minkowski_sum.hpp¶
求两个点集凸包的 Minkowski 和，结果从字典序最小点开始严格逆时针排列，支持点、线段和空集。坐标和、差及行列式运算须在坐标类型范围内。
Complexity: Time: \(O(n \log n + m \log m)\) including hull normalization. Space: \(O(n + m)\).
Implementation¶
当前头文件，省略 include guard；依赖见 #include。
/// @complexity Time: O(n log n + m log m) including hull normalization.
/// Space: O(n + m).
#include "noya/geometry/geometry_base.hpp"
#include <algorithm>
#include <limits>
#include <vector>
namespace noya {
/// @brief Minkowski sum of two point sets as a strict counter-clockwise convex
/// polygon in O((n+m) log(n+m)), including degenerate point and segment cases.
/// Coordinate sums/differences and determinant arithmetic must fit in T.
template <class T>
std::vector<point<T>> minkowski_sum(std::vector<point<T>> lhs,
std::vector<point<T>> rhs) {
lhs = convex_hull(std::move(lhs));
rhs = convex_hull(std::move(rhs));
if (lhs.empty() || rhs.empty()) {
return {};
}
if (lhs.size() == 1) {
for (point<T> &val : rhs) {
val += lhs[0];
}
return rhs;
}
if (rhs.size() == 1) {
for (point<T> &val : lhs) {
val += rhs[0];
}
return lhs;
}
auto rot = [](std::vector<point<T>> &pg) {
auto it = std::min_element(pg.begin(), pg.end(),
[](const point<T> &a, const point<T> &b) {
return a.y < b.y || (a.y == b.y && a.x < b.x);
});
std::rotate(pg.begin(), it, pg.end());
};
rot(lhs);
rot(rhs);
int n = int(lhs.size()), m = int(rhs.size());
lhs.push_back(lhs[0]);
rhs.push_back(rhs[0]);
std::vector<point<T>> res;
res.reserve(n + m);
int i = 0, j = 0;
while (i < n || j < m) {
res.push_back(lhs[i] + rhs[j]);
if (j == m) {
i++;
} else if (i == n) {
j++;
} else {
T rot = cross(lhs[i + 1] - lhs[i], rhs[j + 1] - rhs[j]);
if (rot >= T{}) i++;
if (rot <= T{}) j++;
}
}
if constexpr (std::numeric_limits<T>::is_integer) {
// Exact edge merging is already strict CCW; only restore the start vertex.
std::rotate(res.begin(), std::min_element(res.begin(), res.end()), res.end());
return res;
} else {
return convex_hull(std::move(res));
}
}
} // namespace noya
// NOYA_MINKOWSKI_SUM_HPP
#ifndef NOYA_MINKOWSKI_SUM_HPP
#define NOYA_MINKOWSKI_SUM_HPP 1
/// @complexity Time: O(n log n + m log m) including hull normalization.
/// Space: O(n + m).
#include "noya/geometry/geometry_base.hpp"
#include <algorithm>
#include <limits>
#include <vector>
namespace noya {
/// @brief Minkowski sum of two point sets as a strict counter-clockwise convex
/// polygon in O((n+m) log(n+m)), including degenerate point and segment cases.
/// Coordinate sums/differences and determinant arithmetic must fit in T.
template <class T>
std::vector<point<T>> minkowski_sum(std::vector<point<T>> lhs,
std::vector<point<T>> rhs) {
lhs = convex_hull(std::move(lhs));
rhs = convex_hull(std::move(rhs));
if (lhs.empty() || rhs.empty()) {
return {};
}
if (lhs.size() == 1) {
for (point<T> &val : rhs) {
val += lhs[0];
}
return rhs;
}
if (rhs.size() == 1) {
for (point<T> &val : lhs) {
val += rhs[0];
}
return lhs;
}
auto rot = [](std::vector<point<T>> &pg) {
auto it = std::min_element(pg.begin(), pg.end(),
[](const point<T> &a, const point<T> &b) {
return a.y < b.y || (a.y == b.y && a.x < b.x);
});
std::rotate(pg.begin(), it, pg.end());
};
rot(lhs);
rot(rhs);
int n = int(lhs.size()), m = int(rhs.size());
lhs.push_back(lhs[0]);
rhs.push_back(rhs[0]);
std::vector<point<T>> res;
res.reserve(n + m);
int i = 0, j = 0;
while (i < n || j < m) {
res.push_back(lhs[i] + rhs[j]);
if (j == m) {
i++;
} else if (i == n) {
j++;
} else {
T rot = cross(lhs[i + 1] - lhs[i], rhs[j + 1] - rhs[j]);
if (rot >= T{}) i++;
if (rot <= T{}) j++;
}
}
if constexpr (std::numeric_limits<T>::is_integer) {
// Exact edge merging is already strict CCW; only restore the start vertex.
std::rotate(res.begin(), std::min_element(res.begin(), res.end()), res.end());
return res;
} else {
return convex_hull(std::move(res));
}
}
} // namespace noya
#endif // NOYA_MINKOWSKI_SUM_HPP
/// @complexity Time: O(n log n + m log m) including hull normalization.
/// Space: O(n + m).
/// @complexity Time: O(1) per point/segment primitive, O(n) for polygon area or
/// point location, O(n log n) for convex hull. Space: O(1), except O(n) for hull.
#include <algorithm>
#include <cmath>
#include <optional>
#include <vector>
namespace noya {
/// @brief Two-dimensional point with vector arithmetic and lexicographic order.
template <class T> struct point {
T x{};
T y{};
point() = default;
point(T x_, T y_) : x(x_), y(y_) {}
point &operator+=(const point &rhs) {
x += rhs.x;
y += rhs.y;
return *this;
}
point &operator-=(const point &rhs) {
x -= rhs.x;
y -= rhs.y;
return *this;
}
point &operator*=(const T &scl) {
x *= scl;
y *= scl;
return *this;
}
point &operator/=(const T &scl) {
x /= scl;
y /= scl;
return *this;
}
friend point operator+(point l, const point &r) { return l += r; }
friend point operator-(point l, const point &r) { return l -= r; }
friend point operator*(point val, const T &scl) { return val *= scl; }
friend point operator*(const T &scl, point val) { return val *= scl; }
friend point operator/(point val, const T &scl) { return val /= scl; }
friend bool operator==(const point &, const point &) = default;
friend bool operator<(const point &l, const point &r) {
return l.x < r.x || (l.x == r.x && l.y < r.y);
}
};
/// @brief Circle represented by a center and a nonnegative radius.
template <class Real> struct circle {
point<Real> o;
Real r1{};
};
/// @brief Return the dot product of two vectors.
template <class T> T dot(const point<T> &a, const point<T> &b) {
return a.x * b.x + a.y * b.y;
}
/// @brief Return the signed cross product of two vectors.
template <class T> T cross(const point<T> &a, const point<T> &b) {
return a.x * b.y - a.y * b.x;
}
/// @brief Return cross(a - o1, b - o1).
template <class T>
T cross(const point<T> &o1, const point<T> &a, const point<T> &b) {
return cross(a - o1, b - o1);
}
/// @brief Return the squared Euclidean norm.
template <class T> T norm2(const point<T> &val) { return dot(val, val); }
/// @brief Compare a value with zero using an optional absolute tolerance.
template <class T> int sign(const T &val, const T &eps = T{}) {
return (val > eps) - (val < -eps);
}
/// @brief Return -1, 0, or 1 for a clockwise, collinear, or counter-clockwise
/// turn.
template <class T>
int orientation(const point<T> &a, const point<T> &b, const point<T> &c,
const T &eps = T{}) {
return sign(cross(a, b, c), eps);
}
/// @brief Test whether p lies on the closed segment [a, b].
template <class T>
bool on_segment(const point<T> &p, const point<T> &a, const point<T> &b,
const T &eps = T{}) {
if (orientation(a, b, p, eps) != 0) {
return false;
}
return std::min(a.x, b.x) - eps <= p.x && p.x <= std::max(a.x, b.x) + eps &&
std::min(a.y, b.y) - eps <= p.y && p.y <= std::max(a.y, b.y) + eps;
}
/// @brief Test whether the closed segments [a, b] and [c, d] intersect.
template <class T>
bool segments_intersect(const point<T> &a, const point<T> &b, const point<T> &c,
const point<T> &d, const T &eps = T{}) {
int s1 = orientation(a, b, c, eps);
int s2 = orientation(a, b, d, eps);
int s3 = orientation(c, d, a, eps);
int s4 = orientation(c, d, b, eps);
if (s1 == 0 && on_segment(c, a, b, eps)) {
return true;
}
if (s2 == 0 && on_segment(d, a, b, eps)) {
return true;
}
if (s3 == 0 && on_segment(a, c, d, eps)) {
return true;
}
if (s4 == 0 && on_segment(b, c, d, eps)) {
return true;
}
return s1 * s2 < 0 && s3 * s4 < 0;
}
/// @brief Intersect the infinite lines through (a, b) and (c, d), returning
/// nullopt when they are parallel or coincident.
template <class T>
std::optional<point<long double>>
line_intersection(const point<T> &a, const point<T> &b, const point<T> &c,
const point<T> &d, long double eps = 0) {
point<long double> lhs{static_cast<long double>(a.x),
static_cast<long double>(a.y)};
point<long double> b1{static_cast<long double>(b.x),
static_cast<long double>(b.y)};
point<long double> z{static_cast<long double>(c.x),
static_cast<long double>(c.y)};
point<long double> d1{static_cast<long double>(d.x),
static_cast<long double>(d.y)};
point<long double> da = b1 - lhs;
point<long double> db = d1 - z;
long double den = cross(da, db);
if (std::abs(den) <= eps) {
return std::nullopt;
}
long double rat = cross(z - lhs, db) / den;
return lhs + da * rat;
}
/// @brief Return the convex hull in counter-clockwise order without repetition.
/// Starts at the lexicographically smallest point; kc retains boundary points.
/// Coordinate differences and determinant arithmetic must fit in T.
template <class T>
std::vector<point<T>> convex_hull(std::vector<point<T>> pt, bool kc = false) {
std::sort(pt.begin(), pt.end());
pt.erase(std::unique(pt.begin(), pt.end()), pt.end());
if (pt.size() <= 1) {
return pt;
}
if (kc) {
bool col = true;
for (int i = 2; i < int(pt.size()) && col; i++) {
col = orientation(pt[0], pt[1], pt[i]) == 0;
}
if (col) return pt;
}
std::vector<point<T>> res;
res.reserve(pt.size() + 1);
for (const point<T> &p : pt) {
while (res.size() >= 2) {
int rot = orientation(res[res.size() - 2], res.back(), p);
if (rot > 0 || (kc && rot == 0)) {
break;
}
res.pop_back();
}
res.push_back(p);
}
auto k = res.size(); // Keep the lower chain while building the upper chain.
for (int i = int(pt.size()) - 2; i >= 0; i--) {
while (res.size() > k) {
int rot = orientation(res[res.size() - 2], res.back(), pt[i]);
if (rot > 0 || (kc && rot == 0)) {
break;
}
res.pop_back();
}
res.push_back(pt[i]);
}
res.pop_back();
return res;
}
/// @brief Return twice the signed area of a polygon.
/// Coordinate differences, products and the accumulated area must fit in T.
template <class T> T polygon_area2(const std::vector<point<T>> &pg) {
T res{};
if (pg.size() < 3) return res;
auto prev = pg[1] - pg[0];
for (int i = 2; i < int(pg.size()); i++) {
auto cur = pg[i] - pg[0];
res += cross(prev, cur);
prev = cur;
}
return res;
}
/// @brief Classify a point relative to a polygon: -1 outside, 0 boundary, 1
/// inside.
template <class T>
int point_in_polygon(const point<T> &p, const std::vector<point<T>> &pg) {
bool in = false;
for (int i = 0; i < int(pg.size()); i++) {
point<T> a = pg[i];
point<T> b = pg[(i + 1) % pg.size()];
if (on_segment(p, a, b)) {
return 0;
}
if (a.y <= p.y && p.y < b.y && orientation(a, b, p) > 0) {
in = !in;
}
if (b.y <= p.y && p.y < a.y && orientation(a, b, p) < 0) {
in = !in;
}
}
return in ? 1 : -1;
}
/// @brief Return the Euclidean distance between two points.
template <class T> long double distance(const point<T> &a, const point<T> &b) {
return std::hypot(
static_cast<long double>(a.x) - static_cast<long double>(b.x),
static_cast<long double>(a.y) - static_cast<long double>(b.y));
}
} // namespace noya
#include <algorithm>
#include <limits>
#include <vector>
namespace noya {
/// @brief Minkowski sum of two point sets as a strict counter-clockwise convex
/// polygon in O((n+m) log(n+m)), including degenerate point and segment cases.
/// Coordinate sums/differences and determinant arithmetic must fit in T.
template <class T>
std::vector<point<T>> minkowski_sum(std::vector<point<T>> lhs,
std::vector<point<T>> rhs) {
lhs = convex_hull(std::move(lhs));
rhs = convex_hull(std::move(rhs));
if (lhs.empty() || rhs.empty()) {
return {};
}
if (lhs.size() == 1) {
for (point<T> &val : rhs) {
val += lhs[0];
}
return rhs;
}
if (rhs.size() == 1) {
for (point<T> &val : lhs) {
val += rhs[0];
}
return lhs;
}
auto rot = [](std::vector<point<T>> &pg) {
auto it = std::min_element(pg.begin(), pg.end(),
[](const point<T> &a, const point<T> &b) {
return a.y < b.y || (a.y == b.y && a.x < b.x);
});
std::rotate(pg.begin(), it, pg.end());
};
rot(lhs);
rot(rhs);
int n = int(lhs.size()), m = int(rhs.size());
lhs.push_back(lhs[0]);
rhs.push_back(rhs[0]);
std::vector<point<T>> res;
res.reserve(n + m);
int i = 0, j = 0;
while (i < n || j < m) {
res.push_back(lhs[i] + rhs[j]);
if (j == m) {
i++;
} else if (i == n) {
j++;
} else {
T rot = cross(lhs[i + 1] - lhs[i], rhs[j + 1] - rhs[j]);
if (rot >= T{}) i++;
if (rot <= T{}) j++;
}
}
if constexpr (std::numeric_limits<T>::is_integer) {
// Exact edge merging is already strict CCW; only restore the start vertex.
std::rotate(res.begin(), std::min_element(res.begin(), res.end()), res.end());
return res;
} else {
return convex_hull(std::move(res));
}
}
} // namespace noya
// NOYA_MINKOWSKI_SUM_HPP