算法学习： 计算几何找凸包及求点线面交点

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前置知识：
计算几何基础

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找凸包：

vector<P> convex(vector<P> l)
{
    vector<P> ans, s;
    P tmp(lim, lim);
    int pos = 0;
    for (int i = 0; i < l.size(); i++)
        if (l[i] < tmp)
            tmp = l[i], pos = i;
    for (int i = pos, cnt_ = 0; cnt_ < l.size(); cnt_++, i = (i + 1) % l.size())
    {
        while (s.size() >= 2 && sgn(cross(s[s.size() - 2], s[s.size() - 1], l[i])) <= 0)
            s.pop_back();
        s.push_back(l[i]);
        pos = i;
    }
    ans = s; s.clear();
    for (int i = pos, cnt_ = 0; cnt_ < l.size(); cnt_++, i = (i - 1) % l.size())
    {
        while (s.size() >= 2 && sgn(cross(s[s.size() - 2], s[s.size() - 1], l[i])) <= 0)
            s.pop_back();
        s.push_back(l[i]);
        pos = i;
    }
    for (int i = 1; i + 1 < s.size(); i++)
        ans.push_back(s[i]);
    return ans;
}

找凸包

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求点是否在多边形内

通过计算点两边的边的个数是否相等，如果相等则说明点在多边形内

同时，计算点是否在直线上，查找点在直线上的情况

bool P_In_S(P a, vector<P> b)
{
    int cnt = b.size();
    int cnt_ = 0;
    b.push_back(b[0]);
    for (int i = 0; i < cnt; i++)
    {
        if (onL(a, L(b[i], b[i + 1])))    return true;
        int k = sgn(cross(a, b[i], b[i + 1]));
        int d1 = sgn(a.y - b[i].y),
            d2 = sgn(a.y - b[i + 1].y);
        if (k < 0 && d1 >= 0 && d2 < 0)cnt_++;
        if (k > 0 && d2 >= 0 && d1 < 0)cnt_--;
    }
    return cnt_ != 0;
}

点是否在多边形内

求两向量是否相交

bool L_is_Inter(L a, L b)
{
    int c1 = sgn(cross(a.s, a.t, b.s)),
        c2 = sgn(cross(a.s, a.t, b.t)),
        c3 = sgn(cross(b.s, b.t, a.s)),
        c4 = sgn(cross(b.s, b.t, a.t));
    if (c1*c2 < 0 && c3*c4 < 0)return true;
    if (!c1  &&  onL(b.s, a))return true;
    if (!c2  &&  onL(b.t, a))return true;
    if (!c3  &&  onL(a.s, a))return true;
    if (!c4  &&  onL(a.t, a))return true;
    return false;
}

两线段是否相交

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通过以上两个知识点：

能够求两个多边形是否相交

bool IF_inter(vector<P> a, vector<P> b)
{
    //printf("1\n");
    for (int i = 0; i < a.size(); i++)
        if (P_In_S(a[i], b))
            return true;
    //printf("2\n");
    for (int i = 0; i < b.size(); i++)
        if (P_In_S(b[i], a))
            return true;
    int cnt1 = a.size(), cnt2 = b.size();
    a.push_back(a[0]), b.push_back(b[0]);
    //printf("3\n");
    for (int i = 0; i < cnt1; i++)
        for (int j = 0; j < cnt2; j++)
            if (L_is_Inter(L(a[i], a[i + 1]), L(b[j], b[j + 1])))
                return true;
    return false;
}

两多边形是否相交