能看到其他所有点的区域就是轮廓线的半平面交。
然后最小高度就是半平面交与轮廓线这两个一次分段函数的差，极值肯定出现在分段点上，分别求一下即可。
#include <bits/stdc++.h>
#define db double
const db eps = 1e-9;
inline int sign(db k) { return k < -eps ? -1 : k > eps; }
inline int cmp(db k1, db k2) { return sign(k1 - k2); }
struct P {
db x, y;
P() {}
P(db x, db y): x(x), y(y) {}
P operator + (const P &rhs) const { return P(x + rhs.x, y + rhs.y); }
P operator - (const P &rhs) const { return P(x - rhs.x, y - rhs.y); }
P operator * (const db &k) const { return P(x * k, y * k); }
P operator / (const db &k) const { return P(x / k, y / k); }
bool operator < (const P &rhs) const { int c = cmp(x, rhs.x); return c ? c == -1 : cmp(y, rhs.y) == -1; }
bool operator == (const P &rhs) const { return !cmp(x, rhs.x) && !cmp(y, rhs.y); }
db distTo(const P &rhs) const { return (*this - rhs).abs(); }
db alpha() { return atan2(y, x); }
void read() { scanf("%lf%lf", &x, &y); }
void print() { printf("%.10f %.10f\n", x, y); }
db abs() { return sqrt(abs2()); }
db abs2() { return x * x + y * y; }
P rot(const db &k) { return P(x * cos(k) - y * sin(k), x * sin(k) + y * cos(k)); }
P rot90() { return P(-y, x); }
P unit() { return *this / abs(); }
P normal() { return rot90() / abs(); }
int quad() { return sign(y) == 1 || (sign(y) == 0 && sign(x) >= 0); }
db dot(const P &p) const { return x * p.x + y * p.y; }
db det(const P &p) const { return x * p.y - y * p.x; }
};
struct L { // ps[0] -> ps[1]
P ps[2];
L() {}
L(const P &p0, const P &p1) {
ps[0] = p0; ps[1] = p1;
}
P &operator[](int i) { return ps[i]; }
P dir() { return ps[1] - ps[0]; }
bool include(const P &p) { return sign((ps[1] - ps[0]).det(p - ps[0])) > 0; }
L push() { // push eps outawrd
const db Eps = 1e-6;
P delta = (ps[1] - ps[0]).normal() * Eps;
return {ps[0] - delta, ps[1] - delta};
}
};
#define cross(p1, p2, p3) ((p2.x - p1.x) * (p3.y - p1.y) - (p3.x - p1.x) * (p2.y - p1.y))
#define crossOp(p1, p2, p3) sign(cross(p1, p2, p3))
// 判断 p1p2 和 q1q2 是否相交
bool chkLL(const P &p1, const P &p2, const P &q1, const P &q2) {
db a1 = cross(q1, q2, p1), a2 = -cross(q1, q2, p2);
return sign(a1 + a2) != 0;
}
// 直线交点
P isLL(const P &p1, const P &p2, const P &q1, const P &q2) {
assert(chkLL(p1, p2, q1, q2));
db a1 = cross(q1, q2, p1), a2 = -cross(q1, q2, p2);
return (p1 * a2 + p2 * a1) / (a1 + a2);
}
P isLL(L l1, L l2) {
return isLL(l1[0], l1[1], l2[0], l2[1]);
}
/***** 线段相交 *****/
bool intersect(db l1, db r1, db l2, db r2) {
if (l1 > r1) std::swap(l1, r2); if (l2 > r2) std::swap(l2, r2);
return !(cmp(r1, l2) == -1 || cmp(r2, l1) == -1);
}
bool isSS(const P &p1, const P &p2, const P &q1, const P &q2) {
return intersect(p1.x, p2.x, q1.x, q2.x) && intersect(p1.y, p2.y, q1.y, q2.y)
&& crossOp(p1, p2, q1) * crossOp(p1, p2, q2) <= 0
&& crossOp(q1, q2, p1) * crossOp(q1, q2, p2) <= 0;
}
bool isSS_strict(const P &p1, const P &p2, const P &q1, const P &q2) {
return crossOp(p1, p2, q1) * crossOp(p1, p2, q2) < 0
&& crossOp(q1, q2, p1) * crossOp(q1, q2, p2) < 0;
}
/***************/
/***** 点在线段上判定 *****/
bool isMiddle(db a, db m, db b) {
return sign(a - m) == 0 || sign(b - m) == 0 || (a < m != b < m);
}
bool isMiddle(const P &a, const P &m, const P &b) {
return isMiddle(a.x, m.x, b.x) && isMiddle(a.y, m.y, b.y);
}
bool onSeg(const P &p1, const P &p2, const P &q) {
return crossOp(p1, p2, q) == 0 && isMiddle(p1, q, p2);
}
bool onSeg_strict(const P &p1, const P &p2, const P &q) {
return crossOp(p1, p2, q) == 0 && sign((q - p1).dot(p1 - p2)) * sign((q - p2).dot(p1 - p2)) < 0;
}
/*******************/
// 投影
P proj(const P &p1, const P &p2, const P &q) {
P dir = p2 - p1;
return p1 + dir * (dir.dot(q - p1) / dir.abs2());
}
// 反射
P reflect(const P &p1, const P &p2, const P &q) {
return proj(p1, p2, q) * 2 - q;
}
// 最近点
db nearest(const P &p1, const P &p2, const P &q) {
P h = proj(p1, p2, q);
if (isMiddle(p1, h, p2)) return q.distTo(h);
return std::min(p1.distTo(q), p2.distTo(q));
}
// 线段距离
db disSS(const P &p1, const P &p2, const P &q1, const P &q2) {
if (isSS(p1, p2, q1, q2)) return 0;
return std::min(std::min(nearest(p1, p2, q1), nearest(p1, p2, q2)), std::min(nearest(q1, q2, p1), nearest(q1, q2, p2)));
}
// 夹角
db rad(const P &p1, const P &p2) {
return atan2l(p1.det(p2), p1.dot(p2));
}
// 多边形面积
db area(const std::vector<P> &ps) {
db ans = 0;
for (int i = 0, n = ps.size(); i < n; i++)
ans += ps[i].det(ps[(i + 1) % n]);
return ans;
}
// 点包含 2: inside 1: onSeg 0: outside
int contain(const std::vector<P> &ps, const P &p) {
int n = ps.size(), ret = 0;
for (int i = 0; i < n; i++) {
P u = ps[i], v = ps[(i + 1) % n];
if (onSeg(u, v, p)) return 1;
if (cmp(u.y, v.y) <= 0) std::swap(u, v);
if (cmp(p.y, u.y) > 0 || cmp(p.y, v.y) <= 0) continue;
ret ^= crossOp(p, u, v) > 0;
}
return ret * 2;
}
// 凸包
std::vector<P> convexHull(std::vector<P> ps) {
int n = ps.size(); if (n <= 1) return ps;
std::sort(ps.begin(), ps.end());
std::vector<P> qs(n * 2); int k = 0;
for (int i = 0; i < n; qs[k++] = ps[i++])
while (k > 1 && crossOp(qs[k - 2], qs[k - 1], ps[i]) <= 0) --k;
for (int i = n - 2, t = k; i >= 0; qs[k++] = ps[i--])
while (k > t && crossOp(qs[k - 2], qs[k - 1], ps[i]) <= 0) --k;
qs.resize(k - 1);
return qs;
}
std::vector<P> convexHullNonStrict(std::vector<P> ps) {
int n = ps.size(); if (n <= 1) return ps;
std::sort(ps.begin(), ps.end());
std::vector<P> qs(n * 2); int k = 0;
for (int i = 0; i < n; qs[k++] = ps[i++])
while (k > 1 && crossOp(qs[k - 2], qs[k - 1], ps[i]) < 0) --k;
for (int i = n - 2, t = k; i >= 0; qs[k++] = ps[i--])
while (k > t && crossOp(qs[k - 2], qs[k - 1], ps[i]) < 0) --k;
qs.resize(k - 1);
return qs;
}
// 点集直径
db convexDiameter(const std::vector<P> &ps) {
int n = ps.size(); if (n <= 1) return 0;
int is = 0, js = 0;
for (int k = 1; k < n; k++)
is = ps[k] < ps[is] ? k : is, js = ps[js] < ps[k] ? k : js;
int i = is, j = js;
db ret = ps[i].distTo(ps[j]);
do {
if ((ps[(i + 1) % n] - ps[i]).det(ps[(j + 1) % n] - ps[j]) >= 0)
(++j) %= n;
else
(++i) %= n;
ret = std::max(ret, ps[i].distTo(ps[j]));
} while (i != is || j != js);
return ret;
}
// convecCut
std::vector<P> convexCut(const std::vector<P> &ps, const P &q1, const P &q2) {
std::vector<P> qs;
int n = ps.size();
for (int i = 0; i < n; i++) {
P p1 = ps[i], p2 = ps[(i + 1) % n];
int d1 = crossOp(q1, q2, p1), d2 = crossOp(q1, q2, p2);
if (d1 >= 0) qs.push_back(p1);
if (d1 * d2 < 0) qs.push_back(isLL(p1, p2, q1, q2));
}
return qs;
}
// min_dis
db min_dis(const std::vector<P> &ps, int l, int r) {
if (r - l <= 5) {
db ret = 1e100;
for (int i = l; i < r; i++)
for (int j = l; j < i; j++)
ret = std::min(ret, ps[i].distTo(ps[j]));
return ret;
}
int mid = l + r >> 1;
db ret = std::min(min_dis(ps, l, mid), min_dis(ps, mid, r));
std::vector<P> qs;
for (int i = l; i < r; i++)
if (cmp(fabs(ps[i].x - ps[mid].x), ret) <= 0) qs.push_back(ps[i]);
std::sort(qs.begin(), qs.end(), [](const P & a, const P & b) -> bool { return cmp(a.y, b.y) < 0; });
for (int i = 1; i < qs.size(); i++)
for (int j = i - 1; j >= 0 && cmp(qs[j].y, qs[i].y - ret) >= 0; j--)
ret = std::min(ret, qs[j].distTo(qs[i]));
return ret;
}
// 圆的关系
int type(const P &o1, db r1, const P &o2, db r2) {
db d = o1.distTo(o2);
if (cmp(d, r1 + r2) == 1) return 4; // 相离
if (cmp(d, r1 + r2) == 0) return 3; // 外切
if (cmp(d, fabs(r1 - r2)) == 1) return 2; // 相交
if (cmp(d, fabs(r1 - r2)) == 0) return 1; // 内切
return 0;
}
bool parallel(L l0, L l1) {
return sign(l0.dir().det(l1.dir())) == 0;
}
bool sameDir(L l0, L l1) {
return parallel(l0, l1) && sign(l0.dir().dot(l1.dir())) == 1;
}
bool cmp(P a, P b) {
if (a.quad() != b.quad()) {
return a.quad() < b.quad();
} else {
return sign(a.det(b)) > 0;
}
}
bool operator < (L l0, L l1) {
if (sameDir(l0, l1)) {
return l1.include(l0[0]);
} else {
return cmp(l0.dir(), l1.dir());
}
}
bool check(L u, L v, L w) {
return w.include(isLL(u, v));
}
const int N = 1e3 + 7;
L que[N];
std::vector<L> halfPlaneIS(std::vector<L> &l) {
std::sort(l.begin(), l.end());
int head = 0, tail = 0;
for (int i = 0; i < l.size(); i++) {
if (i && sameDir(l[i], l[i - 1])) continue;
while (tail - head > 1 && !check(que[tail - 2], que[tail - 1], l[i])) tail--;
while (tail - head > 1 && !check(que[head + 1], que[head], l[i])) head++;
que[tail++] = l[i];
}
while (tail - head > 2 && !check(que[tail - 2], que[tail - 1], que[0])) tail--;
while (tail - head > 2 && !check(que[1], que[0], que[tail - 1])) head++;
std::vector<L> ans;
for (int i = head; i < tail; i++)
ans.push_back(que[i]);
return ans;
}
db gety(P p, std::vector<P> point, std::vector<L> line) {
int n = point.size();
if (sign(p.x - point[0].x) <= 0) return isLL(line[0], L(p, P(p.x, p.y + 10))).y;
if (sign(p.x - point[n - 1].x) >= 0) return isLL(line[n], L(p, P(p.x, p.y + 10))).y;
for (int i = 0; i < n - 1; i++) {
if (isMiddle(point[i].x, p.x, point[i + 1].x))
return isLL(line[i + 1], L(p, P(p.x, p.y + 10))).y;
}
return 1e11;
}
db getyy(P p, std::vector<P> point) {
for (int i = 0, sz = point.size(); i < sz - 1; i++) {
if (isMiddle(point[i].x, p.x, point[i + 1].x))
return isLL(point[i], point[i + 1], p, P(p.x, p.y + 10)).y;
}
return 1e11;
}
int main() {
int n;
scanf("%d", &n);
if (n <= 2) {
puts("0");
return 0;
}
std::vector<P> p(n);
for (int i = 0; i < n; i++)
scanf("%lf", &p[i].x);
for (int i = 0; i < n; i++)
scanf("%lf", &p[i].y);
std::vector<L> l;
for (int i = 0; i < n - 1; i++)
l.push_back(L(p[i], p[i + 1]));
std::vector<L> half = halfPlaneIS(l);
db ans = 1e10;
std::vector<P> ss;
for (int i = 0, sz = half.size(); i < sz - 1; i++)
ss.push_back(isLL(half[i], half[i + 1]));
for (int i = 0; i < n; i++) {
ans = std::min(ans, std::fabs(gety(p[i], ss, half) - p[i].y));
}
for (int i = 0, sz = half.size(); i < sz - 1; i++) {
P pp = ss[i];
ans = std::min(ans, std::fabs(getyy(pp, p) - pp.y));
}
printf("%.3f\n", ans);
return 0;
}
来源：https://www.cnblogs.com/Mrzdtz220/p/12233059.html