给定凸包，我们需要在凸包中添加给定数量的点，并在每次添加点后打印凸包。添加每个点后，这些点应按逆时针顺序排列。
例子：
Input :
Convex Hull : (0, 0), (3, -1), (4, 5), (-1, 4)
Point to add : (100, 100)
Output :
New convex hull : (-1, 4) (0, 0) (3, -1) (100, 100)我们首先检查该点是否在给定的凸包内。如果是，则无需执行任何操作，我们将直接返回给定的凸包。如果该点在凸包之外，我们将找到上下切线，然后将该点与给定的凸包合并以找到新的凸包，如图所示。
红色轮廓线显示了合并该点和给定凸包后的新凸包。
要找到上切线，我们首先在船体上选择一个最接近给定点的点。然后，当连接凸包上的点和给定点的线穿过凸包时，我们逆时针移动，直到得到切线。
该图显示了凸包上的点的移动，以找到上切线。
注意：这里假定初始凸包的输入是逆时针顺序的，否则我们必须首先以逆时针顺序对其进行排序，然后应用以下代码。
代码：
CPP
// C++ program to add given a point p to a given
// convext hull. The program assumes that the
// point of given convext hull are in anti-clockwise
// order.
#include
using namespace std;
// checks whether the point crosses the convex hull
// or not
int orientation(pair a, pair b,
pair 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;
}
// Returns the square of distance between two input points
int sqDist(pair p1, pair p2)
{
return (p1.first-p2.first)*(p1.first-p2.first) +
(p1.second-p2.second)*(p1.second-p2.second);
}
// Checks whether the point is inside the convex hull or not
bool inside(vector> a, pair p)
{
// Initialize the centroid of the convex hull
pair mid = {0, 0};
int n = a.size();
// Multiplying with n to avoid floating point
// arithmetic.
p.first *= n;
p.second *= n;
for (int i=0; i> &a, pair p)
{
// If point is inside p
if (inside(a, p))
return;
// point having minimum distance from the point p
int ind = 0;
int n = a.size();
for (int i=1; i=0)
up = (up + 1) % n;
// Find the lower tangent
int low = ind;
while (orientation(p, a[low], a[(n+low-1)%n])<=0)
low = (n+low - 1) % n;
// Initialize result
vector>ret;
// making the final hull by traversing points
// from up to low of given convex hull.
int curr = up;
ret.push_back(a[curr]);
while (curr != low)
{
curr = (curr+1)%n;
ret.push_back(a[curr]);
}
// Modify the original vector
ret.push_back(p);
a.clear();
for (int i=0; i > a;
a.push_back({0, 0});
a.push_back({3, -1});
a.push_back({4, 5});
a.push_back({-1, 4});
int n = a.size();
pair p = {100, 100};
addPoint(a, p);
// Print the modified Convex Hull
for (auto e : a)
cout << "(" << e.first << ", "
<< e.second << ") ";
return 0;
} Java
// Java program to add given a point p to a given
// convext hull. The program assumes that the
// point of given convext hull are in anti-clockwise
// order.
import java.io.*;
import java.util.*;
class GFG
{
// checks whether the point crosses the convex hull
// or not
static int orientation(ArrayList a,
ArrayList b,
ArrayList c)
{
int res = (b.get(1) - a.get(1)) * (c.get(0) - b.get(0)) -
(c.get(1) - b.get(1)) * (b.get(0)-a.get(0));
if (res == 0)
return 0;
if (res > 0)
return 1;
return -1;
}
// Returns the square of distance between two input points
static int sqDist(ArrayListp1, ArrayListp2)
{
return (p1.get(0) - p2.get(0)) * (p1.get(0) - p2.get(0)) +
(p1.get(1) - p2.get(1)) * (p1.get(1) - p2.get(1));
}
// Checks whether the point is inside the convex hull or not
static boolean inside(ArrayList> A,ArrayListp)
{
// Initialize the centroid of the convex hull
ArrayList mid = new ArrayList(Arrays.asList(0,0));
int n = A.size();
for (int i = 0; i < n; i++)
{
mid.set(0,mid.get(0) + A.get(i).get(0));
mid.set(1,mid.get(1) + A.get(i).get(1));
}
// if the mid and the given point lies always
// on the same side w.r.t every edge of the
// convex hull, then the point lies inside
// the convex hull
for (int i = 0, j; i < n; i++)
{
j = (i + 1) % n;
int x1 = A.get(i).get(0)*n, x2 = A.get(j).get(0)*n;
int y1 = A.get(i).get(1)*n, y2 = A.get(j).get(1)*n;
int a1 = y1 - y2;
int b1 = x2 - x1;
int c1 = x1 * y2 - y1 * x2;
int for_mid = a1 * mid.get(0) + b1 * mid.get(1) + c1;
int for_p = a1 * p.get(0) * n + b1 * p.get(1) * n + c1;
if (for_mid*for_p < 0)
return false;
}
return true;
}
// Adds a point p to given convex hull a[]
static void addPoint(ArrayList> a,ArrayList p)
{
// If point is inside p
if (inside(a, p))
return;
// point having minimum distance from the point p
int ind = 0;
int n = a.size();
for (int i = 1; i < n; i++)
{
if (sqDist(p, a.get(i)) < sqDist(p, a.get(ind)))
{
ind = i;
}
}
// Find the upper tangent
int up = ind;
while (orientation(p, a.get(up), a.get((up+1)%n))>=0)
up = (up + 1) % n;
// Find the lower tangent
int low = ind;
while (orientation(p, a.get(low), a.get((n+low-1)%n))<=0)
low = (n+low - 1) % n;
// Initialize result
ArrayList> ret = new ArrayList>();
// making the final hull by traversing points
// from up to low of given convex hull.
int curr = up;
ret.add(a.get(curr));
while (curr != low)
{
curr = (curr + 1) % n;
ret.add(a.get(curr));
}
// Modify the original vector
ret.add(p);
a.clear();
for (int i = 0; i < ret.size(); i++)
{
a.add(ret.get(i));
}
}
// Driver code
public static void main (String[] args)
{
// the set of points in the convex hull
ArrayList> a = new ArrayList>();
a.add(new ArrayList(Arrays.asList(0, 0)));
a.add(new ArrayList(Arrays.asList(3, -1)));
a.add(new ArrayList(Arrays.asList(4, 5)));
a.add(new ArrayList(Arrays.asList(-1, 4)));
int n = a.size();
ArrayList p = new ArrayList(Arrays.asList(100,100));
addPoint(a, p);
// Print the modified Convex Hull
for(ArrayList e:a )
{
System.out.print("(" + e.get(0) + ", " + e.get(1) + ") ");
}
}
}
// This code is contributed by rag2127 Python3
# Python 3 program to add given a point p to a given
# convext hull. The program assumes that the
# point of given convext hull are in anti-clockwise
# order.
import copy
# checks whether the point crosses the convex hull
# or not
def orientation(a, b, c):
res = ((b[1] - a[1]) * (c[0] - b[0]) -
(c[1] - b[1]) * (b[0] - a[0]))
if (res == 0):
return 0;
if (res > 0):
return 1;
return -1;
# Returns the square of distance between two input points
def sqDist(p1, p2):
return ((p1[0] - p2[0]) * (p1[0] - p2[0]) +
(p1[1] - p2[1]) * (p1[1] - p2[1]));
# Checks whether the point is inside the convex hull or not
def inside( a, p ):
# Initialize the centroid of the convex hull
mid = [0, 0]
n = len(a)
# Multiplying with n to avoid floating point
# arithmetic.
p[0] *= n;
p[1] *= n;
for i in range(n):
mid[0] += a[i][0];
mid[1] += a[i][1];
a[i][0] *= n;
a[i][1] *= n;
# if the mid and the given point lies always
# on the same side w.r.t every edge of the
# convex hull, then the point lies inside
# the convex hull
for i in range( n ):
j = (i + 1) % n;
x1 = a[i][0]
x2 = a[j][0]
y1 = a[i][1]
y2 = a[j][1]
a1 = y1 - y2;
b1 = x2 - x1;
c1 = x1 * y2 - y1 * x2;
for_mid = a1 * mid[0] + b1 * mid[1] + c1;
for_p = a1 * p[0] + b1*p[1]+c1;
if (for_mid*for_p < 0):
return False;
return True;
# Adds a point p to given convex hull a[]
def addPoint( a, p):
# If point is inside p
arr= copy.deepcopy(a)
prr =p.copy()
if (inside(arr, prr)):
return;
# point having minimum distance from the point p
ind = 0;
n = len(a)
for i in range(1, n):
if (sqDist(p, a[i]) < sqDist(p, a[ind])):
ind = i
# Find the upper tangent
up = ind;
while (orientation(p, a[up], a[(up + 1) % n]) >= 0):
up = (up + 1) % n;
# Find the lower tangent
low = ind;
while (orientation(p, a[low], a[(n + low - 1) % n]) <= 0):
low = (n + low - 1) % n
# Initialize result
ret = []
# making the final hull by traversing points
# from up to low of given convex hull.
curr = up;
ret.append(a[curr]);
while (curr != low):
curr = (curr + 1) % n;
ret.append(a[curr]);
# Modify the original vector
ret.append(p);
a.clear();
for i in range(len(ret)):
a.append(ret[i]);
# Driver code
if __name__ == "__main__":
# the set of points in the convex hull
a = []
a.append([0, 0]);
a.append([3, -1]);
a.append([4, 5]);
a.append([-1, 4]);
n = len(a)
p = [100, 100]
addPoint(a, p);
# Print the modified Convex Hull
for e in a :
print("(" , e[0], ", ",
e[1] , ") ",end=" ")
# This code is contributed by chitranayalC#
// C# program to add given a point p to a given
// convext hull. The program assumes that the
// point of given convext hull are in anti-clockwise
// order.
using System;
using System.Collections.Generic;
public class GFG{
// checks whether the point crosses the convex hull
// or not
static int orientation(List a,List b,List c)
{
int res=(b[1]-a[1]) * (c[0]-b[0]) - (c[1]-b[1]) * (b[0]-a[0]);
if (res == 0)
return 0;
if (res > 0)
return 1;
return -1;
}
// Returns the square of distance between two input points
static int sqDist(Listp1, Listp2)
{
return (p1[0] - p2[0]) * (p1[0] - p2[0]) +
(p1[1] - p2[1]) * (p1[1] - p2[1]);
}
// Checks whether the point is inside the convex hull or not
static bool inside(List> A,Listp)
{
// Initialize the centroid of the convex hull
List mid = new List(){0,0};
int n = A.Count;
for (int i = 0; i < n; i++)
{
mid[0]+=A[i][0];
mid[1]+=A[i][1];
}
// if the mid and the given point lies always
// on the same side w.r.t every edge of the
// convex hull, then the point lies inside
// the convex hull
for (int i = 0, j; i < n; i++)
{
j = (i + 1) % n;
int x1 = A[i][0]*n, x2 = A[j][0]*n;
int y1 = A[i][1]*n, y2 = A[j][1]*n;
int a1 = y1 - y2;
int b1 = x2 - x1;
int c1 = x1 * y2 - y1 * x2;
int for_mid = a1 * mid[0] + b1 * mid[1] + c1;
int for_p = a1 * p[0] * n + b1 * p[1] * n + c1;
if (for_mid*for_p < 0)
return false;
}
return true;
}
// Adds a point p to given convex hull a[]
static void addPoint(List> a,List p)
{
// If point is inside p
if (inside(a, p))
return;
// point having minimum distance from the point p
int ind = 0;
int n = a.Count;
for (int i = 1; i < n; i++)
{
if (sqDist(p, a[i]) < sqDist(p, a[ind]))
{
ind = i;
}
}
// Find the upper tangent
int up = ind;
while (orientation(p, a[up], a[(up+1)%n])>=0)
up = (up + 1) % n;
// Find the lower tangent
int low = ind;
while (orientation(p, a[low], a[(n+low-1)%n])<=0)
low = (n+low - 1) % n;
// Initialize result
List> ret = new List>();
// making the final hull by traversing points
// from up to low of given convex hull.
int curr = up;
ret.Add(a[curr]);
while (curr != low)
{
curr = (curr + 1) % n;
ret.Add(a[curr]);
}
// Modify the original vector
ret.Add(p);
a.Clear();
for (int i = 0; i < ret.Count; i++)
{
a.Add(ret[i]);
}
}
// Driver code
static public void Main (){
// the set of points in the convex hull
List> a = new List>();
a.Add(new List(){0,0});
a.Add(new List(){3,-1});
a.Add(new List(){4,5});
a.Add(new List(){-1,4});
int n=a.Count;
List p = new List(){100,100};
addPoint(a, p);
// Print the modified Convex Hull
foreach(List e in a)
{
Console.Write("(" + e[0] + ", " + e[1] + ") ");
}
}
}
// This code is contributed by avanitrachhadiya2155 输出：
(-1, 4) (0, 0) (3, -1) (100, 100)