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.

• May paint the same pixel several times and computing the color of a pixel may be expensive. So might compute the color only if it passes the z_buffer test. Might sort the polygons and scan front to back (reverse of painter's algorithm) - this still tests all the polygons but avoids the expense of computing the intensity and writing it to the frame buffer.
• Large memory requirements: if used real (4 bytes) then for 640 x 480 resolution: 4bytes/pixel = 1,228,000 bytes. We usually use a 24-bit z-buffer so 900,000 bytes or 16 - bit z-buffer = 614,000 bytes. Note: For VGA mode 19 (320 x 200 and only use 240 x 200) then need only 96,000 bytes for a 16-bit z_buffer. However, may need additional z - buffers for special effects, e.g. shadows.

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.

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