The Camera - Essentials of Interactive Computer Graphics - page 387

In this way,

W dc W dc

2

W dc
2

x dc = x ndc ×

2 +

=( x ndc + 1 ) × ,

H d 2 + H dc

2

H dc
2

y dc = y ndc × =( y ndc + 1 ) × .

2. Resolve visibility/occlusion along the NDC z -axis. To determine which ob-
ject is visible on the drawing area, their relative front-to-back ordering must
be resolved. For example, from Figure 14.21, we can see that the teapot oc-
cludes part of the floor. The graphics API uses the Z-buffer hardware to
record the z -value for each pixel. When the Z-buffer depth stenciling is
enabled, before a color is assigned to a pixel, the corresponding z -value is
compared. A pixel's color will be modified only when the corresponding
primitive's z -value is closer to the origin.

The NDC-to-DC transformation described here is the final stage of both the or-
thographic projection (of Figure 14.8 ) and the perspective projection (of Fig-
ure 14.12 ) . The difference between these two projections is in the EC-to-NDC
transform. Orthographic projection proportionally scales all vertices in the EC
space into the NDC cube. As discussed in Section 14.6, perspective projection
scales vertices in EC space disproportionately according to their z -distance from
the origin. In both cases, the NDC-to-DC transformations are identical.

14.8 Re-Examining Tutorial 13.1

Recall that Tutorial 13.1 is our fist encounter with a 3D program (presented in the
previous chapter). We are now finally ready to examine Step 2 of Listing 13.2
where we programmed the matrix processors. Listing 14.4 shows the details of
Step 2 from Listing 13.2. We see that the matrix processors are set up to transform
vertices from object coordinate (OC) space to the NDC:

Label A Label B Label C

C M W C M V C M P O −→ W −→ E −→ NDC .

Because they are set up to transform coordinates, we will refer to the contents of
the WORLD , VIEW ,and PROJECTION matrix processors as M o 2 w (content of the top
of matrix stack), M w 2 e (Equation (14.4) on p. 401), and M e 2 n , respectively. In
this way, with the input vertex in OC space V oc , the output from the WORLD matrix
processor V wc is a vertex position in WC space:

= =

V oc M o 2 w .

V wc V oc M W