Difference between revisions of "Homothety"
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| − | A transformation of Euclidean space with respect to a certain point | + | {{TEX|done}} |
| + | 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 | + | 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|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 | + | 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|affine transformation]] with one (and only one) fixed point. | 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|affine transformation]] with one (and only one) fixed point. | ||
| − | In | + | 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 | + | 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. |
Revision as of 14:09, 12 April 2014
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.
Comments
A homotopy is also called a central dilatation (cf. also Dilatation).
References
| [a1] | M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French) |
| [a2] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961) |
| [a3] | E. Artin, "Geometric algebra" , Interscience (1957) |
Homothety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homothety&oldid=11584