Difference between a similarity and a homothetic transformation Can someone explain to me the difference between similarities and homothetic transformations?
I understand that scalings are affine transformations of the plane that can be characterized by their factors $(r,s)$, and that if $r=s$, we have a similarity. In this case the scaling factor is the same in both directions of the axes of the plane. Isn't that a homothetic transformation?
Are similarities homothetic transformations?
 A: In general affine spaces, homotheties are similarities,but the converse is not true. A homotheity is a specific kind of similarity. In older geometry books and papers, homotheties refer to the mappings in Euclidean spaces that in more modern jargon are called magnifications,scalings or stretches. A homotheity $h:V\rightarrow V$ where V is the affine space,is defined as the following linear transformation: 
  g(x) = $x_0 + \lambda$(x-$x_0$)  
where $\lambda\in\mathbb R$. $x_0$ is called the center of the space V and it is determined by a choice of coordinate system (i.e. a basis). The definition of a similarity, by comparison, only requires a distance function and is more general then a homothiety: A similarity f:$X \rightarrow X$ on a metric space X is defined for any $x,y\in X$ : 
  d(f(x),f(y)) =  $\lambda$d(x,y) 
So the transformation doesn't have to be a linear transformation-indeed, the space can be coordinate free. In a Euclidean space V , it's clear that any homothiety is a similarity in V.  
A: In geometry, a homothety (homothetic transformation) is a special case of a similarity.  A similarity admits arbitrary translation and rotation; a homothety is a similarity with a homothetic center.
