Before the transformations become too complicated, we want to caution you about interpreting them correctly. Figures 3.4(a) and 3.4(b) show an example in which a triangle is translated by $ x_t= -8$ and $ y_t= -7$. The vertex coordinates are the same in Figures 3.4(b) and 3.4(c). Figure 3.4(b) shows the case we are intended to cover so far: The triangle is interpreted as having moved in the virtual world. However, Figure 3.4(c) shows another possibility: The coordinates of the virtual world have been reassigned so that the triangle is closer to the origin. This is equivalent to having moved the entire world, with the triangle being the only part that does not move. In this case, the translation is applied to the coordinate axes, but they are negated. When we apply more general transformations, this extends so that transforming the coordinate axes results in an inverse of the transformation that would correspondingly move the model. Negation is simply the inverse in the case of translation.

Thus, we have a kind of ``relativity'': Did the object move, or did the whole world move around it? This idea will become important in Section 3.4 when we want to change viewpoints. If we were standing at the origin, looking at the triangle, then the result would appear the same in either case; however, if the origin moves, then we would move with it. A deep perceptual problem lies here as well. If we perceive ourselves as having moved, then VR sickness might increase, even though it was the object that moved. In other words, our brains make their best guess as to which type of motion occurred, and sometimes get it wrong.

Steven M LaValle 2020-01-06