Bresenham's 3-D Line Drawing Algorithm

Bresenham's 3-D Line Drawing Algorithm is an extension of Bresenham's algorithm for drawing lines in 2-D space. It is used to draw lines in 3-D space by taking into account the depth or Z-axis.

Algorithm

The algorithm works by dividing the given line into small steps and then using Bresenham's algorithm to draw each step. The Z-axis is taken into consideration by modifying the decision parameter based on the relative distance between two consecutive points.

Let's assume that we want to draw a line from point (x0, y0, z0) to point (x1, y1, z1). We can start by calculating the differences between the two points in each axis:

dx = x1 - x0 dy = y1 - y0 dz = z1 - z0

Next, we need to determine the axis with the largest difference. This is necessary because we need to step along the largest axis to ensure minimum distortion of the line. We can determine this as:

if (abs(dx) > abs(dy) && abs(dx) > abs(dz)) largest = dx else if (abs(dy) > abs(dx) && abs(dy) > abs(dz)) largest = dy else largest = dz

We then calculate the steps to take in each direction for the given line:

x_step = dx / largest y_step = dy / largest z_step = dz / largest

Finally, we use Bresenham's algorithm to draw the line. The decision parameter is modified based on the Z-axis distance between two consecutive pixels:

//Bresenham's algorithm with Z-axis modification x = x0 y = y0 z = z0 p = 2 * abs(largest) - abs(dx) for i from 0 to abs(largest) plot(x, y, z) //draw the pixel if (p < 0) p += 2 * abs(dy) x += x_step y += y_step else p += 2 * abs(dz) - 2 * abs(dy) x += x_step y += y_step z += z_step

Code

def draw_line_3d(x0, y0, z0, x1, y1, z1):
    dx = x1 - x0
    dy = y1 - y0
    dz = z1 - z0
    if (abs(dx) > abs(dy) && abs(dx) > abs(dz)):
        largest = dx
    elif (abs(dy) > abs(dx) && abs(dy) > abs(dz)):
        largest = dy
    else:
        largest = dz
    x_step = dx / largest
    y_step = dy / largest
    z_step = dz / largest
    x = x0
    y = y0
    z = z0
    p = 2 * abs(largest) - abs(dx)
    for i in range(abs(largest)):
        plot(x, y, z) #draw the pixel
        if (p < 0):
            p += 2 * abs(dy)
            x += x_step
            y += y_step
        else:
            p += 2 * abs(dz) - 2 * abs(dy)
            x += x_step
            y += y_step
            z += z_step

Conclusion

Bresenham's 3-D Line Drawing Algorithm is a useful algorithm for drawing lines in 3-D space, taking into account the Z-axis. It is based on Bresenham's algorithm for drawing lines in 2-D space and extends it to 3-D. The algorithm involves dividing the given line into small steps and drawing each step using Bresenham's algorithm modified to account for the Z-axis. The resulting code is efficient and produces nice-looking lines, making it a great choice for 3-D graphics programming.