Dilation Transformation Calculator – Scale Points & Shapes
A dilation (also called scaling) is a transformation that changes the size of a figure
while keeping it “similar” (same shape) when the scaling is uniform.
Every point moves along a ray starting at a fixed point called the center of dilation.
1) Dilation about the origin
If the center is the origin \(C=(0,0)\) and the scale factor is \(k\), then:
\[
(x,y)\;\to\;(x',y')=(kx,\;ky).
\]
Example: \(P=(2,3)\) with \(k=2\) gives \(P'=(4,6)\).
2) Dilation about an arbitrary center \((h,k_0)\)
If the center is \(C=(h,k_0)\) and the scale factor is \(k\), then:
\[
\begin{aligned}
x' &= h + k(x-h),\\
y' &= k_0 + k(y-k_0).
\end{aligned}
\]
This says: shift to the center, scale, then shift back.
3) Uniform scale factor and what it does
-
If \(k>1\), the figure enlarges.
-
If \(0<k<1\), the figure shrinks.
-
If \(k<0\), the figure is scaled by \(|k|\) and also flipped through the center.
4) Length, perimeter, and area scaling (uniform dilation)
A key reason dilation is important is that it rescales measurements in predictable ways:
-
Lengths scale by \(|k|\).
\[
L' = |k|\,L
\]
-
Perimeter scales by \(|k|\).
\[
P' = |k|\,P
\]
-
Area scales by \(k^2\).
\[
A' = k^2\,A
\]
5) Matrix form (homogeneous coordinates)
Dilation about a center \((h,k_0)\) can be written as a matrix multiplication:
\[
\begin{bmatrix}x'\\y'\\1\end{bmatrix}
=
\begin{bmatrix}
k & 0 & (1-k)h\\
0 & k & (1-k)k_0\\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}x\\y\\1\end{bmatrix}.
\]
6) Advanced option: separate \(k_x\) and \(k_y\)
Sometimes you want different scaling in the horizontal and vertical directions:
\[
x' = h + k_x(x-h),\qquad y' = k_0 + k_y(y-k_0).
\]
This is not a similarity transformation in general (it can distort angles),
but it is very useful in applications.
The area scaling factor becomes:
\[
A' = |k_x k_y|\,A.
\]
7) Reading the diagram in this tool
- The diagram uses square units so scales match on both axes.
- You can pan by dragging and zoom with the wheel/trackpad or pinch on touch.
- Play animates the motion from the original figure to the dilated one.
- Optional rays show how every point lies on a line from the center.
8) Common mistakes
- Confusing the center’s \(k_0\) with the scale factor \(k\).
- Setting \(k=0\) (everything collapses to the center, which is usually not intended).
- For \(k<0\), forgetting that the figure is also flipped through the center.
