Bresenham's Algorithm for 3-D Line Drawing

Bresenham’s Algorithm for 3-D Line Drawing

Last Updated : 15 Jul, 2018

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:-

Input the two endpoints and store the initial point as $(x_{0}, y_{0}, z_{0})$
Plot $(x_{0}, y_{0}, z_{0})$
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
Let’s suppose that X-axis is the driving axis, then
$py_{0} = 2dy - dx\\ pz_{0} = 2dz - dx$
At each 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 $py_{k+1}=py_{k}+2dy-2dx, pz_{k+1}=pz_{k}+2dz-2dx$ >
Repeat step 5 $dx-1$ times

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() 

Output:

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

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