Affine Transformation

Affine Transformation

A transformation which involves any combination of translations, reflections, stretches, shrinks, or rotations. Note: Collinearity and concurrency are invariant under affine transformations.

Key Formula

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} e \\ f \end{pmatrix}

Where:

$(x, y)$ = Original coordinates of a point
$(x', y')$ = Transformed coordinates of the point
$a, b, c, d$ = Entries of the transformation matrix, controlling rotation, scaling, reflection, and shear
$e, f$ = Translation components that shift the result horizontally and vertically

Worked Example

Problem: Apply an affine transformation to the point (2, 3) using the matrix [[2, 0], [0, 1]] with a translation vector (5, −1). This combines a horizontal stretch by factor 2 with a shift.

Step 1: Write the transformation formula and substitute the values.

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 5 \\ -1 \end{pmatrix}

Step 2: Perform the matrix multiplication.

\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 \cdot 2 + 0 \cdot 3 \\ 0 \cdot 2 + 1 \cdot 3 \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}

Step 3: Add the translation vector to get the final transformed point.

\begin{pmatrix} 4 \\ 3 \end{pmatrix} + \begin{pmatrix} 5 \\ -1 \end{pmatrix} = \begin{pmatrix} 9 \\ 2 \end{pmatrix}

Answer: The point (2, 3) maps to (9, 2). The x-coordinate was stretched by 2 (giving 4) then shifted right by 5, while the y-coordinate stayed at 3 then shifted down by 1.

Another Example

Problem: Show that three collinear points remain collinear after applying the affine transformation with matrix [[1, 2], [0, 1]] (a shear) and translation (0, 0). Use the points A = (0, 0), B = (1, 1), and C = (2, 2).

Step 1: Verify the original points are collinear. All three lie on the line y = x.

Step 2: Transform each point using the shear matrix (no translation needed here).

A' = \begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}\begin{pmatrix}0\\0\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}, \quad B' = \begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}\begin{pmatrix}1\\1\end{pmatrix} = \begin{pmatrix}3\\1\end{pmatrix}, \quad C' = \begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}\begin{pmatrix}2\\2\end{pmatrix} = \begin{pmatrix}6\\2\end{pmatrix}

Step 3: Check collinearity of the transformed points. The slope from A' to B' is 1/3, and from B' to C' is (2−1)/(6−3) = 1/3. Equal slopes confirm collinearity.

Answer: The shear distorted the triangle of points, but A'(0,0), B'(3,1), and C'(6,2) still lie on a single line (y = x/3). This illustrates that affine transformations preserve collinearity.

Frequently Asked Questions

What is the difference between an affine transformation and a linear transformation?

A linear transformation maps the origin to itself and can be written as multiplication by a matrix alone: (x', y') = A·(x, y). An affine transformation adds a translation vector, so it has the form (x', y') = A·(x, y) + (e, f). Every linear transformation is affine (with zero translation), but not every affine transformation is linear, because translation shifts the origin.

What properties does an affine transformation preserve?

Affine transformations preserve collinearity (points on a line stay on a line), parallelism (parallel lines remain parallel), concurrency (lines meeting at a point still meet at a point), and ratios of distances along any line. They do not necessarily preserve angles, distances, or areas, though special cases like rigid motions do.

Affine Transformation vs. Rigid Transformation (Isometry)

A rigid transformation (isometry) preserves distances and angles — it only allows translations, rotations, and reflections. An affine transformation is more general: it allows everything a rigid transformation does, plus scaling, stretching, and shearing. Every rigid transformation is affine, but affine transformations can change lengths and angles while still keeping lines straight and parallel lines parallel.

Why It Matters

Affine transformations are foundational in computer graphics, where every time you move, resize, rotate, or skew an image on screen, an affine transformation is applied. They also appear in coordinate geometry when changing between coordinate systems, such as converting between different map projections. Understanding which properties survive an affine transformation (like collinearity) and which do not (like distances) is a core idea in geometry.

Common Mistakes

Mistake: Assuming affine transformations preserve angles and distances.

Correction: Affine transformations preserve straight lines and parallelism, but not necessarily angles or distances. Only the special subset of rigid transformations (isometries) preserves those. A shear, for instance, keeps lines straight but distorts angles.

Mistake: Thinking that a translation is a linear transformation.

Correction: A translation shifts every point by a fixed vector, which moves the origin. Linear transformations must fix the origin. Translation is affine but not linear — this is precisely why affine transformations are defined as a matrix multiplication plus a translation vector.

Related Terms

Transformations — General category that includes affine transformations
Shift — Translation component of an affine transformation
Reflection — A specific type of affine transformation
Dilation — Scaling is an affine transformation
Rotation — A rigid and affine transformation
Collinear — Property preserved by affine transformations
Concurrent — Property preserved by affine transformations
Invariant — Describes properties unchanged by a transformation