What Is an Isometry?

An isometry is a transformation that lies about how far things moved, but never about how far things are. The original points might have travelled halfway across the plane, but the distance between any two of them is what it always was. That single condition turns out to capture every notion of “rigid motion” we care about and forbids every notion of distortion. It is the load-bearing definition of the whole chapter.

The definition, stated cleanly

A mapping $f: \mathbb{R}^2 \to \mathbb{R}^2$ is an isometry if for every pair of points $P, Q$,

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

where $d$ is Euclidean distance. Notice: nothing is required about how individual points move, only that the spacing between every pair stays the same. From this single property the entire structure of plane geometry survives: angles, areas, congruences, all of it.

The three motions are all isometries

Translation. If $f(P) = P + v$, then $f(P) - f(Q) = (P + v) - (Q + v) = P - Q$, so $|f(P) - f(Q)| = |P - Q|$. The translation cancels itself in the difference.

Rotation. A rotation about any point is a rigid spin; computing distances using the rotation formula and the identity $\cos^2\theta + \sin^2\theta = 1$ gives the same answer as before rotating. Or, more abstractly: rotation is an orthogonal transformation, and orthogonal transformations preserve dot products and hence lengths.

Reflection. Reflecting across a line preserves the perpendicular distance from the line, and the component along the line is unchanged, so the distance between any two reflected points is unchanged.

Non-isometries: what gets in the way

Many natural transformations are not isometries. A dilation $f(P) = kP$ with $k \neq \pm 1$ multiplies every distance by $|k|$. A shear $f(x, y) = (x + y, y)$ keeps horizontal distances but stretches diagonal ones; pick $P = (0, 0)$ and $Q = (0, 1)$, then $d(P, Q) = 1$ but $d(f(P), f(Q)) = \sqrt{2}$. A squeeze like $f(x, y) = (2x, y/2)$ preserves area but mangles distance. The lesson: lots of transformations are interesting and useful; only the isometries are rigid.

• Physics: The laws of physics are isometry-invariant: experiments give the same result whether you translate, rotate, or reflect your lab, and Noether's theorem tells you each such invariance corresponds to a conservation law (energy, momentum, angular momentum).
• Computer Vision: Object recognition often demands isometry invariance: a chair is still a chair if rotated or shifted in the image. CNNs achieve translation invariance for free, and equivariant networks extend this to rotations too.
• Origami: A flat-foldable paper crease pattern must satisfy local angle conditions because folds are reflections (isometries of the paper); Erik Demaine's algorithms exploit exactly this isometry property.

Use the widget to apply rotation, translation, or reflection. The dashed lines connect corresponding vertices of the original and the image. Their lengths in |P - Q| readouts confirm the isometry preserves distance.

Try it

• Predict first: a rotation is an isometry, so should $|PQ|$ change under it? In rotation mode, set the angle to 60^\circ and read off both distances to verify.
• Construct a non-isometry by hand: pick the map $f(x, y) = (2x, y)$ and verify that $P = (0, 0)$, $Q = (1, 0)$ are pushed apart by a factor of $2$.
• Prove that the composition of any isometry with the identity is the same isometry. (One line.)
• If $f$ is an isometry that fixes the origin ($f(O) = O$), what must $|f(P)|$ equal for every point $P$? Why?

Pause: the definition only says “preserves pairwise distance.” Why does that force isometries to also preserve angles? Sketch the argument before reading on.

A trap to watch for

Two non-isometries are easy to mistake for the real thing. The first is the dilation $f(P) = kP$: it preserves shape and angles, so figures end up looking “the same,” but distances are scaled by $|k|$, so it is not an isometry. The right name for this is a similarity, which is a strictly larger class than isometries. The second is any area-preserving map that happens not to preserve distance, squeeze maps and shears both preserve area. Area-preserving is a real and useful property (it is what physicists call volume-preserving in phase space), but it is not the same as distance-preserving. The fix: when you read a problem, ask precisely which quantity is preserved, distance, angle, area, shape, and never let one promise the others.

What you now know

You can state the isometry condition (preserves pairwise distance), recognise translations, rotations, and reflections as isometries, and distinguish them from dilations and shears. The next section composes isometries: stack two together, get a third. That stacking is what eventually turns the isometries into a group, the doorway into modern algebra.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 6, §2, the distance-preserving definition and its immediate consequences.
• Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. §3.1, isometries as the building blocks of plane symmetry.
• Artin, E. (1957). Geometric Algebra. Interscience. Chapter 3 frames isometries as orthogonal transformations of a real inner-product space.
• Stillwell, J. (1992). Geometry of Surfaces. Springer. Chapter 1 motivates the distance-preserving definition by classifying the symmetries of the plane.