Talk:Homothety

Unfortunate formulations

I agree that this entry is poorly written. The sign of c is confusing. The slip between dilation and dilatation is unfortunate. I wouldn't say that: "Typical examples of dilatations are translations, ...". More to the point, typical examples of dilatations are dilations (which therefore need to be defined in this context). The last sentence is misleading since it implies that all homotheties have centers.

On the other hand, the previous criticism is off the mark. It is nonsense that all affine maps send lines to parallel lines. As to multiplication by a negative number: this is a homothety, according to the definition of "homothety" given in this entry. Although the literature isn't entirely consistent, the definition given here is the one most commonly adopted. —Preceding unsigned comment added by GrantCairns (talk • contribs) 03:00, 3 August 2009 (UTC)

line?

"image of every line L is a line parallel to L". Is it appropriate to use the term "line"?? Jackzhp (talk) 04:14, 9 April 2011 (UTC)

Definitely. Lines are well defined in affine geometry and this is also how it is introduced in college courses. BertSeghers (talk) 12:01, 15 March 2013 (UTC)

Intuition for artists / any graduate of an 8th grade art class

Isn't this basically 'perspective' where S is the vanishing point? Can someone with more pedagogical writing chops put this into the article intelligently? 50.181.211.34 (talk) 21:39, 27 March 2014 (UTC)

simple explanation?

Is it true that any Homothetic transformation is a composition of a simple scaling transformation, and a translation? I suspect it's not, but if it is true the article should certainly say this, and if not the article could equally stand to have a "comparison with scaling and translation" section explaining how homothety is (I presume) more general. —Steve Summit (talk) 18:52, 9 July 2015 (UTC)

@Scs: Yes, that's true. Actually a homothety with a center point $S$ and a ratio $\lambda$ is simply a scaling with respect to $S$ by a scale $\lambda$.

Do scaling by $\lambda$ with repect to the origin $O$, so that each point $M$ is transformed to $M'=\lambda M$. Then $S$ gets transformed to $S'=\lambda S$.

Next do translation so that $S'$ falls back onto $S$. That is equivalent to adding ${\overrightarrow {S'S}}$ to each point. As a result we obtain
$M''=M'+{\overrightarrow {S'S}}=M'+(S-S')=\lambda M+(S-\lambda S)=S+\lambda (M-S)=S+\lambda \,{\overrightarrow {SM}}$
which is exactly a definition of homothety, given in the lead section. --CiaPan (talk) 09:50, 16 February 2018 (UTC)

P.S.
You can also do a translation first (shift everything by ${\overrightarrow {SS_{0}}}$ for $S_{0}={\tfrac {1}{\lambda }}\,S$, so that $S_{0}$ will fall back onto $S$ when scaled) and then scale by $\lambda$ with respect to the origin. --CiaPan (talk)

k = 1/k ?

Please verify this part

...which is a uniform scaling and shows the meaning of special choices for $k$:

for $k=1$ one gets the identity mapping,

for $k=-1$ one gets the reflection at the center,

for $k=1/k$ one gets the inverse mapping.

in the edit Special:Diff/1103639049 by User:Ag2gaeh.

Defining a value of a constant $k$ with an expression dependent on $k$ seems quite strange to me... --CiaPan (talk) 18:40, 11 August 2022 (UTC)

Thanks for Your hint. I changed it.--Ag2gaeh (talk) 05:39, 12 August 2022 (UTC)