# Homothety

A transformation of Euclidean space with respect to a certain point \$O\$ that brings each point \$M\$ in a one-to-one correspondence with a point \$M'\$ on the straight line \$OM\$ in accordance with the rule

\$\$OM'=kOM,\$\$

where \$k\$ is a constant number, not equal to zero, which is known as the homothety ratio. The point \$O\$ is said to be the centre of the homothety. If \$k>0\$, the points \$M\$ and \$M'\$ lie on the same ray; if \$k<0\$, on different sides from the centre. The point \$O\$ corresponds to itself. A homothety is a special case of a similarity. Two figures called homothetic (similar or similarly situated) if each one consists of points obtained from the other figure by a homothety with respect to some centre.

Simplest properties of a homothety. A homothety with \$k\neq1\$ is a one-to-one mapping of the Euclidean space onto itself, with one fixed point. If \$k=1\$, the homothety is the identity transformation. A homothety maps a straight line (a plane) passing through its centre into itself; a straight line (a plane) not passing through its centre into a straight line (a plane) parallel to it; the angles between straight lines (planes) are preserved under this transformation. Under a homothety segments are mapped into parallel segments with a length which is \$|k|\$ times the original length, i.e. a homothety is a contraction (expansion) of the Euclidean space at the point \$O\$. Under a homothety a sphere is mapped into another sphere, and the centre of the former is mapped to the centre of the latter.

A homothety is most often specified (geometrically) by the homothety centre and a pair of corresponding points or by two pairs of corresponding points. A homothety is an affine transformation with one (and only one) fixed point.

In \$n\$-dimensional Euclidean space a homothety leaves the set of all \$k\$-dimensional subspaces invariant, \$k<n\$.

A homothety is defined in a similar manner in pseudo-Euclidean spaces. A homothety in Riemannian spaces and in pseudo-Riemannian spaces is defined as a transformation that transforms the metric of the space into itself, up to a constant factor. The set of homotheties forms a Lie group of transformations, and the \$r\$-parameter homothety group of a Riemannian space contains the \$(r-1)\$-parameter normal subgroup of displacements.