Shortest distance between a Line and a Point in a 3-D plane

Given a line passing through two points A and B and an arbitrary point C in a 3-D plane, the task is to find the shortest distance between the point C and the line passing through the points A and B.

Examples:

Input: A = (5, 2, 1), B = (3, 1, -1), C = (0, 2, 3)
Output: Shortest Distance is 5

Input: A = (4, 2, 1), B = (3, 2, 1), C = (0, 2, 0)
Output: Shortest Distance is 1

Consider a point C and a line that passes through A and B as shown in the below figure.

Now Consider the vectors, AB and AC and the shortest distance as CD. The Shortest Distance is always the perpendicular distance. The point D is taken on AB such that CD is perpendicular to AB.



Construct BP and CP as shown in the figure to form a Parallelogram. Now C is a vertex of parallelogram ABPC and CD is perpendicular to Side AB. Hence CD is the height of the parallelogram.

Note: In the case when D does not fall on line segment AB there will be a point D’ such that PD’ is perpendicular to AB and D’ lies on line segment AB with CD = PD’.

The magnitude of cross product AB and AC gives the Area of the parallelogram. Also, the area of a parallelogram is Base * Height = AB * CD. So,

CD = |ABxAC| / |AB|

Below is the CPP program to find the shortest distance:

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// C++ program to find the Shortest
// Distance Between A line and a
// Given point.
#include<bits/stdc++.h>
using namespace std;
  
class Vector {
private:
    int x, y, z;
    // 3D Coordinates of the Vector
  
public:
    Vector(int x, int y, int z)
    {
        // Constructor
        this->x = x;
        this->y = y;
        this->z = z;
    }
    Vector operator+(Vector v); // ADD 2 Vectors
    Vector operator-(Vector v); // Subtraction
    int operator^(Vector v); // Dot Product
    Vector operator*(Vector v); // Cross Product
    float magnitude()
    {
        return sqrt(pow(x, 2) + pow(y, 2) + pow(z, 2));
    }
    friend ostream& operator<<(ostream& out, const Vector& v);
    // To output the Vector
};
  
// ADD 2 Vectors
Vector Vector::operator+(Vector v)
{
    int x1, y1, z1;
    x1 = x + v.x;
    y1 = y + v.y;
    z1 = z + v.z;
    return Vector(x1, y1, z1);
}
  
// Subtract 2 vectors
Vector Vector::operator-(Vector v)
{
    int x1, y1, z1;
    x1 = x - v.x;
    y1 = y - v.y;
    z1 = z - v.z;
    return Vector(x1, y1, z1);
}
  
// Dot product of 2 vectors
int Vector::operator^(Vector v)
{
    int x1, y1, z1;
    x1 = x * v.x;
    y1 = y * v.y;
    z1 = z * v.z;
    return (x1 + y1 + z1);
}
  
// Cross product of 2 vectors
Vector Vector::operator*(Vector v)
{
    int x1, y1, z1;
    x1 = y * v.z - z * v.y;
    y1 = z * v.x - x * v.z;
    z1 = x * v.y - y * v.x;
    return Vector(x1, y1, z1);
}
  
// Display Vector
ostream& operator<<(ostream& out,
                    const Vector& v)
{
    out << v.x << "i ";
    if (v.y >= 0)
        out << "+ ";
    out << v.y << "j ";
    if (v.z >= 0)
        out << "+ ";
    out << v.z << "k" << endl;
    return out;
}
  
// calculate shortest dist. from point to line
float shortDistance(Vector line_point1, Vector line_point2,
                    Vector point)
{
    Vector AB = line_point2 - line_point1;
    Vector AC = point - line_point1;
    float area = Vector(AB * AC).magnitude();
    float CD = area / AB.magnitude();
    return CD;
}
  
// Driver program
int main()
{
    // Taking point C as (2, 2, 2)
    // Line Passes through A(4, 2, 1)
    // and B(8, 4, 2).
    Vector line_point1(4, 2, 1), line_point2(8, 4, 2);
    Vector point(2, 2, 2);
  
    cout << "Shortest Distance is : "
         << shortDistance(line_point1, line_point2, point);
  
  return 0;
}

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Output:

Shortest Distance is : 1.63299


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