Page 101 - Computer Graphics Handout

In the same frame, any vector w can be written as
w = δ1v1+ δ2v2+ δ3v3

Thus, w can be represented by the column matrix

There are numerous ways to interpret this formulation geometrically. We simply note that we can carry out operations on points
and vectors using their homogeneous-coordinate representations and ordinary matrix algebra. Consider, for example, a change of
frames—a problem that caused difficulties when we used threedimensional representations. If (v1, v2, v3, P0) and (u1, u2, u3, Q0) are
two frames, then we can express the basis vectors and reference point of the second frame in terms of the first as
u1= γ11v1+ γ12v2+ γ13v3,
u2= γ21v1+ γ22v2+ γ23v3,
u3= γ31v1+ γ32v2+ γ33v3,
Q0= γ41v1+ γ42v2+ γ43v3+ P0.
These equations can be written in the form

where now M is the 4 × 4 matrix

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