hdu 5027 Help!（计算几何 三分求极值）

hdu 5027 Help!

根据题意，在陆地上的路线解分为两部分：①从起点直线到达；②无法直线到达。救到人后返回陆地的时间可以先求出来。

可以求出起点到凸多边形的相切点，在两点间的边可直线到达。找出使总时间最小的入水点，可以用三分搜索求极值（列出公式求导后可发现是一个单峰函数）。

考虑入水点无法直线到达的情况，原理与上类似

计算几何的实现还是相当蛋疼的。有几个比较烦人的地方：

1、找切点。对于每个点 p 求出 sz[p] = ( vector(p, p+1) ^ vector(p, point(Xo, Yo)） )>=0，若sz[p] != sz[p+1]，则说明点 p+1 是切点。而且，若起点就在多边形的边上，则切点为其所处的边的两端点。

2、水点无法直线到达的情况下，应先到达切点，再沿着边寻找入水点

3、对于求出的切点 a，b。最好改变其位置使 a->b 之间的点属于①，b->a间的点属于②。

#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
#include <cmath>
using namespace std;
const int MAXN = 50010;
const double eps = 1e-4, inf = 1e100;
double Ts, Vr, Vs, rres, aans, len;
int n, sz[MAXN];
inline int cmp(double a, double b)
{
    return ((a-b)>eps) - ((a-b)<-eps);
}
struct point
{
    double x, y;
    point() {}
    point(double x, double y)
    {
        this->x = x;
        this->y = y;
    }
    double operator ^ (const point & a)
    {
        return x*a.y-y*a.x;
    }
    double operator * (const point & a)
    {
        return x*a.x + y*a.y;
    }
    double dis2()
    {
        return (x*x + y*y);
    }
    double dis()
    {
        return sqrt(x*x + y*y);
    }
    point operator - (const point &a )
    {
        return point(x-a.x, y-a.y);
    }
    point operator + (const point &a )
    {
        return point(x+a.x, y+a.y);
    }
    point operator / (const double &a )
    {
        return point(x/a, y/a);
    }
    point operator * (const double a )
    {
        return point(x*a, y*a);
    }
    bool operator == (const point &a) const
    {
        return cmp(x,a.x)==0 && cmp(y,a.y) == 0;
    }
    point trunc(double r)
    {
        double l=dis();
        if (!cmp(l,0))return *this;
        return point(x*r/l,y*r/l);
    }
} da[MAXN], OO, PP, frm;
double persToedge(point o, point p1, point p2)
{
    if (p1 == p2)
    {
        return (o-p1).dis();
    }
    point l1 = o-p1, l2 = o-p2, l3 = p1-p2;
    if (cmp(l1*l3, 0)==1)
    {
        return l1.dis();
    }
    if (cmp(l2*l3, 0)==-1)
    {
        return l2.dis();
    }
    return fabs(l1^l2)/(l3.dis());
}
double dis_Point_to_Line(point o, point a, point b)
{
    return (fabs((a-o)^(b-o)))/((a-b).dis());
}
void circle_X_line(point o, double r, point a, point b, int &nc, point *res)
{
    int k ;
    double dis = dis_Point_to_Line(o, a, b);
    k = cmp(r, dis);
    if (k==-1)
    {
        nc = 0;
        return;
    }
    point xx = a+((b-a)*(((b-a)*(o-a))/ ((b-a).dis2()) ));
    if (k == 0)
    {
        nc = 1;
        res[0]=xx;
        return;
    }
    dis = r*r - dis*dis;
    nc = 2;
    res[0] = xx-((b-a).trunc(dis));
    res[1] = xx+((b-a).trunc(dis));
    if (res[0].x > res[1].x) swap(res[0], res[1]);
}
inline bool isPointOnSeg(point o, point a, point b)
{
    if (a.x > b.x) swap(a, b);
    return cmp(b.x, o.x)>=0 && cmp(o.x, a.x)>=0;
}
double getNum(point x, int cs = -1)
{
    double tp = (PP-x).dis()/Vs;
    if (cs > 0)
        return tp += ((len + (x-frm).dis()) / Vr);
    return (OO-x).dis()/Vr + tp;
}
double tripleSearch(point bg, point ed, int cs=-1)
{
    point ml, mr;
    while ((bg-ed).dis() > eps)
    {
        ml = (bg*2+ed)/3.0;
        mr = (bg+ed*2)/3.0;
        if (getNum(ml, cs) < getNum(mr, cs)) ed = mr;
        else bg = ml;
    }
    return getNum(bg, cs);
}

void cal(int bg, int ed, int cs)
{
    int nc, i = bg;
    point ap[2], p1, p2;
    while (true)
    {
        nc = 0;
        p1 = da[(i+n)%n];
        p2 = da[(i+1+n)%n];
        if (cs == 1) frm = p1;
        else frm = p2;
        circle_X_line(PP, (Ts-aans)*Vs, p1, p2, nc, ap);
        if (nc == 1)
        {
            if (isPointOnSeg(ap[0], p1, p2))
                rres = min(rres, (len + (ap[0]-p1).dis())/Vr + (PP-ap[0]).dis()/Vs);
        }
        else if (nc == 2)
        {
            point e1, e2;
            if (cmp(ap[0].x, max(p1.x, p2.x))>0 ||
                    cmp(ap[1].x, min(p1.x, p2.x))<0) continue;
            if (isPointOnSeg(p1, ap[0], ap[1])) e1 = p1;
            else e1 = ap[0];
            if (isPointOnSeg(p2, ap[0], ap[1])) e2 = p2;
            else e2 = ap[1];
            rres = min(rres, tripleSearch(e1, e2, 1));
        }
        len += (p1-p2).dis();
        if (i == ed) break;
        i = (i+cs+n)%n;
    }
}
void solve()
{
    aans = inf;
    rres = inf;
    int bg, ed, nc, ssa[2], hc = 0;
    point ap[2], p1, p2, ssp[2];
    if (cmp( (da[1]-da[0]) ^ (da[2]-da[1]), 0) == -1)
        reverse(da, da+n);
    for (int i = 0; i< n; ++i)
        aans = min(aans, persToedge(PP, da[i], da[(i+1)%n])/Vs*2);
    if (aans > Ts)
    {
        puts("-1");
        return;
    }
    for (int i = 0; i< n; ++i)
        sz[i] = cmp( (da[(i+1)%n]-da[i]) ^ (OO-da[i]), 0)>=0;
    for (int i = 0, j = 0; i< n; ++i)
        if (sz[i] != sz[(i+1)%n])
            ssa[j] = (i+1)%n, ssp[j] = da[ssa[j]], ++j, hc = 1;
    if (!hc)    //
    {
        // rres = 0;
        for (int i = 0; i< n; ++i)
            if (cmp(dis_Point_to_Line(OO, da[i], da[(i+1)%n]), 0) == 0)
            {
                ssa[0] = i;
                ssa[1] = (i+1)%n;
                ssp[0] = da[i];
                ssp[1] = da[(i+1)%n];
                break;
            }
    }
    else
    {
        if (sz[ssa[0]] != 0) swap(ssa[0], ssa[1]), swap(ssp[0], ssp[1]);
        bg = ssa[0], ed = ssa[1] < ssa[0]?ssa[1]+n:ssa[1];
        for (int i = bg; i<ed; ++i)
        {
            nc = 0;
            p1 = da[i%n];
            p2 = da[(i+1)%n];
            if (p1.x > p2.x) swap(p1, p2);
            circle_X_line(PP, (Ts-aans)*Vs, p1, p2, nc, ap);
            if (nc == 0) continue;
            else if (nc == 1)
            {
                if (isPointOnSeg(ap[0], p1, p2))
                    rres = min(rres, (ap[0]-OO).dis()/Vr+(ap[0]-PP).dis()/Vs);
            }
            else
            {
                point e1, e2;
                if (cmp(ap[0].x, p2.x)>0 || cmp(ap[1].x, p1.x) < 0) continue;
                if (isPointOnSeg(p1, ap[0], ap[1])) e1 = p1;
                else e1 = ap[0];
                if (isPointOnSeg(p2, ap[0], ap[1])) e2 = p2;
                else e2 = ap[1];
                rres = min(rres, tripleSearch(e1, e2));
            }
        }
    }
    len = (ssp[1]-OO).dis();
    cal(ssa[1], (ssa[0]-1+n)%n, 1);
    len = (ssp[0]-OO).dis();
    cal((ssa[0]-1+n)%n, ssa[1], -1);
    if (rres == inf) puts("-1");
    else printf("%.2lf\n", rres + aans);
}
int main()
{
#ifndef ONLINE_JUDGE
    freopen("in.txt", "r", stdin);
    //freopen("out.txt", "w", stdout);
#endif // ONLINE_JUDGE
    int t;
    scanf("%d", &t);
    while (t--)
    {
        scanf("%lf%lf%lf",  &Ts, &Vr, &Vs);
        scanf("%lf%lf%lf%lf", &OO.x, &OO.y, &PP.x, &PP.y);
        scanf("%d", &n);
        for (int i = 0; i< n; ++i)
            scanf("%lf%lf", &da[i].x, &da[i].y);
        solve();
    }
    return 0;
}