Where is the difference between an "transform" and an "affine transform"?

Question

There is a function like:

CATransform3DGetAffineTransform

Returns the affine transform represented by 't'. If 't' can not be exactly represented as an affine transform the returned value is undefined.

I'm not so math-orientated, so a easy to understand description would be very nice. Wikipedia was no big help here.

Answer 1

Affine transformations are transformations that do not "overly distort" your geometry in the sense that:

• points that lay on a line before still lie on a line after the transformation
• the relative distances between points stay the same (so a square might not be square anymore, but still has equal side lengths)

So rotation and scaling are affine - projection, for example, is not.

Answer 2

Affine transformations are transformations, but transformations need not be Affine. For example, a shear of the plane is not Affine because it doesn't send lines to lines.

Affine transformations are by definition those transformations that preserve ratios of distances and send lines to lines (preserving "colinearity"). In finite-dimensional Euclidean geometry, these act by a linear transformation followed by a translation i.e. x -> Ax + b where x is a vector, A is a linear transformation and b is a vector.

Affine transformations are great for changing co-ordinate systems, perhaps from one that is fairly hard to visualise back to the usual co-ordinates.

The Affine transformations you're likely to come across in practice are translations, dilation (i.e. scaling), rotation and reflection. If you're developing for iOS, say, check out the CGAffineTransform sections of the developer documentation. These will explain how to create Affine transformations using several convenience methods, and how to apply these to UIView instances with the -setTransform message.

Last, an important point about Affine transformations is that they're not commutative under composition: Applying transform T1 followed by transform T2 is in general different from applying T2 followed by T1. That is, the order in which you apply these matters.