Maths - Affine Transformations - Forum discussion

By: s_ludwig ( Sven Ludwig )
file special affine transformations  
2003-08-31 23:27

Hi, on

you say
"Affine Transformations can represent _an_ combination of rotations and translations, but not other _transformatons_ like scaling, shearing and reflection."

I think this sounds a little bit like affine transformations can never represent scaling, shear or reflection, which might be misleading for beginners.

I suggest something like

The combination of translations and rotations can be represented by special affine transformations that perform no scaling, shear or reflection.

Best Regards,


By: martinbaker ( Martin Baker )
file RE: special affine transformations  
2003-09-01 08:13

Hi Sven,

Thanks very much for pointing this out. I would like to work out some generally accepted terminology for these transforms, I had a look at some other websites and have come up with the following definitions, do these look right to you?

Orthogonal transform - can represent rotations only. positive determinant.
Affine transform - can represent translations, rotations, scaling (may be different in x,y and z dimensions), shearing and reflections. Lines remain straight and parallel lines remain parallel, but the angle between intersecting lines can change.
Euclidean transformations - transformations preserve both angles and lengths (ie translations, rotations and reflections). Determinant negative if there is a reflection.
Rigid transformation - transformation which can represent the movement of a solid object (ie translations and rotations only).
Special Affine transform - an affine transform where determinant of 3x3 part equals unity.


By: s_ludwig ( Sven Ludwig )
file RE: special affine transformations  
2003-09-01 21:33


as I am not an expert I cannot say
yes or no to these definitions,
but I can give my findings.

I read that orthogonal matrices can represent reflections.
Further more on
it is stated that orthogonal transformations correspond to orthogonal matrices.

I am not sure if Euclidean geometry allows for reflections.

I think parts of the following document may be helpful here
as it talks about the four geometries Euclidean, similar,
affine and projective geometry:


By: martinbaker ( Martin Baker )
file RE: special affine transformations  
2003-09-02 18:26

Hi Sven,

Thanks very much for this. I'll read these and update the web page. Also I'll include a link to your messages if that's alright.


By: martinbaker ( Martin Baker )
file RE: special affine transformations  
2003-09-06 15:01

I have updated this page:
To reflect this thread. Thanks for the corrections so far.


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