Two-dimensional transforms
We've already used simple two-dimensional transforms. Here we
will simply develop a consistent framework for them.
Two-dimensional transforms are nothing but functions used
to map points from one place to another. Transforms can be
applyed to any of the drawing primitives that we've discussed
to this point. Transforms can also be applied to the pixel
coordinates of an image.
We'll begin with a very simple transfrom and generalize it
until we arrive at a very powerful representation.
Translations
Translations are a simple family of two-dimensional transforms.
Tranlations were at the heart of our Sprite implementations in
Project #1.
Translations have the following form
x' = x + tx
y' = y + ty
For every translation there exists an inverse function which undoes
the translation. In our case the inverse looks like:
x = x' - tx
y = y' - ty
There also exists a special translation, called the identity,
that leaves every point unchanged.
x' = x + 0
y' = y + 0
These properties might seem trivial at first glance, but they are
actually very important, because when these conditions are shown
for any class of functions it can be proven that such a class is
closed under composition (i.e. any series of translations can be
composed to a single translation). In mathematical parlance this
the same as saying that translations form an algebraic group.
Rotations
Another interesting class of 2-transforms involve rotations about
the origin. Rotations are a group.
Euclidean Transforms
Euclidean transforms combine both rotation and translation.
Note: Discuss order
Simlitude Transforms
Affine Transforms
This page was last modified Wednesday, October 02, 1996