2D transformations: affine & projective, interactive demo

Parametric transformations

A transformation $T$ is a machine that changes coordinates: every point $p$ goes to $p' = T(p)$ . It is global (or parametric) when the same rule applies to every point and a few parameters describe it: 2 for a translation, 1 for a rotation, 2 for a scaling. Aligning two images, for example from matched features, means finding these parameters. The plane here uses image coordinates, with $x$ to the right and $y$ down; the image covers $[-1, 1]^2$ with its center at the origin. The faint grid and "F" are the original, the solid ones their image under the transformation. The two colored segments start out parallel; on the teal one, the dot is the image of its midpoint and the ring is the midpoint of its image.

Linear transformations

Scaling, rotation, mirroring and shear are linear: a 2 × 2 matrix times the point.

\underbrace{\begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}}_{\text{scaling}} \quad \underbrace{\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}}_{\text{rotation}} \quad \underbrace{\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}}_{\text{mirror at the } y\text{-axis}} \quad \underbrace{\begin{bmatrix} 1 & a_x \\ a_y & 1 \end{bmatrix}}_{\text{shear}}

Because $y$ points down, a positive angle $\theta$ turns clockwise on the screen. Mirroring over the origin, $x' = -x$ , $y' = -y$ , is the same as a rotation by 180°. Every combination of these maps is linear again, and linear maps share a set of properties:

the origin maps to the origin,
lines map to lines,
parallel lines remain parallel,
ratios along a line are preserved, e.g. the midpoint stays the midpoint,
they are closed under composition.

The determinant tells how areas change: a rotation has $\det = 1$ , a scaling $s_x s_y$ , a shear $1 - a_x a_y$ . A negative determinant means the plane is flipped over, as by the mirror at the $y$ -axis: the "F" then reads backwards.

Translation and homogeneous coordinates

A translation $x' = x + t_x$ , $y' = y + t_y$ is not linear: it moves the origin, and no 2 × 2 matrix can do that. Homogeneous coordinates solve this with a third coordinate 1, which turns the translation into a matrix product as well:

\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}

All other maps get the 2 × 2 matrix in the top left corner of a 3 × 3 matrix, and a sequence of transformations becomes a single product. The order matters: in $H = T\,R\,\mathit{Sh}\,S\,\mathit{Mi}\,P$ the rightmost factor acts first. Here the point is mirrored and scaled first, then sheared, rotated about the origin and finally translated.

Affine transformations

Linear maps followed by a translation are affine, with six free entries:

\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} a & b & c \\ d & e & f \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}

They keep all the properties of linear maps except that the origin may move: lines stay lines, parallel lines stay parallel and ratios along a line are preserved.

Projective transformations

A projective transformation, or homography, combines an affine map with a perspective part that fills the last row:

M_\text{affine} \cdot M_\text{perspective} = \begin{bmatrix} a & b & c \\ d & e & f \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ g & h & 1 \end{bmatrix}, \qquad \begin{bmatrix} x' \\ y' \\ w' \end{bmatrix} = \begin{bmatrix} a' & b' & c' \\ d' & e' & f' \\ g & h & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}

Now $w' = g x + h y + 1$ is no longer 1, and the point is $(x'/w', y'/w')$ . Points with a larger $w'$ are pulled towards the origin, so the image looks like a plane seen at an angle. Multiplying the whole matrix by a factor $k \ne 0$ multiplies $x'$ , $y'$ and $w'$ alike and gives the same points, so $H$ is only defined up to scale: 9 entries, but 8 degrees of freedom.

Lines still map to lines, and projective maps are closed under composition.
Parallel lines do not necessarily remain parallel. All lines with direction $d$ contain the point at infinity $(d_x, d_y, 0)^\top$ , so their images meet at $H (d_x, d_y, 0)^\top$ . Its $w' = g d_x + h d_y$ is 0 for affine maps, but otherwise it is a finite point: the vanishing point of that direction.
Ratios are not preserved: the image of the midpoint is no longer the midpoint of the image segment.

A photo of a flat object, such as a document or a facade, is related to its front view by such a transformation. Knowing $H$ , the photo can be unwarped (rectified) into the front view.

Try this

Choose the identity, then set $\theta = 90^\circ$ and $t_x = 0.5$ , and pick the point $(1, 0)$ . $T\,R$ rotates it to $(0, 1)$ and then moves it to $(0.5, 1)$ . In the order $R\,T$ it would be moved to $(1.5, 0)$ first and land at $(0, 1.5)$ : the order of composition matters.
Choose the mirror at the $y$ -axis: the "F" reads backwards and $\det H$ turns negative. The mirror over the origin only turns the "F" upside down without flipping it, because it is a rotation by 180°.
Set only $a_x$ : the square becomes a parallelogram, but $\det H = 1$ , so its area is unchanged. Add $a_y$ and the determinant becomes $1 - a_x a_y$ .
Starting from the identity, set only $g = 0.45$ : the vertical grid lines stay parallel, since their direction $(0, 1)$ gives $w' = g \cdot 0 + h \cdot 1 = 0$ , while the horizontal ones now point at $H (1, 0, 0)^\top$ , i.e. $(1/g, 0) \approx (2.2, 0)$ on the $x$ -axis. Then try only $h$ .
In the projective preset, compare the dot (image of the midpoint) and the ring (midpoint of the image segment) on the teal line: they separate, so ratios are not preserved.
Pick a point and follow the panel: $H$ turns $(x, y, 1)$ into $(x', y', w')$ , and only the division by $w'$ gives the point. Scaling all of $H$ would scale $x', y', w'$ together and change nothing.