Page 119 - Computer Graphics Handout

it is not possible to find a 3×3 matrixDsuch that q_ = Dq for the given displacement vector d. For this reason, the use of homogeneous
coordinates is often seen as a clever trick that allows us to convert the addition of column matrices in three dimensions to matrix–
matrix multiplication in four dimensions.
We can obtain the inverse of a translation matrix either by applying an inversion algorithm or by noting that if we displace a point
by the vector d, we can return to the original position by a displacement of −d. By either method, we find that

3.9.2 Scaling
For both scaling and rotation, there is a fixed point that is unchanged by the transformation. We let the fixed point be the origin,
and we show how we can concatenate transformations to obtain the transformation for an arbitrary fixed point.
A scaling matrix with a fixed point of the origin allows for independent scaling along the coordinate axes. The three equations are
x_ = βxx,
y_ = βyy,
z_ = βzz.
These three equations can be combined in homogeneous form as
,
P = Sp,
Where

As is true of the translation matrix and, indeed, of all homogeneous coordinate transformations, the final row of the matrix does
not depend on the particular transformation, but rather forces the fourth component of the transformed point to retain
the value 1. We obtain the inverse of a scaling matrix by applying the reciprocals of the scale factors:

3.9.3 Rotation
We first look at rotation with a fixed point at the origin. There are three degrees of freedom corresponding to our ability to rotate
independently about the three coordinate axes. We have to be careful, however, because matrix multiplication is not a commutative
operation (Appendix C). Rotation about the x-axis by an angle θ followed by rotation about the y-axis by an angle φ does not give
us the same result as the one that we obtain if we reverse the order of the rotations.

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