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Bresenham’s Algorithm for 3-D Line Drawing

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Given two 3-D co-ordinates we need to find the points on the line joining them. All points have integer co-ordinates. Examples:

Input  : (-1, 1, 1), (5, 3, -1)
Output : (-1, 1, 1), (0, 1, 1), (1, 2, 0),
          (2, 2, 0), (3, 2, 0), (4, 3, -1), 
          (5, 3, -1)
 
 
 
Input  : (-7, 0, -3), (2, -5, -1)
Output : (-7, 0, -3), (-6, -1, -3), (-5, -1, -3),
         (-4, -2, -2), (-3, -2, -2), (-2, -3, -2),
         (-1, -3, -2), (0, -4, -1), (1, -4, -1),
          (2, -5, -1)

Bresenham’s Algorithm is efficient as it avoids floating point arithmetic operations. As in the case of 2-D Line Drawing, we use a variable to store the slope-error i.e. the error in slope of the line being plotted from the actual geometric line. As soon as this slope-error exceeds the permissible value we modify the digital to negate the error.

The driving axis of the line to be plotted is the one along which the line travels the farthest i.e. the difference in axes co-ordinates is greatest. Thus the co-ordinate values increase linearly by 1 along the driving axis and the slope-error variable is used to determine the change in the co-ordinate values of the other axis.

In case of a 2-D line we use one slope-error variable but in case of a 3-D line we need two (py, pz        ) of them for each of the non-driving axes. If current point is P_{k}        (x, y, z) and the driving axis is the positive X-axis, then the next point P_{k+1}        could be

  • (x+1, y, z)
  • (x+1, y+1, z)
  • (x+1, y, z+1)
  • (x+1, y+1, z+1)

Candidate points for next point on a 3D Line

The value of slope-error variables are determined according to the following equations:- py_{k+1} = py_{k} + 2dy - 2dx(y_{k+1} - y_{k})\\ pz_{k+1} = pz_{k} + 2dz - 2dx(z_{k+1} - z_{k})

The initial value of slope-error variables are given by the following equations:- py_{0} = 2dy - dx\\ pz_{0} = 2dz - dx        Here dx, dy, dz        denote the difference in co-ordinates of the two end points along the X, Y, Z axes.

Algorithm:-

  1. Input the two endpoints and store the initial point as (x_{0}, y_{0}, z_{0})
  2. Plot (x_{0}, y_{0}, z_{0})
  3. Calculate constants dx, dy, dz        and determine the driving axis by comparing the absolute values of dx, dy, dz        If abs(dx        ) is maximum, then X-axis is the driving axis If abs(dy        ) is maximum, then Y-axis is the driving axis If abs(dz        ) is maximum, then Z-axis is the driving axis
  4. Let’s suppose that X-axis is the driving axis, then py_{0} = 2dy - dx\\ pz_{0} = 2dz - dx
  5. At each x_{k}        along the line, starting at k = 0, check the following conditions and determine the next point:-
    • If py_{k} < 0        AND pz_{k} < 0        , then plot (x_{k}+1, y_{k}, z_{k})        and set py_{k+1}=py_{k}+2dy, pz_{k+1}=pz_{k}+2dz
    • Else If py_{k} > 0        AND pz_{k} < 0        , then plot (x_{k}+1, y_{k}+1, z_{k})        and set py_{k+1}=py_{k}+2dy-2dx, pz_{k+1}=pz_{k}+2dz
    • Else If py_{k} 0        , then plot (x_{k}+1, y_{k}, z_{k}+1)        and set py_{k+1}=py_{k}+2dy, pz_{k+1}=pz_{k}+2dz-2dx
    • Else then plot (x_{k}+1, y_{k}+1, z_{k}+1)        and set py_{k+1}=py_{k}+2dy-2dx, pz_{k+1}=pz_{k}+2dz-2dx        >
  6. Repeat step 5 dx-1        times

C++

#include <cmath>
#include <iostream>
#include <vector>
using namespace std;
 
vector<vector<int> > Bresenham3D(int x1, int y1, int z1,
                                 int x2, int y2, int z2)
{
  vector<vector<int> > ListOfPoints;
  ListOfPoints.push_back({ x1, y1, z1 });
  int dx = abs(x2 - x1);
  int dy = abs(y2 - y1);
  int dz = abs(z2 - z1);
  int xs;
  int ys;
  int zs;
  if (x2 > x1)
    xs = 1;
  else
    xs = -1;
  if (y2 > y1)
    ys = 1;
  else
    ys = -1;
  if (z2 > z1)
    zs = 1;
  else
    zs = -1;
 
  // Driving axis is X-axis"
  if (dx >= dy && dx >= dz) {
    int p1 = 2 * dy - dx;
    int p2 = 2 * dz - dx;
    while (x1 != x2) {
      x1 += xs;
      if (p1 >= 0) {
        y1 += ys;
        p1 -= 2 * dx;
      }
      if (p2 >= 0) {
        z1 += zs;
        p2 -= 2 * dx;
      }
      p1 += 2 * dy;
      p2 += 2 * dz;
      ListOfPoints.push_back({ x1, y1, z1 });
    }
 
    // Driving axis is Y-axis"
  }
  else if (dy >= dx && dy >= dz) {
    int p1 = 2 * dx - dy;
    int p2 = 2 * dz - dy;
    while (y1 != y2) {
      y1 += ys;
      if (p1 >= 0) {
        x1 += xs;
        p1 -= 2 * dy;
      }
      if (p2 >= 0) {
        z1 += zs;
        p2 -= 2 * dy;
      }
      p1 += 2 * dx;
      p2 += 2 * dz;
      ListOfPoints.push_back({ x1, y1, z1 });
    }
 
    // Driving axis is Z-axis"
  }
  else {
    int p1 = 2 * dy - dz;
    int p2 = 2 * dx - dz;
    while (z1 != z2) {
      z1 += zs;
      if (p1 >= 0) {
        y1 += ys;
        p1 -= 2 * dz;
      }
      if (p2 >= 0) {
        x1 += xs;
        p2 -= 2 * dz;
      }
      p1 += 2 * dy;
      p2 += 2 * dx;
      ListOfPoints.push_back({ x1, y1, z1 });
    }
  }
  return ListOfPoints;
}
 
int main()
{
  int x1 = -1;
  int y1 = 1;
  int z1 = 1;
  int x2 = 5;
  int y2 = 3;
  int z2 = -1;
  vector<vector<int> > ListOfPoints
    = Bresenham3D(x1, y1, z1, x2, y2, z2);
  cout << "The generated points are:" << endl;
  for (auto it : ListOfPoints)
    cout << it[0] << " " << it[1] << " " << it[2]
    << endl;
  return 0;
}
 
// This code is contributed by ishankhandelwals.

                    

Java

import java.util.ArrayList;
import java.util.List;
 
public class Bresenham3D {
 
  public static List<List<Integer>> Bresenham3D(int x1, int y1, int z1, int x2, int y2, int z2) {
    List<List<Integer>> ListOfPoints = new ArrayList<>();
    ListOfPoints.add(List.of(x1, y1, z1));
    int dx = Math.abs(x2 - x1);
    int dy = Math.abs(y2 - y1);
    int dz = Math.abs(z2 - z1);
    int xs;
    int ys;
    int zs;
    if (x2 > x1) {
      xs = 1;
    } else {
      xs = -1;
    }
    if (y2 > y1) {
      ys = 1;
    } else {
      ys = -1;
    }
    if (z2 > z1) {
      zs = 1;
    } else {
      zs = -1;
    }
 
    // Driving axis is X-axis"
    if (dx >= dy && dx >= dz) {
      int p1 = 2 * dy - dx;
      int p2 = 2 * dz - dx;
      while (x1 != x2) {
        x1 += xs;
        if (p1 >= 0) {
          y1 += ys;
          p1 -= 2 * dx;
        }
        if (p2 >= 0) {
          z1 += zs;
          p2 -= 2 * dx;
        }
        p1 += 2 * dy;
        p2 += 2 * dz;
        ListOfPoints.add(List.of(x1, y1, z1));
      }
 
      // Driving axis is Y-axis"
    } else if (dy >= dx && dy >= dz) {
      int p1 = 2 * dx - dy;
      int p2 = 2 * dz - dy;
      while (y1 != y2) {
        y1 += ys;
        if (p1 >= 0) {
          x1 += xs;
          p1 -= 2 * dy;
        }
        if (p2 >= 0) {
          z1 += zs;
          p2 -= 2 * dy;
        }
        p1 += 2 * dx;
        p2 += 2 * dz;
        ListOfPoints.add(List.of(x1, y1, z1));
      }
 
      // Driving axis is Z-axis"
    } else {
      int p1 = 2 * dy - dz;
      int p2 = 2 * dx - dz;
      while (z1 != z2) {
        z1 += zs;
        if (p1 >= 0) {
          y1 += ys;
          p1 -= 2 * dz;
        }
        if (p2 >= 0) {
          x1 += xs;
          p2 -= 2 * dz;
        }
        p1 += 2 * dy;
        p2 += 2 * dx;
        ListOfPoints.add(List.of(x1, y1, z1));
      }
    }
    return ListOfPoints;
  }
 
  public static void main(String[] args) {
    int x1 = -1;
    int y1 = 1;
    int z1 = 1;
    int x2 = 5;
    int y2 = 3;
    int z2 = -1;
    List<List<Integer>> ListOfPoints = Bresenham3D(x1, y1, z1, x2, y2, z2);
    System.out.println("The generated points are:");
    for (List<Integer> it : ListOfPoints) {
      System.out.println(it.get(0) + " " + it.get(1) + " " + it.get(2));
    }
  }
}
 
// This code is contributed by ishankhandelwals.

                    

Python3

# Python3 code for generating points on a 3-D line
# using Bresenham's Algorithm
 
def Bresenham3D(x1, y1, z1, x2, y2, z2):
    ListOfPoints = []
    ListOfPoints.append((x1, y1, z1))
    dx = abs(x2 - x1)
    dy = abs(y2 - y1)
    dz = abs(z2 - z1)
    if (x2 > x1):
        xs = 1
    else:
        xs = -1
    if (y2 > y1):
        ys = 1
    else:
        ys = -1
    if (z2 > z1):
        zs = 1
    else:
        zs = -1
 
    # Driving axis is X-axis"
    if (dx >= dy and dx >= dz):       
        p1 = 2 * dy - dx
        p2 = 2 * dz - dx
        while (x1 != x2):
            x1 += xs
            if (p1 >= 0):
                y1 += ys
                p1 -= 2 * dx
            if (p2 >= 0):
                z1 += zs
                p2 -= 2 * dx
            p1 += 2 * dy
            p2 += 2 * dz
            ListOfPoints.append((x1, y1, z1))
 
    # Driving axis is Y-axis"
    elif (dy >= dx and dy >= dz):      
        p1 = 2 * dx - dy
        p2 = 2 * dz - dy
        while (y1 != y2):
            y1 += ys
            if (p1 >= 0):
                x1 += xs
                p1 -= 2 * dy
            if (p2 >= 0):
                z1 += zs
                p2 -= 2 * dy
            p1 += 2 * dx
            p2 += 2 * dz
            ListOfPoints.append((x1, y1, z1))
 
    # Driving axis is Z-axis"
    else:       
        p1 = 2 * dy - dz
        p2 = 2 * dx - dz
        while (z1 != z2):
            z1 += zs
            if (p1 >= 0):
                y1 += ys
                p1 -= 2 * dz
            if (p2 >= 0):
                x1 += xs
                p2 -= 2 * dz
            p1 += 2 * dy
            p2 += 2 * dx
            ListOfPoints.append((x1, y1, z1))
    return ListOfPoints
 
 
def main():
    (x1, y1, z1) = (-1, 1, 1)
    (x2, y2, z2) = (5, 3, -1)
    ListOfPoints = Bresenham3D(x1, y1, z1, x2, y2, z2)
    print(ListOfPoints)
 
main()

                    

Javascript

// JS code for generating points on a 3-D line
// using Bresenham's Algorithm
function Bresenham3D(x1, y1, z1, x2, y2, z2) {
    let ListOfPoints = [];
    ListOfPoints.push([x1, y1, z1]);
    let dx = Math.abs(x2 - x1);
    let dy = Math.abs(y2 - y1);
    let dz = Math.abs(z2 - z1);
    let xs;
    let ys;
    let zs;
    if (x2 > x1) {
        xs = 1;
    } else {
        xs = -1;
    }
    if (y2 > y1) {
        ys = 1;
    } else {
        ys = -1;
    }
    if (z2 > z1) {
        zs = 1;
    } else {
        zs = -1;
    }
 
    // Driving axis is X-axis"
    if (dx >= dy && dx >= dz) {
        let p1 = 2 * dy - dx;
        let p2 = 2 * dz - dx;
        while (x1 != x2) {
            x1 += xs;
            if (p1 >= 0) {
                y1 += ys;
                p1 -= 2 * dx;
            }
            if (p2 >= 0) {
                z1 += zs;
                p2 -= 2 * dx;
            }
            p1 += 2 * dy;
            p2 += 2 * dz;
            ListOfPoints.push([x1, y1, z1]);
        }
 
        // Driving axis is Y-axis"
    } else if (dy >= dx && dy >= dz) {
        let p1 = 2 * dx - dy;
        let p2 = 2 * dz - dy;
        while (y1 != y2) {
            y1 += ys;
            if (p1 >= 0) {
                x1 += xs;
                p1 -= 2 * dy;
            }
            if (p2 >= 0) {
                z1 += zs;
                p2 -= 2 * dy;
            }
            p1 += 2 * dx;
            p2 += 2 * dz;
            ListOfPoints.push([x1, y1, z1]);
        }
 
        // Driving axis is Z-axis"
    } else {
        let p1 = 2 * dy - dz;
        let p2 = 2 * dx - dz;
        while (z1 != z2) {
            z1 += zs;
            if (p1 >= 0) {
                y1 += ys;
                p1 -= 2 * dz;
            }
            if (p2 >= 0) {
                x1 += xs;
                p2 -= 2 * dz;
            }
            p1 += 2 * dy;
            p2 += 2 * dx;
            ListOfPoints.push([x1, y1, z1]);
        }
    }
    return ListOfPoints;
}
 
function main() {
    let x1 = -1;
    let y1 = 1;
    let z1 = 1;
    let x2 = 5;
    let y2 = 3;
    let z2 = -1;
    let ListOfPoints = Bresenham3D(x1, y1, z1, x2, y2, z2);
    console.log(ListOfPoints);
}
 
main();
 
// This code is contributed by ishankhandelwals.

                    

C#

using System;
using System.Collections.Generic;
 
namespace Bresenham3D
{
    public class Program
    {
        public static List<List<int>> Bresenham3D(int x1, int y1, int z1, int x2, int y2, int z2)
        {
            List<List<int>> ListOfPoints = new List<List<int>>();
            ListOfPoints.Add(new List<int> { x1, y1, z1 });
            int dx = Math.Abs(x2 - x1);
            int dy = Math.Abs(y2 - y1);
            int dz = Math.Abs(z2 - z1);
            int xs;
            int ys;
            int zs;
            if (x2 > x1)
                xs = 1;
            else
                xs = -1;
            if (y2 > y1)
                ys = 1;
            else
                ys = -1;
            if (z2 > z1)
                zs = 1;
            else
                zs = -1;
 
            // Driving axis is X-axis"
            if (dx >= dy && dx >= dz)
            {
                int p1 = 2 * dy - dx;
                int p2 = 2 * dz - dx;
                while (x1 != x2)
                {
                    x1 += xs;
                    if (p1 >= 0)
                    {
                        y1 += ys;
                        p1 -= 2 * dx;
                    }
                    if (p2 >= 0)
                    {
                        z1 += zs;
                        p2 -= 2 * dx;
                    }
                    p1 += 2 * dy;
                    p2 += 2 * dz;
                    ListOfPoints.Add(new List<int> { x1, y1, z1 });
                }
 
                // Driving axis is Y-axis"
            }
            else if (dy >= dx && dy >= dz)
            {
                int p1 = 2 * dx - dy;
                int p2 = 2 * dz - dy;
                while (y1 != y2)
                {
                    y1 += ys;
                    if (p1 >= 0)
                    {
                        x1 += xs;
                        p1 -= 2 * dy;
                    }
                    if (p2 >= 0)
                    {
                        z1 += zs;
                        p2 -= 2 * dy;
                    }
                    p1 += 2 * dx;
                    p2 += 2 * dz;
                    ListOfPoints.Add(new List<int> { x1, y1, z1 });
                }
 
                // Driving axis is Z-axis"
            }
            else
            {
                int p1 = 2 * dy - dz;
                int p2 = 2 * dx - dz;
                while (z1 != z2)
                {
                    z1 += zs;
                    if (p1 >= 0)
                    {
                        y1 += ys;
                        p1 -= 2 * dz;
                    }
                    if (p2 >= 0)
                    {
                        x1 += xs;
                        p2 -= 2 * dz;
                    }
                    p1 += 2 * dy;
                    p2 += 2 * dx;
                    ListOfPoints.Add(new List<int> { x1, y1, z1 });
                }
            }
            return ListOfPoints;
        }
 
        public static void Main(string[] args)
        {
            int x1 = -1;
            int y1 = 1;
            int z1 = 1;
            int x2 = 5;
            int y2 = 3;
            int z2 = -1;
            List<List<int>> ListOfPoints
                = Bresenham3D(x1, y1, z1, x2, y2, z2);
            Console.WriteLine("The generated points are:");
            foreach (var it in ListOfPoints)
                Console.WriteLine($"{it[0]} {it[1]} {it[2]}");
        }
    }
}

                    
Output:
[(-1, 1, 1), (0, 1, 1), (1, 2, 0), (2, 2, 0), (3, 2, 0), (4, 3, -1), (5, 3, -1)]


Last Updated : 20 Feb, 2023
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