Z-Buffer Algorithm

The basic idea is to test the z - depth of each surface to determine the closest (visible) surface. Declare an array z buffer(x, y) with one entry for each pixel position. Initialize the array to the maximum depth. Note: if have performed a perspective depth transformation then all z values 0.0 <= z (x, y) <="1.0". So initialize all values to 1.0. Then the algorithm is as follows:

for each polygon P
  for each pixel (x, y) in P 
    compute z_depth at x, y
    if z_depth < z_buffer (x, y) then 
       set_pixel (x, y, color) 
       z_buffer (x, y) <= z_depth 

The polyscan procedure can easily be modified to add the z-buffer test. The computation of the z_depth (x, y) is done using coherence calculations similar to the x-intersection calculations. It is actually a bi-linear interpolation, i.e., interpolation both down (y) and across (x) scan lines.

Advantages of z-buffer algorithm: It always works and is simple to implement.

Disadvantages:

Alternative method for computing z depth values using plane equations.

  1. Perform a perspective depth transformation (to maintain planes).
  2. Compute plane equations (not the same as before the perspective transformation)
  3. For each pixel in PDC (xp, yp) find the x , y wdc values (using an inverse point transform)
  4. Put x, y into plane equation to find z in wdc
  5. Perform z buffer test on z.

Now remember from 2D viewing transformation in the procedure Point Viewing Transform we had:

Sx (VTScaleX) Sy(VTScaleY)
Cx (VTConstX) Cy (VTConstY)

all of which are functions of the window, viewport, and PDC.

Now in Point Viewing Transform we did:

xp <-- round(x* Sx + Cx)
yp <-- round(y * Sy + Cy)
 
or x = (xp - Cx) / Sx = xp/Sx - Cx/ Sx we can compute Cx/Sx, Cy/Sy once and
y = (yp - Cy) / Sy = yp/Sy - Cy/Sy call them Ax, Ay

Now as we scan across a polygon Xp+1 = Xp + 1

So X w = xp/Sx - Ax and Xw+1 = Xp+1/Sx - Ax = Xp/Sx + 1/Sx - Ax = Xw + 1/Sx

Similarly if Yp+1 = Yp + 1 then Yw+1 = Yw + 1/Sy

Now from the plane equations Z(Xw, Yw) = (- A*Xw - B*Yw + D) / C

So Z(Xw+1, Yw) = (-A * (X w + 1/Sx) - B*Yw + D) / C

= Z(Xw, Yw) - (A * (1/Sx)) / C

Similarly Z(Xw, Yw+1) = Z*Xw, Yw) - (B*(1/Sy)) / C

So for each polygon compute the terms (A * (1/Sx)) / C and (B*(1/Sy)) / C

So can find Xw, Yw, Zw at the polygon vertices and use the above to compute rest of Zw values.

Visible Surface Determination
HyperGraph home page.

Last changed May 13, 1998, G. Scott Owen, owen@siggraph.org