Talk:Scaling (geometry)

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Is this the same as Dilation (mathematics)? Should the articles be merged? Rojomoke (talk) 15:58, 24 April 2009 (UTC)

Dilation, or enlargement is a special case of scaling. If we don't merge, then we certainly should have links. Dbfirs 12:09, 18 April 2010 (UTC)
I agree that dilation is a special case of scaling. Strictly, the term dilation should not be used as a synonym of scaling (as stated in the introduction). Etimologically, dilation means magnification, enlargement. Its antonym is contraction. Paolo.dL (talk) 15:29, 20 February 2012 (UTC)

Matrix or vector[edit]

I've always used the terminology of a scaling matrix multiplying a vector, not a scaling vector multiplying a point, but is the terminology used in this article common elsewhere? Dbfirs 12:09, 18 April 2010 (UTC)

vector space or affine space?[edit]

Although this article mentions Euclidean geometry, it in fact assumes an origin, so is in fact about vector spaces. I'm very doubtful that this is implied by the notion of dilation. In fact this article is confused when it talks about affine transformations as given by a matrix, and I'm pretty sure that one does not call any diagonalizable linear transformation (i.e., nearly all of them over the complex numbers) a scaling. If the notion is widened to allow an arbitrary center of scaling, then this article is to be merged with Homothetic transformation (but it should be shaped up first). Marc van Leeuwen (talk) 10:58, 1 March 2011 (UTC)

I believe that scaling is a linear transformation, while a homothetic transformation or homothety is a generalization of scaling and not a linear, but an affine transformation. Hence, there is a clear distinction between the two concepts. This is not clearly stated in the article, in which the term homothety is defined as a synonym of scaling. But I think that this article should be kept separated from Homothetic transformation. Paolo.dL (talk) 11:27, 20 February 2012 (UTC)

Scope of definition[edit]

If possible, please discuss the following points separately in the appropriate subsections. Paolo.dL (talk) 14:22, 20 February 2012 (UTC)

Negative scaling factor[edit]

As mentioned above, I believe that homothety should not be considered to be a synonym of scaling or uniform scaling. Uniform scaling is a special case of homothety (see Homothetic transformation#Homothety and scaling).

I have seen homotheties with negative scale factors, but I am not sure whether a scaling with negative scaling factor should be considered a pure scaling (scaling proper) or just a special case of homothety. Shouldn't this be made clear in this article?

Notice that (although most homotheties are non-linear trasformations) a scaling with negative scale factor is still a linear transformation.

Paolo.dL (talk) 14:22, 20 February 2012 (UTC)

Scaling with negative scale factors is on some GCSE syllabuses in the UK. Dbfirs 08:09, 14 November 2012 (UTC)

Scaling along orthogonal axes not coinciding with the standard basis[edit]

If R is a n×n rotation matrix representing the orientation in space of a rotated basis, and R' is its transpose, then a non-uniform scaling with respect to the rotated basis can be represented by

w = R' S R v

where S is a n×n diagonal matrix containing the n scaling factors. This transformation is a linear transformation, as it is a composition of linear transformations (rotation and scaling).

Does this composition meet the (strict) definition of scaling? I am not sure. Shouldn't this be made clear in this article?

(By the way, notice that a uniform scaling by a scale factor s with respect to the rotated basis can be represented by w = R' s R v = R' R (s v) = s v

Paolo.dL (talk) 14:22, 20 February 2012 (UTC)

Scaling along non-orthogonal axes[edit]

The article describes scaling along non-orthogonal axes as a generalization of scaling. It also classifies it as an affine transformation. In 3-D Euclidean space, scaling along 3 (or even more) non-orthogonal axes a1, a2, a3, ... by scale factors s1, s2, s3, ... can be obtained as a composition. Let's define three (or more) non-uniform scaling matrices:

S_1 = \begin{bmatrix}s_1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, 
        S_2 = \begin{bmatrix}s_2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix},
        S_3 = \begin{bmatrix}s_3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, etc. ...


w = ...(R_3' S_3 R_3) (R_2' S_2 R_2) (R_1' S_1 R_1) v,

where the three axes a1, a2, a3 are represented by the first column of each of the three (orthogonal) rotation matrices R1, R2, R3. This is a composition of linear transformations (rotation and scaling), and therefore it is a linear transformation. Although linear transformations are a special case of affine transformations, in my opinion it is misleading to classify scaling as an affine transformation. Namely, the statement "scaling is an affine transformation" is true but useless, and might be interpreted by laymen as "scaling is not a linear trasformation", which is wrong.

Does this composition meet the (strict) definition of scaling? I am not sure. Shouldn't this be made clear in this article?

Paolo.dL (talk) 14:22, 20 February 2012 (UTC)

redirection from different names[edit]

(as: enlargement, dilation, contraction, compression... ) I reached the article with difficulty. — Preceding unsigned comment added by (talk) 18:48, 7 May 2012 (UTC)