इस ट्यूटोरियल में, हम दिए गए बिंदुओं के उत्तल पतवार को खोजने के लिए एक कार्यक्रम पर चर्चा करेंगे।
उत्तल पतवार सबसे छोटी बहुभुज उत्तल आकृति है जिसमें दिए गए सभी बिंदु या तो आकृति के अंदर की सीमा पर होते हैं।
इस कार्यक्रम में, हम दिए गए बिंदुओं को छोटे खंडों में विभाजित करने के लिए पाशविक बल का उपयोग करेंगे और फिर अंत में उत्तल पतवार का निर्माण करने के लिए उन बिंदुओं को मिला देंगे।
उदाहरण
#include<bits/stdc++.h>
using namespace std;
//storing the center point of polygon
pair<int, int> mid;
//calculating the quadrant of
//a particular point
int quad(pair<int, int> p){
if (p.first >= 0 && p.second >= 0)
return 1;
if (p.first <= 0 && p.second >= 0)
return 2;
if (p.first <= 0 && p.second <= 0)
return 3;
return 4;
}
//if line is touching the polygon
int calc_line(pair<int, int> a, pair<int, int> b,
pair<int, int> c){
int res = (b.second-a.second)*(c.first-b.first) -
(c.second-b.second)*(b.first-a.first);
if (res == 0)
return 0;
if (res > 0)
return 1;
return -1;
}
//comparing functions for sorting
bool compare(pair<int, int> p1, pair<int, int> q1){
pair<int, int> p = make_pair(p1.first - mid.first,
p1.second - mid.second);
pair<int, int> q = make_pair(q1.first - mid.first,
q1.second - mid.second);
int one = quad(p);
int two = quad(q);
if (one != two)
return (one < two);
return (p.second*q.first < q.second*p.first);
}
//finding the upper tangent for both polygons
vector<pair<int, int>> merger(vector<pair<int, int> > a,
vector<pair<int, int> > b){
int n1 = a.size(), n2 = b.size();
int ia = 0, ib = 0;
for (int i=1; i<n1; i++)
if (a[i].first > a[ia].first)
ia = i;
//calculating leftmost point of b
for (int i=1; i<n2; i++)
if (b[i].first < b[ib].first)
ib=i;
int inda = ia, indb = ib;
bool done = 0;
while (!done){
done = 1;
while (calc_line(b[indb], a[inda], a[(inda+1)%n1]) >=0)
inda = (inda + 1) % n1;
while (calc_line(a[inda], b[indb], b[(n2+indb-1)%n2]) <=0){
indb = (n2+indb-1)%n2;
done = 0;
}
}
int uppera = inda, upperb = indb;
inda = ia, indb=ib;
done = 0;
int g = 0;
//calculating the lower tangent
while (!done){
done = 1;
while (calc_line(a[inda], b[indb], b[(indb+1)%n2])>=0)
indb=(indb+1)%n2;
while (calc_line(b[indb], a[inda], a[(n1+inda-1)%n1])<=0){
inda=(n1+inda-1)%n1;
done=0;
}
}
int lowera = inda, lowerb = indb;
vector<pair<int, int>> ret;
//merging the two polygons to get convex hull
int ind = uppera;
ret.push_back(a[uppera]);
while (ind != lowera){
ind = (ind+1)%n1;
ret.push_back(a[ind]);
}
ind = lowerb;
ret.push_back(b[lowerb]);
while (ind != upperb){
ind = (ind+1)%n2;
ret.push_back(b[ind]);
}
return ret;
}
//brute force algo to find convex hull
vector<pair<int, int>> bruteHull(vector<pair<int, int>> a){
set<pair<int, int> >s;
for (int i=0; i<a.size(); i++){
for (int j=i+1; j<a.size(); j++){
int x1 = a[i].first, x2 = a[j].first;
int y1 = a[i].second, y2 = a[j].second;
int a1 = y1-y2;
int b1 = x2-x1;
int c1 = x1*y2-y1*x2;
int pos = 0, neg = 0;
for (int k=0; k<a.size(); k++){
if (a1*a[k].first+b1*a[k].second+c1 <= 0)
neg++;
if (a1*a[k].first+b1*a[k].second+c1 >= 0)
pos++;
}
if (pos == a.size() || neg == a.size()){
s.insert(a[i]);
s.insert(a[j]);
}
}
}
vector<pair<int, int>>ret;
for (auto e:s)
ret.push_back(e);
//moving through anti clockwise direction
mid = {0, 0};
int n = ret.size();
for (int i=0; i<n; i++){
mid.first += ret[i].first;
mid.second += ret[i].second;
ret[i].first *= n;
ret[i].second *= n;
}
sort(ret.begin(), ret.end(), compare);
for (int i=0; i<n; i++)
ret[i] = make_pair(ret[i].first/n, ret[i].second/n);
return ret;
}
//returning the value of convex hull
vector<pair<int, int>> divide(vector<pair<int, int>> a){
if (a.size() <= 5)
return bruteHull(a);
// left contains the left half points
// right contains the right half points
vector<pair<int, int>>left, right;
for (int i=0; i<a.size()/2; i++)
left.push_back(a[i]);
for (int i=a.size()/2; i<a.size(); i++)
right.push_back(a[i]);
vector<pair<int, int>>left_hull = divide(left);
vector<pair<int, int>>right_hull = divide(right);
//merging the convex hulls
return merger(left_hull, right_hull);
}
int main(){
vector<pair<int, int> > a;
a.push_back(make_pair(0, 0));
a.push_back(make_pair(1, -4));
a.push_back(make_pair(-1, -5));
a.push_back(make_pair(-5, -3));
a.push_back(make_pair(-3, -1));
a.push_back(make_pair(-1, -3));
a.push_back(make_pair(-2, -2));
a.push_back(make_pair(-1, -1));
a.push_back(make_pair(-2, -1));
a.push_back(make_pair(-1, 1));
int n = a.size();
sort(a.begin(), a.end());
vector<pair<int, int> >ans = divide(a);
cout << "Convex Hull:\n";
for (auto e:ans)
cout << e.first << " "<< e.second << endl;
return 0;
} आउटपुट
Convex Hull: -5 -3 -1 -5 1 -4 0 0 -1 1