Homothety (Dilation from a Center)
Big Idea: A homothety is a transformation that enlarges or shrinks a figure from a fixed point, preserving its shape but changing its size. It's one of the most elegant transformations in geometry!
Formal Definition
A homothety (also called a dilation) with center $O$ and ratio $k$ (where $k \neq 0$) is a transformation that maps every point $P$ to a point $P'$ such that:
$$\vec{OP'} = k \cdot \vec{OP}$$
Equivalently, in coordinates, if the center is $O(a, b)$ and the point is $P(x, y)$, then the image is:
$$P'\big(a + k(x - a),\; b + k(y - b)\big)$$
Key Parameters
- Center $O$: The fixed point from which scaling occurs
- Ratio $k$: The scale factor
- $|k| > 1$: Enlargement
- $0 < |k| < 1$: Reduction
- $k < 0$: Image is on the opposite side of $O$
- $k = 1$: Identity (no change)