More on transformations

[Angel p189; Foley p208-210, 222-226]

A series of transformations can be accumulated into a single transformation matrix.

Suppose our object is defined as follows:

Suppose we wish to place the object in our scene like this:

This can be accomplished in several ways, one of which is:

Substituting (1) in (2) gives:

compositions of transformations are non-commutative
e.g., trans(2,1,0)rot(z,-90) != rot(z,-90)trans(2,1,0)
Transformations can be thought of as a change in coordinate system.
transformations matrices can be multiplied together, giving a single composite transformation matrix
Inverting a transformation matrix produces the 'inverse' of a transformation.
e.g., if M = translate(a,b,c) then M^-1 = translate(-a,-b,-c).
The transformations necessary to convert a point from CS₂ to CS₁ are given by the transformations necessary to align the CS₁ frame with the CS₂ frame.
Premultiplication effects a transformation in 'world' or 'fixed' coordinates.
Postmultiplication effects a transformation in 'local' coodinates.
Open GL postmultiplies transformation, i.e., applies them in 'local' coordinates.