Figure 3.14 shows a virtual eye that is looking down the negative axis. It is placed in this way so that from the eye's perspective, increases to the right and is upward. This corresponds to familiar Cartesian coordinates. The alternatives would be: 1) to face the eye in the positive direction, which makes the coordinates appear backwards, or 2) reverse the axis, which would unfortunately lead to a left-handed coordinate system. Thus, we have made an odd choice that avoids worse complications.
Suppose that the eye is an object model that we want to place into the virtual world at some position and orientation given by the matrix
This does not, however, solve the problem of how the virtual world should appear to the eye. Rather than moving the eye in the virtual world, we need to move all of the models in the virtual world to the eye's frame of reference. This means that we need to apply the inverse transformation. The inverse rotation is , the transpose of . The inverse of is . Applying (3.26) results in the appropriate transform:
Following Figure 3.4, there are two possible interpretations of (3.36). As stated, this could correspond to moving all of the virtual world models (corresponding to Figure 3.4(b)). A more appropriate interpretation in the current setting is that the virtual world's coordinate frame is being moved so that it matches the eye's frame from Figure 3.14. This corresponds to the case of Figure 3.4(c), which was not the appropriate interpretation in Section 3.2.
Steven M LaValle 2020-11-11