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#include <algorithm>
#include <vector>
template<typename T>
struct Point{
T x,y;
Point(T x=0,T y=0):x(x),y(y){}
Point operator+(const Point& o) const{return Point(x+o.x,y+o.y);}
Point operator-(const Point& o) const{return Point(x-o.x,y-o.y);}
bool operator==(const Point& o) const{return x==o.x&&y==o.y;}
T operator*(const Point& o) const{return x*o.x+y*o.y;}
T operator^(const Point& o) const{return x*o.y-y*o.x;}
bool operator<(const Point& o) const{
if(x==o.x){
return y<o.y;
}
return x<o.x;
}
};
template<typename T>
vector<Point<T>> convexHull(vector<Point<T>> pts){
int n=pts.size();
sort(pts.begin(),pts.end());
if(n<=2){
return pts;
}
vector<Point<T>> hull;
for(auto& p:pts){
while(hull.size()>=2&&((hull.back()-hull[hull.size()-2])^(p-hull.back()))<=0){
hull.pop_back();
}
hull.push_back(p);
}
size_t lwr=hull.size();
for(int i=pts.size()-2;i>=0;i--){
Point<T> p=pts[i];
while(hull.size()>lwr&&((hull.back()-hull[hull.size()-2])^(p-hull.back()))<=0){
hull.pop_back();
}
hull.push_back(p);
}
if(hull.size()>1){
hull.pop_back();
}
return hull;
}
template<typename T>
vector<Point<T>> standardize(const vector<Point<T>>& poly){
if(poly.empty()){
return poly;
}
int idx=0;
for(int i=1;i<(int)poly.size();i++){
if(poly[i].y<poly[idx].y||(poly[i].y==poly[idx].y&&poly[i].x<poly[idx].x)){
idx = i;
}
}
vector<Point<T>> ret;
for(int i=idx;i<(int)poly.size();i++){
ret.push_back(poly[i]);
}
for(int i=0;i<idx;i++){
ret.push_back(poly[i]);
}
return ret;
}
template<typename T>
vector<Point<T>> minkowskiSum(const vector<Point<T>>& a,const vector<Point<T>>& b){
vector<Point<T>> polyA=standardize(a),polyB=standardize(b);
int n=polyA.size(),m=polyB.size();
vector<Point<T>> edA,edB;
for(int i=0;i<n;i++){
edA.push_back(polyA[(i+1)%n]-polyA[i]);
}
for(int i=0;i<m;i++){
edB.push_back(polyB[(i+1)%m]-polyB[i]);
}
vector<Point<T>> merged;
int i=0,j=0;
while(i<n&&j<m){
ll cr=edA[i]^edB[j];
if(cr>0){
merged.push_back(edA[i]);
i++;
}
else if(cr<0){
merged.push_back(edB[j]);
j++;
}
else{
Point<T> sum=edA[i]+edB[j];
if(sum.x!=0||sum.y!=0){
merged.push_back(sum);
}
i++;
j++;
}
}
while(i<n){
merged.push_back(edA[i]);
i++;
}
while(j<m){
merged.push_back(edB[j]);
j++;
}
Point<T> cur=polyA[0]+polyB[0];
vector<Point<T>> c={cur};
for(auto& pt:merged){
cur = cur+pt;
c.push_back(cur);
}
if(c.size()>1&&c.back()==c[0]){
c.pop_back();
}
return c;
}
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