Classifying Isometries

Part 8, Chapter 8: Plane Isometries, Rigid Motions

Learning objectives

State the four-type classification: translation, rotation, reflection, glide reflection
Use the orientation diagnostic to distinguish orientation-preserving from orientation-reversing isometries
Use the fixed-point diagnostic to distinguish the four types
Classify a given coordinate map as one of the four isometry types

You have met four kinds of plane isometry: translation, rotation, reflection, glide reflection. The startling fact is that there are no others. Every distance-preserving map of the plane, no matter how complicated it looks, is exactly one of these four. The proof is constructive: given any isometry, you can read off which type it is from two diagnostics, orientation and fixed points. This is the chapter's climactic theorem, and it is the gateway into the modern theory of symmetry groups.

The classification theorem

Every plane isometry $f$ is exactly one of:

Identity: $f(P) = P$ for all $P$ . Fixes everything. (Counted as the trivial case below.)
Translation $T_v$ v with $v \neq 0$ : shifts every point by $v$ . Preserves orientation. No fixed points.
Rotation $R_{\theta, C}$ theta,C with $\theta \neq 0$ : spins everything around $C$ . Preserves orientation. Exactly one fixed point (the centre $C$ ).
Reflection $\sigma_\ell$ ell: flips across line $\ell$ . Reverses orientation. A whole line of fixed points (namely $\ell$ ).
Glide reflection: reflection across a line $\ell$ followed by a non-zero translation along $\ell$ . Reverses orientation. No fixed points.

The classification by orientation and fixed points is exhaustive: any combination outside this table is impossible for an isometry. If you have a candidate transformation and want to know which kind it is, you only need two questions.

The two diagnostic questions

1. Does the isometry preserve orientation? Apply it to a counterclockwise triangle and check whether the image is still counterclockwise (preserves) or now clockwise (reverses). Computationally, this is the sign of the determinant of the linear part. Preserves: translation or rotation. Reverses: reflection or glide reflection.

2. What are the fixed points? Solve $f(P) = P$ . No fixed points means translation or glide reflection. One fixed point means rotation. A whole line of fixed points means reflection. Every point fixed means identity.

The two diagnostics together pin down a unique cell of the $4 \times 4$ table, that is what makes the classification work.

Where this shows up

Crystallography: The 17 wallpaper groups, the complete classification of repeating 2D patterns, are built entirely from the four types of plane isometry you are studying: translations, rotations, reflections, and glide reflections.
Computer Animation: Motion-capture data is decomposed into a translation of the centre of mass plus rotations about each joint; the orientation/fixed-point classification is what distinguishes "walking" from "walking-while-mirrored."
Art Conservation: Frieze patterns on ancient pottery and architecture are classified into 7 frieze groups, each defined by which combination of translation, glide reflection, and rotation generates the pattern.

Use the widget to apply each kind of isometry to the F. Try to see the orientation flip on reflection and the lack of one for translation. Set the rotation centre on a point of the F and notice that point stays fixed; the rest spins.

Try it

Classify $f(x, y) = (-x, y + 2)$ . Step 1: orientation, the $x$ -flip reverses, the $y$ -shift does not affect orientation, so the composition reverses. Step 2: fixed points, solve $x = -x$ and $y = y + 2$ ; the first gives $x = 0$ , the second is impossible, so no fixed points. Reverses orientation, no fixed points → glide reflection.
Classify $f(x, y) = (y, x)$ . Try orientation by computing the image of a small CCW triangle; the result is clockwise. The fixed-point equation gives $x = y$ , a whole line. Reverses orientation, line of fixed points → reflection across $y = x$ .
Classify $f(x, y) = (-y, x)$ . Orientation: preserves (rotations always do). Fixed points: only the origin. Rotation by $90^\circ$ about the origin.
Without computing anything, can you describe an isometry that fixes two points but is not the identity? Why is the answer no?

Pause: an isometry that preserves orientation and has no fixed points must be a translation. Can a translation be the identity? Under what condition?

A trap to watch for

Two diagnostics, two traps. First: students sometimes treat the axis of symmetry of the shape being moved as if it were the axis of reflection of the isometry. These are different objects. The shape's symmetry axis is a property of the figure; the reflection's axis is determined by the isometry, which you discover by solving $f(P) = P$ . They can coincide, but they need not. Second: do not confuse fixes no points with nothing special. A translation has no fixed points, but it is not a generic transformation, it is a very specific kind (rigid shift). “No fixed points” combined with “preserves orientation” forces the isometry to be a translation; combined with “reverses orientation” it forces a glide reflection. Both diagnostics are needed; one alone is not enough.

What you now know

You can classify any plane isometry into one of four types using two questions: does it preserve orientation, and what are its fixed points? You know the result is exhaustive, no other types exist. The final section of the chapter takes the classification and turns it outward: two figures in the plane are called congruent if some isometry maps one onto the other. With the four-type classification, the notion of congruence is finally fully understood.

Quick check

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References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 6, §5, the four-type classification theorem.
Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. §3.6, classification of orientation-preserving and orientation-reversing isometries.
Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer. Chapter 17: classification proved within the axiomatic Euclidean setting.
Stillwell, J. (1992). Geometry of Surfaces. Springer. §1.4, classification of plane isometries with the orientation diagnostic.
Artin, M. (2011). Algebra, 2nd ed. Pearson. Chapter 6: the same classification, recast in the language of orthogonal matrices.