How to Find the Inverse of a Rigid Transformation

A rigid transformation moves points without changing distances. To find its inverse, undo the operations in reverse order. In Euclidean coordinates, this is especially simple: a rigid transformation consists of an orthogonal linear map (a rotation, reflection, or combination) and a translation.

The inverse formula

Write a rigid transformation of $\mathbb{R}^n$ as

$$T(x)=Qx+b,$$

where $Q$ is an orthogonal matrix and $b$ is a fixed vector. Orthogonality means $Q^TQ=I$, so $Q^{-1}=Q^T$. Given the transformed point $y=T(x)$, first subtract the translation and then undo the orthogonal map:

$$T^{-1}(y)=Q^T(y-b).$$

Equivalently, the inverse is the affine map

$$T^{-1}(y)=Q^Ty-Q^Tb.$$

The order matters. The original map applies $Q$ first and then adds $b$; its inverse subtracts $b$ first and then applies $Q^T$.

Worked example in the plane

Consider a counterclockwise rotation by $90^\circ$ followed by a translation by $(2,-1)$. The rotation matrix and translation vector are

$$Q=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \qquad b=\begin{pmatrix}2\\-1\end{pmatrix}.$$

For a point $(x,y)$, the transformation is

$$T\begin{pmatrix}x\\y\end{pmatrix} =Q\begin{pmatrix}x\\y\end{pmatrix}+b =\begin{pmatrix}-y+2\\x-1\end{pmatrix}.$$

To invert it, transpose $Q$ and subtract $b$ from the output point $(u,v)$:

$$\begin{aligned} T^{-1}\begin{pmatrix}u\\v\end{pmatrix} &=Q^T\left(\begin{pmatrix}u\\v\end{pmatrix}-\begin{pmatrix}2\\-1\end{pmatrix}\right)\\ &=\begin{pmatrix}0&1\\-1&0\end{pmatrix} \begin{pmatrix}u-2\\v+1\end{pmatrix}\\ &=\begin{pmatrix}v+1\\2-u\end{pmatrix}. \end{aligned}$$

For a check, $T(3,4)=(-2,2)$. Applying the inverse gives $T^{-1}(-2,2)=(3,4)$, as required.

Why the inverse is also rigid

A rigid transformation preserves distances: for any points $x_1$ and $x_2$,

$$\|T(x_1)-T(x_2)\|=\|x_1-x_2\|.$$

Let $y_1=T(x_1)$ and $y_2=T(x_2)$. Then

$$\|T^{-1}(y_1)-T^{-1}(y_2)\| =\|x_1-x_2\| =\|y_1-y_2\|.$$

Thus the inverse preserves distances too. Since a rigid transformation is bijective, its inverse is defined on the whole space. In matrix terms, the inverse has orthogonal part $Q^T$ and translation part $-Q^Tb$, so it has the same rigid-transformation form.

Inverting a sequence of transformations

If a transformation is a composition, reverse the order when taking its inverse. For example, if $T=T_2\circ T_1$ (apply $T_1$ first, then $T_2$), then

$$T^{-1}=T_1^{-1}\circ T_2^{-1}.$$

You can see this by undoing the last operation first: $T_2^{-1}$ removes $T_2$, and then $T_1^{-1}$ removes $T_1$. This rule applies whether the individual steps are translations, rotations, or reflections.

Common mistakes

• Negating the translation without accounting for the rotation. The inverse translation is $-Q^Tb$, not generally $-b$. It is $-b$ only when $Q^Tb=b$.
• Using $Q$ instead of $Q^T$. A rotation is undone by the inverse rotation; for an orthogonal matrix, that inverse is its transpose.
• Keeping the original operation order. For a composition, the last operation performed must be the first one undone.
• Confusing a rigid transformation with a distance-preserving map into a smaller subset. The formula above assumes the usual Euclidean rigid transformation, which is a bijection from the space to itself. A distance-preserving map that is not onto has an inverse only on its image.