long double area(polygon_t &polygon)
{
if (polygon.size() < 3) return 0;
long positive_term = 0, negative_term = 0;
for (unsigned i = 0; i < polygon.size() - 1; i++) {
positive_term += polygon[i].first * polygon[i + 1].second;
negative_term += polygon[i].second * polygon[i + 1].first;
}
positive_term += polygon.back().first * polygon[0].second;
negative_term += polygon.back().second * polygon[0].first;
return fabsl((positive_term - negative_term) / 2.0);
}
polygon_t merge(const polygon_t &polygon1, const polygon_t &polygon2) {
int i = 0;
int j = 0;
polygon_t merged;
while (i < polygon1.size() | j < polygon2.size()) {
if (POLAR_ORDER(polygon1[i], polygon2[j])) {
merged.push_back(polygon1[i]);
++i;
} else {
merged.push_back(polygon2[j]);
++j;
}
}
return merged;
}
bool inside_polygon(polygon_t &polygon, point_t point)
{
if (polygon.size() == 0) return false;
for (polygon_t::iterator p = polygon.begin(); p < polygon.end(); p++)
if (*p == point) return true;
if (polygon.size() == 1) return false;
long double total_area = area(polygon);
long double sub_area = 0;
for (unsigned i = 0; i < polygon.size(); i++) {
point_t a = point;
point_t b = polygon[i];
point_t c = polygon[(i + 1) % polygon.size()];
sub_area += fabsl((a.first * b.second + b.first * c.second + c.first * a.second) - (a.second * b.first + b.second * c.first + c.second * a.first)) / 2.0;
}
return (fabsl(total_area - sub_area) < 1e-10L);
}
long double minimum_enclosing_circle(polygon_t &convex_hull)
{
if (convex_hull.size() <= 1) return 0;
if (convex_hull.size() == 2) return distance_square(convex_hull[0], convex_hull[1]) / 4.0;
point_t s_a = convex_hull[0];
point_t s_b = convex_hull[1];
while (1) {
long double alpha = 100;
point_t v;
for (polygon_t::iterator p = convex_hull.begin(); p != convex_hull.end(); p++) {
if (*p == s_a || *p == s_b) continue;
long double a = angle(*p, s_a, s_b);
if (a < alpha) {
alpha = a;
v = *p;
}
}
if (alpha >= M_PI / 2) return distance_square(s_a, s_b) / 4.0;
// printf("s_a:(%ld, %ld) s_b:(%ld, %ld) v:(%ld, %ld)\n", s_a.first, s_a.second, s_b.first, s_b.second, v.first, v.second);
// printf("angle v-s_a-s_b: %Lf\n", angle(s_a, v, s_b));
if (angle(s_a, v, s_b) >= M_PI / 2) {
s_a = v;
continue;
}
// printf("angle v-s_b-s_a: %Lf\n", angle(s_b, v, s_a));
if (angle(s_b, v, s_a) >= M_PI / 2) {
s_b = v;
continue;
}
/* v, s_a, s_b */
fpoint_t center = find_center(v, s_a, s_b);
// printf("center: %Lf, %Lf\n", center.first, center.second);
// printf("%Lf = %Lf, %Lf\n", distance_square(center, v), distance_square(center, s_a), distance_square(center, s_b));
return distance_square(center, v);
}
}
bool isInside(polygon_t polygon, point_t p) {
auto n = polygon.size();
if (n < 3) return false;
point_t extreme = {INF, p.second};
int count = 0, i = 0;
do {
int next = (i + 1) % n;
if (doIntersect(polygon[i], polygon[next], p, extreme)) {
if (orientation(polygon[i], p, polygon[next]) == 0)
return onSegment(polygon[i], p, polygon[next]);
count++;
}
i = next;
} while (i != 0);
return count & 1;
}