Image of Point under Central Dilatation Mapping with Fixed Point at Origin

Theorem

Let $f$ be a central dilatation mapping whose center of enlargement $C$ is located at the origin of a coordinate system.

Let the scale factor of $f$ be $k$.

Let the position vector of a point $P$ be $\mathbf r$.

Then the image of $P$ under $f$ has position vector $k \mathbf r$.

Corollary

Let $f$ be a central dilatation mapping whose center of enlargement $C$ is located at the origin of a Cartesian plane.

Let the scale factor of $f$ be $k$.

Let $P$ be the point $\tuple {x, y}$.

Then the image of $P$ under $f$ is $\tuple {k x, k y}$.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): enlargement (central dilatation, homothety, similitude)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): enlargement (central dilatation, homothety, similitude)