Given affine spaces **A** and **B**,
A function **F** from **A** to **B**
is an *affine transformation* if it preserves affine combinations.
Mathematically, this means that

We can define the action of **F** on vectors in
the affine space by defining

Where **P** and **Q** are any two points whose
difference is the vector **v** (*exercise: why is this
definition independent of the particular choice of ***P***
and ***Q***?)*

There are two other important properties of affine transformations for the purposes of computer graphics. They are linear transformations on the underlying vector spaces. That is,

Also, they preserve the representation of affine points with respect to
a given frame. In other words, the transformation of an affine point
in a frame for **A** has the same affine coordinates in the
image of that frame in **B**. In other words,

One nice aspect of this property is that once the image of the frame
has been calculated, the structure of **F** is completely
known. Remembering that an affine transformation is simply a linear
transformation over the corresponding homegeneous coordinate space,
this is simply the statement that a linear transformation can be completely
understood by its action on a basis.

If we impose the usual Cartesian coordinates on the affine plane, any affine transformation can be expressed as a linear transformation followed by a translation. Note that translations cannot be expressed as linear transformations in Cartesian coordinates. Once we move up to the general affine space, all these transformations become linear.

Note, too, that we can express affine transformations as matrix
multiplications (i.e., linear transformations) in homogeneous coordinates.
They are a special case of matrices in homogeneous coordinates; they
map points with **w = 1** to points with **w = 1**.

**Hyperplanes Map to Hyperplanes:**- In particular, points map to points, line map to lines and
planes map to planes. Also, line segments map to line segments.
This can be seen to be true by observing that
One nice consequence of this fact is that one can calculate the image of a polygon by simply computing the images of its vertices.

**Parallelism is Preserved:**- Two vectors which are parallel will have parallel images
under an affine transformation
**F**: - Thus, parallelograms map to parallelograms and ellipses map
to ellipses.
**Ratios are Preserved:**- This does
*not*mean that ratios between lengths of line segments are preserved. Ratios are preserved in the sense of interpolation. That is, if the point**R**is**x**of the way from**P**to Q, and**F**is an affine transformation, then**F(R)**is**x**of the way from**F(P)**to**F(Q)**.One implication of this is that an object moving at a constant speed gets mapped to an object moving at a constant speed.

- Course Textbook, page 1083

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