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#include <bits/stdc++.h>
using namespace std;
const double EPS = 1e-9;
double DEG_to_RAD(double d) { return d*M_PI / 180.0; }
double RAD_to_DEG(double r) { return r*180.0 / M_PI; }
struct point { double x, y; // only used if more precision is needed
point() { x = y = 0.0; } // default constructor
point(double _x, double _y) : x(_x), y(_y) {} // user-defined
bool operator == (point other) const {
return (fabs(x-other.x) < EPS && (fabs(y-other.y) < EPS)); }
bool operator <(const point &p) const {
return x < p.x || (abs(x-p.x) < EPS && y < p.y); } };
struct vec { double x, y; // name: `vec' is different from STL vector
vec(double _x, double _y) : x(_x), y(_y) {} };
vec toVec(point a, point b) { // convert 2 points to vector a->b
return vec(b.x-a.x, b.y-a.y); }
double dist(point p1, point p2) { // Euclidean distance
return hypot(p1.x-p2.x, p1.y-p2.y); } // return double
// returns the perimeter of polygon P, which is the sum of
// Euclidian distances of consecutive line segments (polygon edges)
double perimeter(const vector<point> &P) { // by ref for efficiency
double ans = 0.0;
for (int i = 0; i < (int)P.size()-1; ++i) // note: P[n-1] = P[0]
ans += dist(P[i], P[i+1]); // as we duplicate P[0]
return ans;
}
// returns the area of polygon P
double area(const vector<point> &P) {
double ans = 0.0;
for (int i = 0; i < (int)P.size()-1; ++i) // Shoelace formula
ans += (P[i].x*P[i+1].y - P[i+1].x*P[i].y);
return fabs(ans)/2.0; // only do / 2.0 here
}
double dot(vec a, vec b) { return (a.x*b.x + a.y*b.y); }
double norm_sq(vec v) { return v.x*v.x + v.y*v.y; }
double angle(point a, point o, point b) { // returns angle aob in rad
vec oa = toVec(o, a), ob = toVec(o, b);
return acos(dot(oa, ob) / sqrt(norm_sq(oa) * norm_sq(ob))); }
double cross(vec a, vec b) { return a.x*b.y - a.y*b.x; }
// returns the area of polygon P, which is half the cross products
// of vectors defined by edge endpoints
double area_alternative(const vector<point> &P) {
double ans = 0.0; point O(0.0, 0.0); // O = the Origin
for (int i = 0; i < (int)P.size()-1; ++i) // sum of signed areas
ans += cross(toVec(O, P[i]), toVec(O, P[i+1]));
return fabs(ans)/2.0;
}
// note: to accept collinear points, we have to change the `> 0'
// returns true if point r is on the left side of line pq
bool ccw(point p, point q, point r) {
return cross(toVec(p, q), toVec(p, r)) > 0;
}
// returns true if point r is on the same line as the line pq
bool collinear(point p, point q, point r) {
return fabs(cross(toVec(p, q), toVec(p, r))) < EPS;
}
// returns true if we always make the same turn
// while examining all the edges of the polygon one by one
bool isConvex(const vector<point> &P) {
int n = (int)P.size();
// a point/sz=2 or a line/sz=3 is not convex
if (n <= 3) return false;
bool firstTurn = ccw(P[0], P[1], P[2]); // remember one result,
for (int i = 1; i < n-1; ++i) // compare with the others
if (ccw(P[i], P[i+1], P[(i+2) == n ? 1 : i+2]) != firstTurn)
return false; // different -> concave
return true; // otherwise -> convex
}
// returns 1/0/-1 if point p is inside/on (vertex/edge)/outside of
// either convex/concave polygon P
int insidePolygon(point pt, const vector<point> &P) {
int n = (int)P.size();
if (n <= 3) return -1; // avoid point or line
bool on_polygon = false;
for (int i = 0; i < n-1; ++i) // on vertex/edge?
if (fabs(dist(P[i], pt) + dist(pt, P[i+1]) - dist(P[i], P[i+1])) < EPS)
on_polygon = true;
if (on_polygon) return 0; // pt is on polygon
double sum = 0.0; // first = last point
for (int i = 0; i < n-1; ++i) {
if (ccw(pt, P[i], P[i+1]))
sum += angle(P[i], pt, P[i+1]); // left turn/ccw
else
sum -= angle(P[i], pt, P[i+1]); // right turn/cw
}
return fabs(sum) > M_PI ? 1 : -1; // 360d->in, 0d->out
}
// compute the intersection point between line segment p-q and line A-B
point lineIntersectSeg(point p, point q, point A, point B) {
double a = B.y-A.y, b = A.x-B.x, c = B.x*A.y - A.x*B.y;
double u = fabs(a*p.x + b*p.y + c);
double v = fabs(a*q.x + b*q.y + c);
return point((p.x*v + q.x*u) / (u+v), (p.y*v + q.y*u) / (u+v));
}
// cuts polygon Q along the line formed by point A->point B (order matters)
// (note: the last point must be the same as the first point)
vector<point> cutPolygon(point A, point B, const vector<point> &Q) {
vector<point> P;
for (int i = 0; i < (int)Q.size(); ++i) {
double left1 = cross(toVec(A, B), toVec(A, Q[i])), left2 = 0;
if (i != (int)Q.size()-1) left2 = cross(toVec(A, B), toVec(A, Q[i+1]));
if (left1 > -EPS) P.push_back(Q[i]); // Q[i] is on the left
if (left1*left2 < -EPS) // crosses line AB
P.push_back(lineIntersectSeg(Q[i], Q[i+1], A, B));
}
if (!P.empty() && !(P.back() == P.front()))
P.push_back(P.front()); // wrap around
return P;
}
vector<point> CH_Graham(vector<point> &Pts) { // overall O(n log n)
vector<point> P(Pts); // copy all points
int n = (int)P.size();
if (n <= 3) { // point/line/triangle
if (!(P[0] == P[n-1])) P.push_back(P[0]); // corner case
return P; // the CH is P itself
}
// first, find P0 = point with lowest Y and if tie: rightmost X
int P0 = min_element(P.begin(), P.end())-P.begin();
swap(P[0], P[P0]); // swap P[P0] with P[0]
// second, sort points by angle around P0, O(n log n) for this sort
sort(++P.begin(), P.end(), [&](point a, point b) {
return ccw(P[0], a, b); // use P[0] as the pivot
});
// third, the ccw tests, although complex, it is just O(n)
vector<point> S({P[n-1], P[0], P[1]}); // initial S
int i = 2; // then, we check the rest
while (i < n) { // n > 3, O(n)
int j = (int)S.size()-1;
if (ccw(S[j-1], S[j], P[i])) // CCW turn
S.push_back(P[i++]); // accept this point
else // CW turn
S.pop_back(); // pop until a CCW turn
}
return S; // return the result
}
vector<point> CH_Andrew(vector<point> &Pts) { // overall O(n log n)
int n = Pts.size(), k = 0;
vector<point> H(2*n);
sort(Pts.begin(), Pts.end()); // sort the points by x/y
for (int i = 0; i < n; ++i) { // build lower hull
while ((k >= 2) && !ccw(H[k-2], H[k-1], Pts[i])) --k;
H[k++] = Pts[i];
}
for (int i = n-2, t = k+1; i >= 0; --i) { // build upper hull
while ((k >= t) && !ccw(H[k-2], H[k-1], Pts[i])) --k;
H[k++] = Pts[i];
}
H.resize(k);
return H;
}
int main() {
// 6(+1) points, entered in counter clockwise order, 0-based indexing
vector<point> P;
P.emplace_back(1, 1); // P0
P.emplace_back(3, 3); // P1
P.emplace_back(9, 1); // P2
P.emplace_back(12, 4); // P3
P.emplace_back(9, 7); // P4
P.emplace_back(1, 7); // P5
P.push_back(P[0]); // loop back, P6 = P0
printf("Perimeter = %.2lf\n", perimeter(P)); // 31.64
printf("Area = %.2lf\n", area(P)); // 49.00
printf("Area = %.2lf\n", area_alternative(P)); // also 49.00
printf("Is convex = %d\n", isConvex(P)); // 0 (false)
//// the positions of P_out, P_on, P_in with respect to the polygon
//7 P5------P_on----P4
//6 | \
//5 | \
//4 | P_in P3
//3 | P1___ /
//2 | / P_out \ ___ /
//1 P0 P2
//0 1 2 3 4 5 6 7 8 9 101112
point p_out(3, 2); // outside this (concave) polygon
printf("P_out is inside = %d\n", insidePolygon(p_out, P)); // -1
printf("P1 is inside = %d\n", insidePolygon(P[1], P)); // 0
point p_on(5, 7); // on this (concave) polygon
printf("P_on is inside = %d\n", insidePolygon(p_on, P)); // 0
point p_in(3, 4); // inside this (concave) polygon
printf("P_in is inside = %d\n", insidePolygon(p_in, P)); // 1
// cutting the original polygon based on line P[2] -> P[4] (get the left side)
//7 P5--------------P4
//6 | | \
//5 | | \
//4 | | P3
//3 | P1___ | /
//2 | / \ ___ | /
//1 P0 P2
//0 1 2 3 4 5 6 7 8 9 101112
// new polygon (notice the index are different now):
//7 P4--------------P3
//6 | |
//5 | |
//4 | |
//3 | P1___ |
//2 | / \ ___ |
//1 P0 P2
//0 1 2 3 4 5 6 7 8 9
P = cutPolygon(P[2], P[4], P);
printf("Perimeter = %.2lf\n", perimeter(P)); // smaller now, 29.15
printf("Area = %.2lf\n", area(P)); // 40.00
// running convex hull of the resulting polygon (index changes again)
//7 P3--------------P2
//6 | |
//5 | |
//4 | P_out |
//3 | |
//2 | P_in |
//1 P0--------------P1
//0 1 2 3 4 5 6 7 8 9
P = CH_Graham(P); // now this is a rectangle
printf("Perimeter = %.2lf\n", perimeter(P)); // precisely 28.00
printf("Area = %.2lf\n", area(P)); // precisely 48.00
printf("Is convex = %d\n", isConvex(P)); // true
printf("P_out is inside = %d\n", insidePolygon(p_out, P)); // 1
printf("P_in is inside = %d\n", insidePolygon(p_in, P)); // 1
return 0;
}