Reflections and the Rigid Motions of the Plane

Place up to three mirror lines and reflect the letter F in each in turn. The page works out from the combined transformation alone what it is: a reflection, a rotation, a translation, a glide reflection, or nothing at all. It can also run the other way, taking a rotation, a translation or a glide and building the mirrors that make it. The table under the drawing does the matrix arithmetic.

Worked example — Reflections and the Rigid Motions of the Plane

Mirror offset d1	0.00
Mirror offset d2	0.00
Determinant	1.00
Orientation	keeps orientation
Rotation angle	90.00°
Centre of rotation	(0.00, 0.00)
Fixed points	one point
Kind	Rotation

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Reflections and the Rigid Motions of the Plane

An isometry of the plane is a transformation that preserves every distance. A reflection in a line sends each point to its mirror image across it, and reverses orientation: a clockwise-lettered shape comes out anticlockwise. Composing two reflections gives an orientation-preserving isometry, a rotation by twice the angle between the lines if they cross or a translation by twice the gap if they are parallel. Three reflections give a reflection or a glide reflection, and every isometry of the plane is a composition of at most three reflections.

What each symbol means

α: the angle from the first mirror to the second. The rotation is 2α
d: the gap between two parallel mirrors. The translation is 2d, at right angles to them
det: the determinant of the linear part: +1 if the motion keeps orientation, −1 if it reverses it
g: the slide of a glide reflection, along its axis
M: the combined transformation as a matrix, the product of the reflections in reverse order of application

Two mirrors make a rotation by twice their angle 🖖

Press Two crossing mirrors. The mirrors meet at 45°, and reflecting the F in one and then the other turns it through 90° about the point where they cross: twice the angle between them. Each reflection flips the F over, so two flips leave it the right way round, and an isometry that keeps orientation and fixes a point is a rotation about it. Now press Two parallel mirrors: two mirrors 2 apart slide the F by 4, twice the gap, at right angles to them. A translation behaves like a rotation whose centre has gone off to infinity, and parallel mirrors are crossing mirrors whose meeting point has gone with it.

The order of two mirrors decides the direction 🖖

Press Two crossing mirrors and then Same mirrors, reversed. The mirrors are the same two lines; only which one comes first has changed, and the rotation goes from +90° to −90°. The second composite undoes the first, because each reflection undoes itself. With three mirrors through one point the order stops mattering: press Three through a point and reverse them, and the result is the same single reflection, not a different one. Order matters for two mirrors unless they coincide or are perpendicular, and for three unless all three meet in one point or are all parallel.

Three mirrors can make the one motion two cannot 🖖

Every rigid motion of the plane is one of five kinds: the identity, a reflection, a rotation, a translation, or a glide reflection, a reflection in a line followed by a slide along it. Footprints in sand are the everyday picture. Press Three in a triangle: three lines that neither meet in one point nor are all parallel make a glide in a line at 60°, with a slide of length 6. One mirror cannot slide and two cannot flip, so no smaller number of reflections gives it. Press Build a glide or Build a rotation and the page runs the other way, finding mirrors for the motion you ask for: at most three are ever needed.

Problem solved in full

What two crossing mirrors do to a shape 5 steps
Two mirrors pass through the origin, the first at 10° to the x-axis and the second at 55°. Reflect in the first and then in the second, and find what single motion the pair is equivalent to. Press Two crossing mirrors to check.

Open this problem in the tool
1. A reflection in a line through the origin at angle a has the matrix:
  
  $R_{a} = (cos 2 a sin 2 a sin 2 a - cos 2 a)$
2. Applying the first and then the second multiplies them in that order from the right:
  
  $M = R_{5 5^{\circ}} R_{1 0^{\circ}}$
3. Multiplying out, the product is a rotation matrix:
  
  $M = (cos 9 0^{\circ} sin 9 0^{\circ} - sin 9 0^{\circ} cos 9 0^{\circ}) = (01 - 1 0)$
4. Its angle is twice the angle from the first mirror to the second:
  
  $θ = 2 (5 5^{\circ} - 1 0^{\circ}) = 9 0^{\circ}$
5. Both lines pass through the origin, which neither reflection moves, so the origin is the centre:
  
  $M (00) = (00)$
Answer

The pair is a rotation by 90° about the origin, twice the 45° between the mirrors. Its determinant is +1, so the F comes out the right way round. Swap the order and the angle becomes −90°: the same two lines, the opposite turn.

$θ = 2 α = 9 0^{\circ}, det M = + 1$

By the 42Math team · Reviewed against published sources. Corrections welcome.