polygon convex_intersect(const polygon &P, const polygon &Q) {
const int n = P.size(), m = Q.size();
int a = 0, b = 0, aa = 0, ba = 0;
enum { Pin, Qin, Unknown } in = Unknown;
polygon R;
do {
int a1 = (a+n-1) % n, b1 = (b+m-1) % m;
number C = cross(P[a] - P[a1], Q[b] - Q[b1]);
number A = cross(P[a1] - Q[b], P[a] - Q[b]);
number B = cross(Q[b1] - P[a], Q[b] - P[a]);
point r;
if (intersect_1pt(P[a1], P[a], Q[b1], Q[b], r)) {
if (in == Unknown) aa = ba = 0;
R.push_back( r );
in = B > number(0) ? Pin : A > number(0) ? Qin : in;
}
if (C == number(0) && B == number(0) && A == number(0)) {
if (in == Pin) { b = (b + 1) % m; ++ba; }
else { a = (a + 1) % m; ++aa; }
} else if (C >= number(0)) {
if (A > number(0)) { if (in == Pin) R.push_back(P[a]); a = (a+1)%n; ++aa; }
else { if (in == Qin) R.push_back(Q[b]); b = (b+1)%m; ++ba; }
} else {
if (B > number(0)) { if (in == Qin) R.push_back(Q[b]); b = (b+1)%m; ++ba; }
else { if (in == Pin) R.push_back(P[a]); a = (a+1)%n; ++aa; }
}
} while ( (aa < n || ba < m) && aa < 2*n && ba < 2*m );
if (in == Unknown) {
if (contains(Q, P[0])) return P;
if (contains(P, Q[0])) return Q;
}
return R;
}
double perimeter(polygon p){
double per = 0.0;
for(int i = 0; i < p.size(); i++){
per += dist(p[i], p[(i+1)%p.size()]);
}
return per;
}
double area(polygon& poly){
double ret = 0.0;
for(int i = 0; i < poly.size(); i++){
ret += trap(poly[i], poly[(i+1)%poly.size()]);
}
return fabs(ret);
}
//testa se o ponto esta dentro de um poligono (nao necessariamente convexo)
bool inside_poly(pt p, polygon poly){
poly.push_back(poly[0]);
for(int i = 0; i < poly.size()-1; i++)
if(point_and_seg(poly[i], poly[i+1], p)) return true; //na borda
for(int i = 0; i < poly.size()-1; i++) poly[i] = poly[i] - p;
p = pt(0, 0);
double theta, y;
while(true){
theta = (double)rand()/10000.0;
bool inter = false;
//evita que um ponto fique no eixo x
for(int i = 0; i < poly.size()-1; i++){
poly[i] = rotate(poly[i], theta);
if( !cmp(poly[i].x) ) inter = true;
}
if( !inter ){
poly[poly.size()-1] = poly[0];
//testa as possiveis intersecoes
for(int i = 0; i < poly.size()-1; i++){
if( cmp( poly[i].x * poly[i+1].x ) < 0 ){
y = poly[i+1].y - poly[i+1].x * (poly[i].y - poly[i+1].y) / (poly[i].x - poly[i+1].x);
if( cmp(y) > 0 ) inter = !inter; //se interecao valida
}
}
return inter; //testa a paridade da semi-reta vertical partindo de p
}
}
return true;
}
//testa se o ponto esta no poligono convexo
bool inside_convex_poly(pt p, polygon& poly){
int left = 0, right = 0, side;
for(int i = 0; i < poly.size(); i++){
side = side_sign(p, poly[i], poly[(i+1)%poly.size()]);
if(side < 0) right++;
if(side > 0) left++;
}
return !(left && right);
}
//Determina se o poligono simples eh convexo
bool is_convex(polygon& p){
int left = 0, right = 0, side;
for(int i = 0; i < p.size(); i++){
side = side_sign(p[i], p[(i+1)%p.size()], p[(i+2)%p.size()]);
if(side < 0) right++;
if(side > 0) left++;
}
return !(left && right);
}
double dist(const polygon &a, const polygon &b) {
double ret = 1e100;
for (int i = 0; i < (int)a.size() - 1; ++ i)
for (int j = 0; j < (int)b.size() - 1; ++ j) {
ret = min(ret, dist(a[i], b[j], b[j + 1]));
ret = min(ret, dist(b[j], a[i], a[i + 1]));
}
return ret;
}
bool positiveArea(polygon p) {
for (int i = 0; i < p.size(); ++i) {
for (int j = i + 1; j < p.size(); ++j) {
for (int k = j + 1; k < p.size(); ++k) {
if (collineal(p[0], p[1], p[2])) return false;
}
}
}
return true;
}
bool polygonintersect(const polygon &a, const polygon &b) {
for (int i = 0; i < (int)a.size() - 1; ++ i)
if (inside(a[i], b)) return true;
for (int i = 0; i < (int)b.size() - 1; ++ i)
if (inside(b[i], a)) return true;
for (int i = 0; i < (int)a.size() - 1; ++ i)
for (int j = 0; j < (int)b.size() - 1; ++ j)
if (intersect(a[i], a[i + 1], b[j], b[j + 1]))
return true;
return false;
}
// 두 다각형이 서로 닿거나 겹치는지 여부를 반환한다.
// 한 점이라도 겹친다면 true 를 반환한다.
bool polygonIntersects(const polygon& p, const polygon& q) {
int n = p.size(), m = q.size();
// 우선 한 다각형이 다른 다각형에 포함되어 있는 경우를 확인하자
if(isInside(p[0], q) || isInside(q[0], p)) return true;
// 이외의 경우, 두 다각형이 서로 겹친다면 서로 닿는 두 변이 반드시 존재한다
for(int i = 0; i < n; i++)
for(int j = 0; j < m; j++)
if(segmentIntersects(p[i], p[(i+1)%n], q[j], q[(j+1)%m]))
return true;
return false;
}
pt centroid(polygon p){
double a = area(p);
double xc = 0.0, yc = 0.0;
for(int i = 0; i < p.size(); i++){
int next = (i+1)%p.size();
xc += (p[i].x + p[next].x)*(p[i].x*p[next].y - p[next].x*p[i].y);
yc += (p[i].y + p[next].y)*(p[i].x*p[next].y - p[next].x*p[i].y);
}
return pt(xc/(6.0*a), yc/(6.0*a));
}
int isPointInPolygon(point p, polygon &pg)
{
bool in = false;
for (int i = 0; i < pg.size(); i++)
{
point a = pg[i] - p, b = pg[(i + 1) % pg.size()] - p;
if (abs(cross(a, b)) < EPSILON && dot(a, b) < EPSILON) return ON;
if (a.y > b.y) swap(a, b);
if (a.y < EPSILON && EPSILON < b.y && cross(a, b) > EPSILON) in = !in;
}
return in ? IN : OUT;
}
inline void draw(const polygon<T,2>& polygon)
{
if (polygon.size() < 3) return;
std::size_t j = polygon.size() - 1;
for (std::size_t i = 0; i < polygon.size(); ++i)
{
draw_segment(polygon[i],polygon[j]);
j = i;
}
}
bool isugly(const polygon <float> & pol)
{
if (pol.size() < 3)
return true;
//Add other ugly conditions
return false;
}
int in_poly(point p, polygon& T) {
double a = 0; int N = T.size();
for (int i = 0; i < N; i++) {
if (between(T[i], p, T[(i+1) % N])) return -1;
a += angle(T[i], p, T[(i+1) % N]);
}
return cmp(a) != 0;
}
/**
* @brief Add a polygon to the mesh.
* @param p
*/
void Mesh::add_polygon( polygon& p )
{
polygons.push_back( p );
if ( p.size() == 3 ) {
tris.push_back( p.at(0) );
tris.push_back( p.at(1) );
tris.push_back( p.at(2) );
}
if ( p.size() == 4 ) {
quads.push_back( p.at(0) );
quads.push_back( p.at(1) );
quads.push_back( p.at(2) );
quads.push_back( p.at(3) );
}
}
bool inside(const point &a, const polygon &p) {
double cnt = 0;
for (int i = 0; i < (int)p.size() - 1; ++ i) {
point x = p[i] - a, y = p[i + 1] - a;
if (abs(x) == 0 || abs(y) == 0) return true;
cnt += asin(cross(x, y) / abs(x) / abs(y));
}
return fabs(cnt) >= 2 * pi - eps;
}
double getArea(polygon &pg)
{
double area = 0.0;
int n = pg.size();
if (n < 3) return area;
for (int i = 0, j = 1; i < n; i++, j = (i + 1) % n)
area += (pg[i].x * pg[j].y - pg[j].x * pg[i].y);
return fabs(area / 2.0);
}
int contains(const polygon& P, const point& p) {
bool in = false;
for (int i = 0; i < P.size(); ++i) {
point a = curr(P,i) - p, b = next(P,i) - p;
if (imag(a) > imag(b)) swap(a, b);
if (imag(a) <= number(0) && number(0) < imag(b))
if (cross(a, b) < number(0)) in = !in;
if (cross(a, b) == number(0) && dot(a, b) <= number(0)) return ON;
}
return in ? IN : OUT;
}
int contains(const polygon& G, const P& p) {
bool in = false;
for (int i = 0; i < G.size(); ++i) {
P a = curr(G,i) - p, b = next(G,i) - p;
if (imag(a) > imag(b)) swap(a, b);
if (imag(a) <= 0 && 0 < imag(b))
if (cross(a, b) < 0) in = !in;
if (cross(a, b) == 0 && dot(a, b) <= 0) return ON;
}
return in ? IN : OUT;
}
point center(const polygon& poly) {
double x = 0.0;
double y = 0.0;
for (auto& p : poly) {
x += p.x / poly.size();
y += p.y / poly.size();
}
return { x, y };
}
//Corta o poligono pela reta ab
//pol1 - poligono do lado esquerdo
//pol2 - poligono do lado direito
void cut_polygon(polygon pol, pt a, pt b, polygon& pol1, polygon& pol2){
polygon pp, pn;
pt p1, p2, r;
for(int i = 0; i < pol.size(); i++){
p1 = pol[i]; p2 = pol[(i+1)%pol.size()];
int side = side_sign(a, b, p1);
if(side >= 0) pp.push_back(p1);
if(side <= 0) pn.push_back(p1);
if(intersect(a, b, p1, p2, r) && (r == pt(INF, INF))){
if(point_and_seg(p1, p2, r)){
pp.push_back(r);
pn.push_back(r);
}
}
if(pp.size() > 2) convex_hull(pp, pol1);
if(pn.size() > 2) convex_hull(pn, pol2);
}
}
void convex_hull(polygon in, polygon& hull){
hull.clear();
if(in.size() < 3){ hull = in; return; }
int pos = 0;
for(int i = 1; i < in.size(); i++) if(in[i] < in[pos]) pos = i;
swap(in[0], in[pos]);
refer = in[0];
sort(in.begin() + 1, in.end(), cmp_angle);
in.resize(unique(in.begin(), in.end()) - in.begin());
hull.push_back(in[0]); hull.push_back(in[1]);
in.push_back(in[0]); //isto evita pontos colineares no final do poligono
for(int i = 2; i < in.size(); ){
if(hull.size() > 2 && side_sign(hull[hull.size() - 2], hull[hull.size() - 1], in[i]) <= 0){
hull.pop_back();
}
else hull.push_back(in[i++]);
}
//tira a duplicata
hull.pop_back();
}
polygon poly_intersect(polygon &P, polygon &Q){
int m = Q.size(), n = P.size();
int a = 0, b = 0, aa = 0, ba = 0, inflag = 0;
polygon R;
while( (aa < n || ba < m) && aa < 2*n && ba < 2*m){
Point p1 = P[a], p2 = P[(a+1) % n], q1 = Q[b], q2 = Q[(b+1) % m];
Point A = p2 - p1, B = q2 - q1;
int cross = cmp(A ^ B), ha = turn(p2, q2, p1), hb = turn(q2, p2, q1);
if(cross == 0 && turn(p1, q1, p2) == 0 && cmp(A * B) < 0){
if(between(p1, q1, p2)) R.push_back(q1);
if(between(p1, q2, p2)) R.push_back(q2);
if(between(q1, p1, q2)) R.push_back(p1);
if(between(q1, p2, q2)) R.push_back(p2);
if(R.size() < 2) return polygon();
inflag = 1; break;
}else if(cross != 0 && seg_intersect(p1, p2, q1, q2)) {
if(inflag == 0) aa = ba = 0;
R.push_back(intersection(p1, p2, q1, q2));
inflag = (hb > 0) ? 1 : -1;
}
if(cross == 0 && hb < 0 && ha < 0) return R;
bool t = cross == 0 && hb == 0 && ha == 0;
if(t ? (inflag == 1) : (cross >= 0) ? (ha <= 0) : (hb > 0)){
if(inflag == -1) R.push_back(q2);
ba++; b++; b %= m;
}else{
if(inflag == 1) R.push_back(p2);
aa++; a++; a %= n;
}
}
if(inflag == 0){
if (in_polygon(P[0], Q)) return P;
if (in_polygon(Q[0], P)) return Q;
}
R.erase(unique(R.begin(), R.end()), R.end());
if(R.size() > 1 && R.front() == R.back()) R.pop_back();
return R;
}
point getCentroid1(polygon &pg)
{
double areaOfPolygon = 0, areaOfTriangle = 0, px = 0, py = 0;
for (int i = 2; i < pg.size(); i++)
{
areaOfTriangle = cross(pg[i - 1] - pg[0], pg[i] - pg[0]);
areaOfPolygon += areaOfTriangle;
px += areaOfTriangle * (pg[0].x + pg[i - 1].x + pg[i].x);
py += areaOfTriangle * (pg[0].y + pg[i - 1].y + pg[i].y);
}
return point(px / (3.0 * areaOfPolygon), py / (3.0 * areaOfPolygon));
}
bool polygonsIntersect(const polygon &a, const polygon &b) {
int na = a.size(), nb = b.size();
for (int i = 0; i < na; ++i) {
if (pointInPoly(a[i], b)) return true;
}
for (int i = 0; i < nb; ++i) {
if (pointInPoly(b[i], a)) return true;
}
for (int i = 0; i < na; ++i) {
for (int j = 0; j < nb; ++j) {
int xa1 = a[i].first, ya1 = a[i].second;
int xa2 = a[(i + 1) % na].first, ya2 = a[(i + 1) % na].second;
int xb1 = b[j].first, yb1 = b[j].second;
int xb2 = b[(j + 1) % nb].first, yb2 = b[(j + 1) % nb].second;
if (segment_segment_intersection(xa1, ya1, xa2, ya2, xb1, yb1, xb2, yb2)) {
return true;
}
}
}
return false;
}
point getCentroid2(polygon &pg)
{
double areaOfPolygon = 0, areaOfTriangle = 0, px = 0, py = 0;
int n = pg.size();
for (int i = 0; i < n; i++)
{
areaOfTriangle = cross(pg[i], pg[(i + 1) % n]);
areaOfPolygon += areaOfTriangle;
px += areaOfTriangle * (pg[i].x + pg[(i + 1) % n].x);
py += areaOfTriangle * (pg[i].y + pg[(i + 1) % n].y);
}
return point(px / (3.0 * areaOfPolygon), py / (3.0 * areaOfPolygon));
}
result lineIntersectsPolygon(line & l, polygon & P) {
result points;
point prev = P[0];
for (size_t i = 1; i < P.size(); i++) {
point cur = P[i];
segment s = {prev, cur};
if (lineIntersectsSegment(l, s)) {
points.insert(i - 1);
}
prev = cur;
}
return points;
}
// Returns a list of points on the convex hull in counter-clockwise order.
// NOTE: the last point in the returned list is the same as the first one.
void convex_hull_2(polygon P, polygon& hull) {
hull.clear();
// Sort points lexicographically
sort(P.begin(), P.end());
P.resize(unique(P.begin(), P.end()) - P.begin());
// Build lower hull
for (int i = 0; i < P.size(); i ++) {
while (hull.size() >= 2 && side_sign(hull[hull.size() - 2], hull[hull.size() - 1], P[i]) <= 0)
hull.pop_back();
hull.push_back(P[i]);
};
// Build upper hull
for (int i = P.size()-2, t = hull.size() + 1; i >= 0; i --) {
while (hull.size() >= t && side_sign(hull[hull.size()-2], hull[hull.size()-1], P[i]) <= 0)
hull.pop_back();
hull.push_back(P[i]);
};
}
// (a,b) 를 포함하는 직선으로 다각형 p 를 자른 뒤, (a,b) 의 왼쪽에 있는 부분들을 반환한다
polygon cutPoly(const polygon& p, const vector2& a, const vector2& b) {
int n = p.size();
vector<bool> inside(n);
for(int i = 0; i < n; ++i)
inside[i] = ((b-a).cross(p[i]-a) >= 0);
// 이외의 경우에는 일부는 직선 왼쪽에, 일부는 직선 오른쪽에 떨어진다
polygon ret;
for(int i = 0; i < n; ++i) {
int j = (i + 1) % n;
if(inside[i]) ret.push_back(p[i]);
if(inside[i] != inside[j]) {
vector2 intersection;
assert(lineIntersection(p[i], p[j], a, b, intersection));
ret.push_back(intersection);
}
}
return ret;
}