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Geometry · Rigid transformation

Rigid transformation

A rigid transformation moves the plane without changing any distance: a translation, a rotation, a reflection or a glide reflection. Reflect in two mirrors and you get a rotation, or a translation when they are parallel.

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Rigid transformations in this visualization

A rigid transformation of the plane (an isometry) keeps every distance: ∣f(P)f(Q)∣=∣PQ∣|f(P)f(Q)| = |PQ| for all points P and Q. There are exactly four kinds: translations, rotations, reflections and glide reflections. The first two keep a figure's side up (determinant +1); the last two turn it over (determinant −1), which is why the footprint's mirror image is a left foot, drawn hatched. Some books keep the name rigid motion for the first two only; here it means any isometry.

The page builds them all from mirrors. Reflect in mirror 1, then in mirror 2: if the mirrors cross at C and the directed angle from mirror 1 to mirror 2 is θ2−θ1\theta_2 - \theta_1, the result is a rotation about C by twice that angle. If they are parallel, a distance d apart, it is a translation by 2d, straight across them from mirror 1 towards mirror 2; if they coincide, every point goes back where it was.

Two reflections
crossing: a turn of 2(θ2−θ1) about C,parallel: a slide of 2d\text{crossing: a turn of } 2(\theta_2 - \theta_1) \text{ about } C, \qquad \text{parallel: a slide of } 2d

Why twice: a point P at angle φ from mirror 1, seen from C, is at −φ after the first reflection; mirror 2 sits at θ from mirror 1, so the second reflection puts it at 2θ + φ. Every point turns by 2θ, whatever φ, and neither reflection changes its distance from C. Point P draws these two pairs of equal angles. Only the angle matters, so both mirrors can turn together about C without changing the result; swapping them turns the other way.

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Things to notice

  • Turn mirror 2 about its point towards mirror 1's direction: the crossing point C runs off along mirror 1 and the turn shrinks, and at exactly parallel the rotation has become a translation. Nearly parallel is still a rotation, about a centre far away; an arrow at the edge points to it and gives its coordinates. The page keeps every angle exactly, to a thousandth of a degree, and decides parallel, the same line, a reflection or a glide from those exact values, never from a rounded one.
  • An odd number of reflections turns the figure over. Three mirrors make a glide reflection — a reflection followed by a slide along its own axis, the step from one footprint of a trail to the next — or a plain reflection when the three mirrors meet in one point or are all parallel.
  • Target runs the argument backwards: any rigid transformation needs at most three mirrors. Mirror 1 is the perpendicular bisector that sends A to A′, mirror 2 sends the image of B to B′ (it passes through A′), and mirror 3, if still needed, sends the image of C to C′ (it passes through A′ and B′). A step whose point is already in place is skipped: a rotation or translation takes two mirrors, a reflection one, a glide reflection three.
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Controls

  • Drag a mirror's point or the mirror itself to move it, and its ring to turn it; a turning mirror catches every 15° from the other mirrors, parallel included. The dial turns mirror 2, with mirror 1's direction marked on it; directions can be typed in degrees (40, 22.5, π/4). In Target, drag the dashed footprint, turn it by its ring, and flip it with Flip or a double-click.
  • Presentation mode opens a full-screen view for projectors: ← → turn by 15°, H hides what the mirrors make while the mirrors and the images stay.
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Related

Rotations and reflections as matrices are on the linear transformation page; moving and reflecting graphs of functions on the function transformation page; multiplying by a complex number of modulus 1 is a rotation on the complex plane page.

Further reading: Wikipedia: Rigid transformation; Wikipedia: Reflection (mathematics); Wikipedia: Glide reflection.