Maths - Affine Transformations - Forum discussion

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

Hi, on

http://www.euclideanspace.com/maths/geometry/affine/index.htm

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,

Sven

By: martinbaker ( Martin Baker )
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.

Martin

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

Hi,

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

On
http://www.wikipedia.org/wiki/Orthogonal_matrix
I read that orthogonal matrices can represent reflections.
Further more on
http://mathworld.wolfram.com/OrthogonalTransformation.html
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:
http://robotics.stanford.edu/~birch/projective/

Sven

By: martinbaker ( Martin Baker )
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.

Martin

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

I have updated this page:
http://www.euclideanspace.com/maths/geometry/affine/
To reflect this thread. Thanks for the corrections so far.

Martin