Composing Isometries

Part 8, Chapter 8: Plane Isometries, Rigid Motions

Learning objectives

Compose isometries and recognise that the result is again an isometry
Compute composition of two translations as a single translation by the vector sum
Identify when the composition of two reflections is a rotation versus a translation
Track the order of composition and recognise that order generally matters

If isometries are the verbs of plane geometry, composition is grammar. Doing one rigid motion after another always gives another rigid motion, you cannot escape the world of isometries by stacking them. The interesting question is which one you end up with. The answer is sometimes obvious (two translations make one bigger translation), sometimes surprising (two reflections can make a rotation), and always determined by what the underlying motions did to length and orientation.

Composition preserves the isometry property

If $f$ and $g$ are isometries, then so is $f \circ g$ , defined by $(f \circ g)(P) = f(g(P))$ . The proof is a one-liner:

$d(f(g(P)),\, f(g(Q))) = d(g(P),\, g(Q)) = d(P,\, Q).$

The first equality is because $f$ is an isometry; the second is because $g$ is. So the set of all plane isometries is closed under composition, the first hint that this set has the structure of a group.

The three named cases

Two translations. $T_u \circ T_v = T_{u + v}$ ucircTv=Tu+v. Vectors add. Order does not matter for translations alone.

Two reflections across parallel lines. If two lines are parallel and a distance $d$ apart, reflecting across one and then the other is the same as translating perpendicular to them by $2d$ .

Two reflections across intersecting lines. If two lines meet at angle $\alpha$ at a point $C$ , the composition of the two reflections is a rotation about $C$ by angle $2\alpha$ .

These two-reflection results are the keystone of the whole chapter. They say: every rotation is a product of two reflections; every translation is a product of two reflections. Reflections are the atoms; everything else is built from them.

Where this shows up

Animation: Composing two key-frame rotations to interpolate between poses is the bread and butter of character animation; the non-commutativity of rotations is why the order of joint rotations in a skeleton matters.
Quantum Computing: Quantum gates are unitary maps (the complex generalisation of isometries), and a quantum circuit is literally a composition of these gates, the entire computation is one big composition.
Cinematography: A camera dolly followed by a pan is one composition of isometries; the famous "Vertigo zoom" depends on choreographing two composed motions that exactly cancel one effect while keeping another.

Pick two transforms. The widget shows original (faded) → intermediate (medium) → final (solid), with arrows between centroids. The presets demo “two reflections = rotation,” “two reflections = translation,” and the order-matters case below.

Order matters, usually

Function composition is associative ((f \circ g) \circ h = f \circ (g \circ h)) but not generally commutative. For isometries, commuting holds only in special cases: two translations always commute; two rotations about the same centre commute. But rotate-then-translate is not in general the same as translate-then-rotate; nor is reflect-then-translate the same as translate-then-reflect. Failing to track order is the single most common error in this chapter.

Try it

Before picking the preset: if two mirror lines meet at $45^\circ$ , the composition of the two reflections is a rotation by what angle? Pick the “Two reflections (intersecting) = rotation” preset and verify the final F is rotated by $90^\circ$ .
Predict first: if two parallel mirrors sit a $1.5$ -unit gap apart, what translation distance does the composition produce? Pick “Two reflections (parallel) = translation” and verify the final F shifts by $3$ units.
Pick “T then R is not R then T.” Then in the same widget, swap Step 1 and Step 2. The final positions should disagree.
Compute by hand: apply $R_{90^\circ}$ to $(2, 0)$ , then $T_{(1, 0)}$ (1,0). Now do them in the other order. The results differ.

Pause: two reflections across intersecting lines give a rotation by twice the angle between the lines. If you wanted a rotation by $180^\circ$ , the two mirrors must meet at what angle?

A trap to watch for

The most common composition error is order confusion. The notation $f \circ g$ means “apply $g$ first, then $f$ ”, the function on the right of $\circ$ acts first. Read left-to-right and you have it backwards. A second trap is assuming that “two of a thing” behave like one bigger version of the thing. Two reflections do not give one bigger reflection; they give a rotation or a translation, depending on whether the mirrors intersect. The atoms of the orientation-reversing isometries (reflections, glide reflections) and the orientation-preserving isometries (translations, rotations) sit in different categories, and composition tells you when you have crossed from one to the other.

What you now know

Composition of isometries is closed (you stay in the isometry world), associative (you can regroup), and usually non-commutative (you cannot swap). You also know the keystone identities: parallel reflections compose to a translation, intersecting reflections compose to a rotation. The next section turns the spotlight on inverses, every isometry has one, and finding it is just a matter of running the same motion backwards.

Quick check

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References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 6, §3, composition of isometries and the two-reflection theorems.
Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. §3.4, the “product of two reflections” classification.
Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer. §17.1, reflections as the generators of the isometry group.
Artin, E. (1957). Geometric Algebra. Interscience. Chapter 3 connects composition with the group structure of the orthogonal group $O(2)$ .