Dilation
Geometry & MeasurementA 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. If the scale factor is greater than $1$, the shape gets bigger. If it is between $0$ and $1$, the shape gets smaller.
Example
Dilating point $(2, 3)$ from the origin by scale factor $4$: multiply each coordinate by $4$ to get $(8, 12)$. A triangle with vertices $(1,1)$, $(2,1)$, $(1,3)$ dilated from the origin by factor $2$ becomes $(2,2)$, $(4,2)$, $(2,6)$.
Key Insight
A dilation changes the size of a shape but not its shape. The image is always similar to the original. All corresponding angles stay equal; all corresponding sides grow or shrink by the same factor.
Definition
A dilation $D(C, k)$ centered at point $C$ with scale factor $k$ maps every point $P$ to $P' = C + k(P - C)$. Dilations preserve shape (all angles) and the ratio of lengths (scale factor $k$), producing similar figures. For $k > 0$, the image and preimage are on the same side of $C$; for $k < 0$, they are on opposite sides.
Example
Dilate triangle with vertices $A(2,4)$, $B(6,4)$, $C(4,8)$ by scale factor $1/2$ centered at origin. $A' = (1,2)$, $B' = (3,2)$, $C' = (2,4)$. The new triangle is half as large, and all corresponding sides are in ratio $1:2$.
Key Insight
Dilations centered at the origin simply multiply both coordinates by $k$. This makes them the easiest dilations to compute. For dilations centered elsewhere, translate the center to the origin, dilate, then translate back.
Definition
A dilation $D(C, k)$ is an affine map with linear part $kI$ (scalar matrix). It is a similarity transformation with ratio $|k|$. Dilations do not preserve distance unless $k = \pm 1$; they scale all distances by $|k|$ and all areas by $k^2$. In complex notation, a dilation centered at $z_0$ is $f(z) = k(z - z_0) + z_0 = kz + (1-k)z_0$.
Example
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 new center $C_3 = (k_1 C_2 - k_2 C_1(k_1-1)) / (k_1 - k_2(k_1-1))$ (derived from fixed-point analysis).
Key Insight
The fixed-point analysis of dilation compositions is the basis of the "spiral similarity" technique in competition geometry. Any two similar, similarly-oriented triangles are related by a spiral similarity (dilation composed with rotation), a powerful problem-solving tool.