# Minimum distance from a point to the line segment using Vectors

Given the coordinates of two endpoints A(x1, y1), B(x2, y2) of the line segment and coordinates of a point E(x, y); the task is to find the minimum distance from the point to line segment formed with the given coordinates.

Note that both the ends of a line can go to infinity i.e. a line has no ending points. On the other hand, a line segment has start and end points due to which length of the line segment is fixed.

Examples:

Input: A = {0, 0}, B = {2, 0}, E = {4, 0}
Output: 2
To find the distance, dot product has to be found between vectors AB, BE and AB, AE.
AB = (x2 – x1, y2 – y1) = (2 – 0, 0 – 0) = (2, 0)
BE = (x – x2, y – y2) = (4 – 2, 0 – 0) = (2, 0)
AE = (x – x1, y – y1) = (4 – 0, 0 – 0) = (4, 0)
AB . BE = (ABx * BEx + ABy * BEy) = (2 * 2 + 0 * 0) = 4
AB . AE = (ABx * AEx + ABy * AEy) = (2 * 4 + 0 * 0) = 8
Therefore, nearest point from E to line segment is point B.
Minimum Distance = BE = = 2

Input: A = {0, 0}, B = {2, 0}, E = {1, 1}
Output: 1

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: The idea is to use the concept of vectors to solve the problem since the nearest point always lies on the line segment. Assuming that the direction of vector AB is A to B, there are three cases that arise:

1. The nearest point from the point E on the line segment AB is point B itself, if the dot product of vector AB(A to B) and vector BE(B to E) is positive where E is the given point. Since AB . BE > 0, the given point lies in the same direction as the vector AB is and the nearest point must be B itself because the nearest point lies on the line segment. 2. The nearest point from the point E on the line segment AB is point A itself, if the dot product of vector AB(A to B) and vector BE(B to E) is negative where E is the given point. Since AB . BE < 0, the given point lies on the opposite direction of the line segment AB and the nearest point must be A itself because the nearest point lies on the line segment. 3. If the dot product is 0, then the point E is perpendicular to the line segment AB and the perpendicular distance to the given point E from the line segment AB is the shortest distance. If some arbitrary point F is the point on the line segment which is perpendicular to E, then the perpendicular distance can be calculated as |EF| = |(AB X AE)/|AB|| Below is the implementation of the above approach:

## C++

 // C++ implementation of the approach  #include     // To store the point  #define Point pair  #define F first  #define S second  using namespace std;     // Function to return the minimum distance  // between a line segment AB and a point E  double minDistance(Point A, Point B, Point E)  {         // vector AB      pair<double, double> AB;      AB.F = B.F - A.F;      AB.S = B.S - A.S;         // vector BP      pair<double, double> BE;      BE.F = E.F - B.F;      BE.S = E.S - B.S;         // vector AP      pair<double, double> AE;      AE.F = E.F - A.F,      AE.S = E.S - A.S;         // Variables to store dot product      double AB_BE, AB_AE;         // Calculating the dot product      AB_BE = (AB.F * BE.F + AB.S * BE.S);      AB_AE = (AB.F * AE.F + AB.S * AE.S);         // Minimum distance from      // point E to the line segment      double reqAns = 0;         // Case 1      if (AB_BE > 0) {             // Finding the magnitude          double y = E.S - B.S;          double x = E.F - B.F;          reqAns = sqrt(x * x + y * y);      }         // Case 2      else if (AB_AE < 0) {          double y = E.S - A.S;          double x = E.F - A.F;          reqAns = sqrt(x * x + y * y);      }         // Case 3      else {             // Finding the perpendicular distance          double x1 = AB.F;          double y1 = AB.S;          double x2 = AE.F;          double y2 = AE.S;          double mod = sqrt(x1 * x1 + y1 * y1);          reqAns = abs(x1 * y2 - y1 * x2) / mod;      }      return reqAns;  }     // Driver code  int main()  {      Point A = make_pair(0, 0);      Point B = make_pair(2, 0);      Point E = make_pair(1, 1);         cout << minDistance(A, B, E);         return 0;  }

## Java

 // Java implementation of the approach  class GFG  {     static class pair  {       double F, S;       public pair(double F, double S)       {           this.F = F;           this.S = S;       }      public pair() {      }   }     // Function to return the minimum distance  // between a line segment AB and a point E  static double minDistance(pair A, pair B, pair E)  {         // vector AB      pair AB = new pair();      AB.F = B.F - A.F;      AB.S = B.S - A.S;         // vector BP      pair BE = new pair();      BE.F = E.F - B.F;      BE.S = E.S - B.S;         // vector AP      pair AE = new pair();      AE.F = E.F - A.F;      AE.S = E.S - A.S;         // Variables to store dot product      double AB_BE, AB_AE;         // Calculating the dot product      AB_BE = (AB.F * BE.F + AB.S * BE.S);      AB_AE = (AB.F * AE.F + AB.S * AE.S);         // Minimum distance from      // point E to the line segment      double reqAns = 0;         // Case 1      if (AB_BE > 0)       {             // Finding the magnitude          double y = E.S - B.S;          double x = E.F - B.F;          reqAns = Math.sqrt(x * x + y * y);      }         // Case 2      else if (AB_AE < 0)      {          double y = E.S - A.S;          double x = E.F - A.F;          reqAns = Math.sqrt(x * x + y * y);      }         // Case 3      else      {             // Finding the perpendicular distance          double x1 = AB.F;          double y1 = AB.S;          double x2 = AE.F;          double y2 = AE.S;          double mod = Math.sqrt(x1 * x1 + y1 * y1);          reqAns = Math.abs(x1 * y2 - y1 * x2) / mod;      }      return reqAns;  }     // Driver code  public static void main(String[] args)  {      pair A = new pair(0, 0);      pair B = new pair(2, 0);      pair E = new pair(1, 1);         System.out.print((int)minDistance(A, B, E));  }  }     // This code is contributed by 29AjayKumar

## Python3

 # Python3 implementation of the approach   from math import sqrt     # Function to return the minimum distance   # between a line segment AB and a point E   def minDistance(A, B, E) :          # vector AB       AB = [None, None];       AB = B - A;       AB = B - A;          # vector BP       BE = [None, None];      BE = E - B;       BE = E - B;          # vector AP       AE = [None, None];      AE = E - A;      AE = E - A;          # Variables to store dot product          # Calculating the dot product       AB_BE = AB * BE + AB * BE;       AB_AE = AB * AE + AB * AE;          # Minimum distance from       # point E to the line segment       reqAns = 0;          # Case 1       if (AB_BE > 0) :             # Finding the magnitude           y = E - B;           x = E - B;           reqAns = sqrt(x * x + y * y);          # Case 2       elif (AB_AE < 0) :          y = E - A;           x = E - A;           reqAns = sqrt(x * x + y * y);          # Case 3       else:             # Finding the perpendicular distance           x1 = AB;           y1 = AB;           x2 = AE;           y2 = AE;           mod = sqrt(x1 * x1 + y1 * y1);           reqAns = abs(x1 * y2 - y1 * x2) / mod;              return reqAns;      # Driver code   if __name__ == "__main__" :          A = [0, 0];       B = [2, 0];       E = [1, 1];          print(minDistance(A, B, E));      # This code is contributed by AnkitRai01

## C#

 // C# implementation of the approach  using System;     class GFG  {     class pair  {       public double F, S;       public pair(double F, double S)       {           this.F = F;           this.S = S;       }      public pair() {      }   }     // Function to return the minimum distance  // between a line segment AB and a point E  static double minDistance(pair A, pair B, pair E)  {         // vector AB      pair AB = new pair();      AB.F = B.F - A.F;      AB.S = B.S - A.S;         // vector BP      pair BE = new pair();      BE.F = E.F - B.F;      BE.S = E.S - B.S;         // vector AP      pair AE = new pair();      AE.F = E.F - A.F;      AE.S = E.S - A.S;         // Variables to store dot product      double AB_BE, AB_AE;         // Calculating the dot product      AB_BE = (AB.F * BE.F + AB.S * BE.S);      AB_AE = (AB.F * AE.F + AB.S * AE.S);         // Minimum distance from      // point E to the line segment      double reqAns = 0;         // Case 1      if (AB_BE > 0)       {             // Finding the magnitude          double y = E.S - B.S;          double x = E.F - B.F;          reqAns = Math.Sqrt(x * x + y * y);      }         // Case 2      else if (AB_AE < 0)      {          double y = E.S - A.S;          double x = E.F - A.F;          reqAns = Math.Sqrt(x * x + y * y);      }         // Case 3      else     {             // Finding the perpendicular distance          double x1 = AB.F;          double y1 = AB.S;          double x2 = AE.F;          double y2 = AE.S;          double mod = Math.Sqrt(x1 * x1 + y1 * y1);          reqAns = Math.Abs(x1 * y2 - y1 * x2) / mod;      }      return reqAns;  }     // Driver code  public static void Main(String[] args)  {      pair A = new pair(0, 0);      pair B = new pair(2, 0);      pair E = new pair(1, 1);         Console.Write((int)minDistance(A, B, E));  }  }     // This code is contributed by 29AjayKumar

Output:

1


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