Inverse Isometries

Part 8, Chapter 8: Plane Isometries, Rigid Motions

Learning objectives

Compute the inverse of a translation, rotation, and reflection
Recognise that every reflection is its own inverse
Apply the shoes-and-socks rule to invert compositions of isometries
Connect the inverse property to the group structure of plane isometries

Every isometry is reversible. If you can translate a figure by a vector, you can translate it back. If you can rotate by an angle, you can rotate by the negative of that angle. If you can reflect across a line, you can simply reflect again. The notion of an inverse bundles this reversibility into algebra: for every isometry $f$ there is a unique isometry $f^{-1}$ that “undoes” $f$ . With this in place, the set of plane isometries finally has all the structure of a group.

The definition

The inverse of an isometry $f$ is the isometry $f^{-1}$ satisfying

$f^{-1} \circ f = \mathrm{id} \qquad \text{and} \qquad f \circ f^{-1} = \mathrm{id},$

where $\mathrm{id}$ is the identity map, $\mathrm{id}(P) = P$ . Both equations are needed in principle, but for isometries it turns out either one implies the other.

The three named inverses

Translation. $T_v^{-1} = T_{-v}$ . Undo a forward push by pushing the same distance backwards.

Rotation. $(R_{\theta, C})^{-1} = R_{-\theta, C}$ . Same centre, opposite sense. A rotation of $90^\circ$ counterclockwise is undone by $90^\circ$ clockwise, or equivalently, $270^\circ$ counterclockwise (since $-90^\circ + 360^\circ = 270^\circ$ ).

Reflection. $\sigma_\ell^{-1} = \sigma_\ell$ ell. A reflection is its own inverse. Reflect, then reflect again across the same line: you get back where you started. This makes reflections involutions, which is one of the reasons they are so structurally special.

Inverse of a composition: reverse the order

For any isometries $f$ and $g$ ,

$(f \circ g)^{-1} = g^{-1} \circ f^{-1}.$

The proof is one line of cancellation:

$(g^{-1} \circ f^{-1}) \circ (f \circ g) = g^{-1} \circ (f^{-1} \circ f) \circ g = g^{-1} \circ g = \mathrm{id}.$

The order reversal is the famous “shoes and socks” rule: to undo “put on socks, then put on shoes,” you must first take off the shoes, then the socks.

Where this shows up

Cryptography: Every encryption scheme needs a decryption (the inverse map); a self-inverse cipher like the Caesar shift by 13 (ROT13) is the cryptographic analogue of "every reflection is its own inverse."
Computer Graphics: Undoing a model transformation in a 3D pipeline applies inverse matrices in reverse order, the shoes-and-socks rule shows up as $(AB)^{-1} = B^{-1}A^{-1}$ in linear algebra.
Group Theory: The inverse property is one of the four group axioms; the set of plane isometries forms a group precisely because every isometry has an inverse, and this is the entry point to abstract algebra.

Pick the preset “T then T^{-1} = identity” or “R(90°) then R(-90°) = identity.” The intermediate F appears, then the final F lands back on the original. Compose-with-inverse should leave a figure unchanged.

Try it

Write down the inverse of $T_{(5, -3)} \circ R_{60^\circ}$ . Use the shoes-and-socks rule.
Before picking the preset: what should $T_{(3, 1)}$ (3,1) followed by $T_{(-3, -1)}$ (−3,−1) produce? Run the “T then T^{-1}” preset and verify the final image lands on the original.
Glide reflection: a reflection followed by a translation along the same line. What is its inverse? (Hint: shoes-and-socks again, but the reflection is its own inverse, so the translation reverses first.)
If $f$ is an isometry that fixes the origin, must $f^{-1}$ also fix the origin? Why?

Pause: rotations and translations preserve orientation; reflections and glide reflections reverse it. Does the inverse of an orientation-preserving isometry also preserve orientation? Argue from the “composition reverses order” rule.

A trap to watch for

The inverse of a glide reflection is not just “the same glide reflection in reverse.” A glide reflection is reflect-then-translate, so by shoes-and-socks its inverse is (translate-by-the-negative)-then-reflect. Because the reflection is its own inverse, the order matters: the translation must come first in the inverse. Beginners often think “reflection is its own inverse and translation has a clear inverse, so just apply both inverses” without reversing the order, and they end up with a different isometry. The fix: always reverse the order when inverting a composition, even when one of the factors equals its own inverse.

What you now know

Every isometry has an inverse, and you can compute it from the three named cases plus the shoes-and-socks rule for compositions. With identity, composition, and inverses in place, the plane isometries are now formally a group, the orchestration of all this structure is the topic of abstract algebra. The next section uses the orientation and fixed-point structure we have collected to classify the plane isometries: every one of them turns out to be exactly one of four types.

Quick check

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References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 6, §4, inverses of plane isometries and the shoes-and-socks rule.
Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. §3.5, group structure of plane isometries.
Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer. §17.2, the inverse and the order-reversing identity.
Artin, M. (2011). Algebra, 2nd ed. Pearson. Chapter 6 introduces the orthogonal group of the plane and its inverses in algebraic language.