Homothety (other)

Homothety is a transformation that enlarges or shrinks objects from a fixed point (the center) by a certain ratio. While it is often discussed in the plane, homothety can also be applied in three-dimensional space. In space, homothety transforms every point $P$ to a new point $P'$ such that the center $O$, $P$, and $P'$ are collinear, and $OP' = k imes OP$, where $k$ is the scale factor.

Important properties

• Homothety preserves the shape of objects but changes their size.

• Lines through the center of homothety remain unchanged in direction.

• Parallelism is preserved: images of parallel lines remain parallel.

• Ratios of lengths along lines through the center are multiplied by $k$.

• Volumes are multiplied by $|k|^3$; areas by $|k|^2$.