Dilation
A dilation resizes a figure by a scale factor from a center point, producing a similar figure that is larger or smaller.
Formula
P' = C + k(P - C)
Definition
A dilation resizes a shape by multiplying all distances from a fixed center point by a scale factor: $D(C, k)$ maps every point $P$ to $P' = C + k(P - C)$, enlarging the shape if $k > 1$ and shrinking it if $0 < k < 1$, while preserving all angles and producing a similar figure. For $k > 0$ the image and preimage are on the same side of $C$; for $k < 0$, opposite sides. Formally, a dilation is an affine map with linear part $kI$ (scalar matrix), a similarity transformation with ratio $|k|$; it scales distances by $|k|$ and areas by $k^2$, and in complex notation a dilation centered at $z_0$ is $f(z) = k(z - z_0) + z_0 = kz + (1-k)z_0$.
Example
Dilating point $(2, 3)$ from the origin by scale factor $4$ gives $(8, 12)$, multiplying each coordinate by $4$; a triangle with vertices $(1,1)$, $(2,1)$, $(1,3)$ dilated from the origin by factor $2$ becomes $(2,2)$, $(4,2)$, $(2,6)$. Dilating a triangle with vertices $A(2,4)$, $B(6,4)$, $C(4,8)$ by scale factor $1/2$ centered at the origin gives $A'=(1,2)$, $B'=(3,2)$, $C'=(2,4)$, half as large with all sides in ratio $1:2$. Composing dilations $D(C_1, k_1)$ and $D(C_2, k_2)$ with $k_1 k_2 = 1$ gives a translation; with $k_1 k_2 \neq 1$, the result is a dilation with a new center found from fixed-point analysis.
Key Insight
A dilation changes the size of a shape but never its shape: the image is always similar to the original, with all angles equal and all sides scaled by the same factor. Dilations centered at the origin simply multiply both coordinates by $k$, the easiest case to compute; for dilations centered elsewhere, translate the center to the origin, dilate, then translate back. This fixed-point analysis underlies the "spiral similarity" technique in competition geometry, since any two similar, similarly-oriented triangles are related by a dilation composed with a rotation.