Homothetic transformation

In geometry we mean by a homothety an affinity, ie a bijective affine transformation of an affine space into itself such that for each line the image straight to parallel:

The homotheties accurately count the following figures

• The identity mapping,
• A translation, ie a parallel shift by a constant vector
• A streching by any center with any stretching factor

While all points will naturally fix on the identity, has a true translation no fixed point, and a real central dilation exactly one fixed point, namely the extension center

In the case of a central extension, the associated linear map on the vector space of translations of always of the form Therefore, one sometimes referred to linear maps, stretch each vector by a fixed non-zero scalar, as (linear) homotheties. These linear bijections of the vector space are interchangeable with all linear self-maps of the vector space.

Remark

Using the meaning described here is the notion of a " affine homothety " also defined more generally for an affinity to an affine space over a division ring. These and other generalizations and geometric characteristics of the concept are described in more detail in the article " dilation ".