Page 98 - Computer Graphics Handout

Where

Then, using our representation of the second basis in terms of the first, we find that

Thus,
a = MTb.
The matrix
T = (MT)−1
takes us from a to b, through the simple matrix equation
b = Ta.
Thus, rather than working with our original vectors, typically directed line segments, we can work
instead with their representations, which are 3-tuples or elements of R3. This result is important
because it moves us from considering abstract vectors to working with column matrices of
scalars—the vectors’ representations. The important point to remember is that whenever we
work with columns of real numbers as “vectors,” there is an underlying basis of which we must
not lose track, lest we end up working in the wrong coordinate system.
These changes in basis leave the origin unchanged. We can use them to represent rotation and
scaling of a set of basis vectors to derive another basis set, as shown in Figure 3.24. However, a
simple translation of the origin, or change of frame as shown in Figure 3.25, cannot be
represented in this way. After we complete a simple example, we introduce homogeneous
coordinates, which allow us to change frames yet still use matrices to represent the change.

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