Transformations in Two Dimensions - Computer Graphics: Principles and Practice - page 235

to get a new point, and more generally, affine combinations of points, that is,
combinations of the form

α 1 P 1 + α 2 P 2 + ... + α k P k , (10.61)

were allowed if and only if α k = 1.

We now have a situation in which these distinctions make sense in terms of
familiar mathematics: We can regard points of the plane as being elements of
R 3 whose third coordinate is 1, and vectors as being elements of R 3 whose third
coordinate is 0.
With this convention, it's clear that the difference of points is a vector, the sum
of a vector and a point is a point, and combinations like the one in Equation 10.61
yield a point if and only if the sum of the coefficients is 1 (because the third
coordinate of the result will be exactly the sum of the coefficients; for the sum to
be a point, this third coordinate is required to be 1).
You may ask, “Why, when we're already familiar with vectors in 3-space,
should we bother calling some of them 'points in the Euclidean plane' and others
'two-dimensional vectors'?” The answer is that the distinctions have geometric
significance when we're using this subset of 3-space as a model for 2D transfor-
mations. Adding vectors in 3-space is defined in linear algebra, but adding together
two of our “points” gives a location in 3-space that's not on the w = 1 plane or
the w = 0 plane, so we don't have a name for it at all.
Henceforth we'll use E 2 (for “Euclidean two-dimensional space”) to denote
this w = 1 plane in xyw -space, and we'll write ( x , y ) to mean the point of E 2

α 1 + α 2 + ... +

⎡
⎣

⎤

x
y
1

⎦ . It's conventional to speak of an affine

corresponding to the 3-space vector

transformation as acting on E 2 , even though it's defined by a 3 × 3 matrix.

10.6 Why Use 3 3 Matrices Instead

of a Matrix and a Vector?

×

Students sometimes wonder why they can't just represent a linear transformation
plus translation in the form

T ( x )= Mx + b , (10.62)

where the matrix M represents the linear part (rotating, scaling, and shearing) and
b represents the translation.
First, you can do that, and it works just fine. You might save a tiny bit of
storage (four numbers for the matrix and two for the vector, so six numbers instead
of nine), but since our matrices always have two 0s and a 1 in the third column, we
don't really need to store that column anyhow, so it's the same. Otherwise, there's
no important difference.
Second, the reason to unify the transformations into a single matrix is that it's
then very easy to take multiple transformations (each represented by a matrix)
and compose them (perform one after another): We just multiply their matrices
together in the right order to get the matrix for the composed transformation. You
can do this in the matrix-and-vector formulation as well, but the programming is
slightly messier and more error-prone.