Definition:Central Dilatation Mapping
Definition
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Definition 1
Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $\lambda \in K$.
The central dilatation mapping $c_\lambda : X \to X$ is defined as:
- $\forall x \in X: \map {c_\lambda} x = \lambda x$
where $\lambda x$ denotes the scalar product of $\lambda$ with $x$.
Definition 2
A central dilatation mapping is a linear transformation involving a fixed point $C$ such that the image $P'$ of a point $P$ is the point on the directed line segment $CP$ such that $CP' = k CP$ where $k$ is a real non-zero constant.
Center of Enlargement
Let $f$ be a central dilatation mapping with fixed point $C$.
$C$ is known as the center of enlargement of $f$.
Scale Factor
Let $f$ be a central dilatation mapping with fixed point $C$ such that the image of the directed line segment $CP$ is $k CP$.
The real non-zero constant $k$ is known as the scale factor of $f$.
Examples
Homothetic Triangles
Two triangles are homothetic if and only if they have corresponding sides parallel and proportional in length.
Also known as
A central dilatation mapping may also be called the multiplication operator.
It can also be referred to as a dilation mapping, but that can be interpreted ambiguously, as it can also refer to a general dilatation mapping, which includes the possibility of such being a translation.
Some sources use the term enlargement, but that can be misleading as the word is also used when the effect is to shrink the domain.
The terms homothety and similitude can also be seen on occasion.
The adjective homothetic can be used to describe two geometric figures which can be mapped one to the other by a homothety.
Also see
- Definition:Left Regular Representation: a similar concept in the context of algebraic structures
- Results about central dilatation mappings can be found here.