t = Cross(w, u) / Cross(v, w); return P+v*t; } bool SegmentProperIntersection(Point a1, Point a2, Point b1, Point b2) { double c1 = Cross(a2-a1, b1-a1), c2 = Cross(a2-a1, b2-a1); double c3 = Cross(b2-b1, a1-b1), c4 = Cross(b2-b1, a2-b1); return dcmp(c1) * dcmp(c2) < 0 && dcmp(c3) * dcmp(c4) < 0; } bool OnSegment(Point p, Point a1, Point a2) { return dcmp(Cross(a1-p, a2-p)) == 0 && dcmp(Dot(a1-p, a2-p)) < 0; } double PolygonArea(Point* p, int n) { double area = 0; for(int i = 1; i < n-1; i++) area += Cross(p[i]-p[0], p[i+1]-p[0]); return area/2; } double PointDistanceToLine(Point P, Point A, Point B) { Vector v1 = B-A, v2 = P-A; return fabs(Cross(v1, v2)) / Length(v1); } double PointDistanceToSegment(Point P, Point A, Point B) { 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); } int isPointInPolygon(Point p, Point *poly, int n) { int wn = 0; for(int i = 0; i < n; i++) { const Point& p1 = poly[i], p2 = poly[(i+1)%n]; if(p == p1 || p == p2 || OnSegment(p, p1, p2)) return -1; int k = dcmp(Cross(p2-p1, p-p1)); int d1 = dcmp(p1.y - p.y); int d2 = dcmp(p2.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; } int ConvexHull(Point *p, int n, Point *ch) //凸包 { sort(p, p+n); n = unique(p, p+n) - p; //去重 int m = 0; for(int i = 0; i < n; i++) { while(m > 1 && Cross(ch[m-1]-ch[m-2], p[i]-ch[m-2]) <= 0) m--; ch[m++] = p[i]; } int k = m; for(int i = n-2; i >= 0; i--) { while(m > k && Cross(ch[m-1]-ch[m-2], p[i]-ch[m-2]) <= 0) m--; ch[m++] = p[i]; } if(n > 1) m--; return m; } Vector Normal(Vector A) { double L = Length(A); return Vector (-A.y/L, A.x/L); } double Dist2(Point p1, Point p2) { return (p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y); } // 返回点集直径的平方 double RotatingCalipers(Point *P, int n) //旋转卡壳 { if(n == 1) return 0; if(n == 2) return Dist2(P[0], P[1]); P[n] = P[0]; //避免取模 double ans = 0; for(int u = 0, v = 1; u < n; u++) { //一条直线贴住边p[u]-p[u+1] for(;;) { // 当Area(p[u], p[u+1], p[v+1]) <= Area(p[u], p[u+1], p[v])时停止旋转 // 即Cross(p[u+1]-p[u], p[v+1]-p[u]) - Cross(p[u+1]-p[u], p[v]-p[u]) <= 0 // 根据Cross(A,B) - Cross(A,C) = Cross(A,B-C) // 化简得Cross(p[u+1]-p[u], p[v+1]-p[v]) <= 0 double diff = Cross(P[u+1]-P[u], P[v+1]-P[v]); if(diff <= 0) { ans = max(ans, Dist2(P[u], P[v])); // u和v是对踵点 if(diff == 0) ans = max(ans, Dist2(P[u], P[v+1])); // diff == 0时u和v+1也是对踵点 break; //即边平行 } v = (v + 1) % n; } } return ans; } bool OnLeft(Line L, Point p) { return Cross(L.v, p-L.P) > 0; } Point GetIntersection(Line a, Line b) { Vector u = a.P-b.P; double t = Cross(b.v, u) / Cross(a.v, b.v); return a.P+a.v*t; } int HalfPlaneIntersection(Line* L, int n, Point* poly) { sort(L, L+n); int first, last; Point *p = new Point[n]; Line* q = new Line[n]; q[first=last=0] = L[0]; for(int i = 1; i < n; i++) { while(first < last && !OnLeft(L[i], p[last-1])) last--; while(first < last && !OnLeft(L[i], p[first])) first++; q[++last] = L[i]; if(fabs(Cross(q[last].v, q[last-1].v)) < eps) { last--; if(OnLeft(q[last], L[i].P)) q[last] = L[i]; } if(first < last) p[last-1] = GetIntersection(q[last-1], q[last]); } while(first < last && !OnLeft(q[first], p[last-1])) last--; if(last - first <= 1) return 0; p[last] = GetIntersection(q[last], q[first]); int m = 0; for(int i = first; i <= last; i++) poly[m++] = p[i]; return m; } Point read_point() { Point A; scanf("%lf%lf", &A.x, &A.y); return A; } int n, m; double Max, Min; const int maxn = 60; const double INF = 0x3f3f3f3f; Point P[maxn], Q[maxn]; void init() { Max = -INF; Min = INF; } void read_case() { init(); scanf("%d%d", &n, &m); for(int i = 0; i < n; i++) P[i] = read_point(); for(int i = 0; i < m; i++) Q[i] = read_point(); } void update(Point P, Point A, Point B) { Min = min(Min, PointDistanceToSegment(P, A, B)); Max = max(Max, max(Length(P-A), Length(P-B))); } void solve() { read_case(); int Sa = 0, Sb = 0; double LenA = 0, LenB = 0; for(int i = 0; i < n-1; i++) LenA += Length(P[i+1]-P[i]); for(int i = 0; i < m-1; i++) LenB += Length(Q[i+1]-Q[i]); Point Pa = P[0], Pb = Q[0]; while(Sa < n-1 && Sb < m-1) { double La = Length(P[Sa+1]-Pa); double Lb = Length(Q[Sb+1]-Pb); double T = min(La/LenA, Lb/LenB); Vector Va = ((P[Sa+1]-Pa)/La)*T*LenA; //位移向量 Vector Vb = ((Q[Sb+1]-Pb)/Lb)*T*LenB; //位移向量 update(Pa, Pb, Pb+Vb-Va); //相对位移 Pa = Pa + Va; Pb = Pb + Vb; if(Pa == P[Sa+1]) Sa++; if(Pb == Q[Sb+1]) Sb++; } printf("%.0lf\n", Max-Min); } int main() { int T, times = 0; scanf("%d", &T); while(T--) { printf("Case %d: ", ++times); solve(); } return 0; }