Maths - Affine Transformations - Forum discussion

 By: s_ludwig ( Sven Ludwig ) special affine transformations   2003-08-31 23:27 Hi, on https://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: https://www.euclideanspace.com/maths/geometry/affine/ To reflect this thread. Thanks for the corrections so far. Martin