Bresenham's 3-D Line Drawing Algorithm

Bresenham's 3-D Line Drawing Algorithm is an extension of the Bresenham's Line Drawing Algorithm, which is used to draw straight lines in 2-D space. This algorithm is used to draw straight lines in 3-D space by rasterizing them onto a 2-D plane.

Background

In 2-D space, the Bresenham's Line Drawing Algorithm works by incrementing either the x or y coordinate of a pixel, depending on which one is closest to the true line. It is a highly efficient algorithm and is widely used in computer graphics applications.

In 3-D space, the Bresenham's 3-D Line Drawing Algorithm works similarly, but it involves using the slope of the line in 3-D space to determine which pixel to increment. The algorithm incrementally moves along the line by rasterizing it onto a 2-D plane and then incrementing the x or y coordinate of the pixel closest to the true line.

Algorithm

The Bresenham's 3-D Line Drawing Algorithm takes in two points in 3-D space: (x1, y1, z1) and (x2, y2, z2) and outputs a set of pixels that approximate the line connecting the two points.

The algorithm can be implemented as follows:

1. Determine the slope of the line in 3-D space:

   dx = x2 - x1
   dy = y2 - y1
   dz = z2 - z1
   
   if abs(dx) > abs(dy) and abs(dx) > abs(dz):
       max_dist = abs(dx)
   elif abs(dy) > abs(dx) and abs(dy) > abs(dz):
       max_dist = abs(dy)
   else:
       max_dist = abs(dz)
   
   x_inc = dx / max_dist
   y_inc = dy / max_dist
   z_inc = dz / max_dist
   

2. Initialize the starting point as (x1, y1, z1) and the error as 0:

   x = x1
   y = y1
   z = z1
   error = 0
   

3. Incrementally move along the line by rasterizing it onto a 2-D plane and then incrementing the x, y, or z coordinate of the closest pixel:

   for i in range(max_dist):
       # Rasterize the pixel
       put_pixel(x, y, z)
   
       # Increment the appropriate coordinate
       if abs(error + y_inc) < abs(error + x_inc) and abs(error + y_inc) < abs(error + z_inc):
           error += y_inc
           y += 1
       elif abs(error + x_inc) < abs(error + y_inc) and abs(error + x_inc) < abs(error + z_inc):
           error += x_inc
           x += 1
       else:
           error += z_inc
           z += 1
   

Conclusion

Bresenham's 3-D Line Drawing Algorithm is a highly efficient algorithm used to draw straight lines in 3-D space. By rasterizing the line onto a 2-D plane and incrementally moving along it, the algorithm can output a set of pixels that approximate the line connecting two points in 3-D space.