Slide, Turn, Flip: The Rigid Motions

Moving a shape without changing its shape

You already know what it means for two triangles to be congruent: same size, same shape, so that one could be laid exactly on top of the other. But that phrase hides an action — laid exactly on top of — and on this rung we finally take that action seriously. A rigid motion (also called an isometry) is a way of moving every point of the plane at once so that distances are perfectly preserved: if A and B go to A' and B', then |A'B'| = |AB|, always, for every pair of points. Nothing is bent, nothing is stretched, nothing is torn. The plane is treated like a stiff transparent sheet you can pick up and set back down.

The word 'isometry' says it plainly: iso means same, metry means measure. The single property of preserving distance is astonishingly strong. Because distance is kept, angles are kept too (an angle is fixed once you know the three distances of a tiny triangle), areas are kept, straight lines stay straight, and the original figure and its image are congruent in the old familiar sense. So this rung does not throw away the congruence you learned earlier — it explains it. Two figures are congruent precisely when some rigid motion carries one onto the other.

The slide: translation

The gentlest rigid motion is the translation — a pure slide, with no turning and no flipping. Every single point moves the same distance in the same direction, as though the whole sheet were nudged across the desk. That 'same distance, same direction' is exactly the description of a vector, so a translation is captured by one arrow: its translation vector. If the vector is (3, 1), then the point (x, y) goes to (x + 3, y + 1), and so does every other point — the entire plane shifts in lockstep.

Why does a slide preserve distance? Take two points A and B; after the slide they become A' and B', each shifted by the same vector v. The little arrow from A to B and the little arrow from A' to B' are then the same arrow — sliding both endpoints by v does not change the displacement between them. Same arrow means same length, so |A'B'| = |AB|. A translation has no fixed point at all (unless the vector is zero, the do-nothing 'identity'), and it leaves every line pointing in the same direction it started: a horizontal line stays horizontal, parallels stay parallel.

The turn: rotation

A rotation turns the plane about a fixed point. To specify one you need two things: a centre O, the single point that stays put, and an angle theta with a sense (counterclockwise is the usual positive direction). Then every point P swings to a new point P' so that |OP'| = |OP| and m(angle POP') = theta. Think of pinning the transparent sheet to the desk with a tack at O and rotating it: the tack does not move, and every other point rides around on a circle centred at O, keeping its distance from the centre exactly.

Because every point keeps its distance to O, two points A and B both ride their own circles while the angle of the whole sheet turns by the same theta. The triangle OAB is carried onto the triangle OA'B' with |OA'| = |OA|, |OB'| = |OB|, and the angle at O between the two arms unchanged — so by side-angle-side the two triangles are congruent and |A'B'| = |AB|. A rotation has exactly one fixed point, its centre (again, unless theta is a whole turn, which does nothing). The special case theta = 180 degrees is worth a name: a half-turn, or point reflection, which sends each point straight through the centre to the opposite side an equal distance away.

The flip: reflection

The third move is the reflection, a flip across a line. The line m is the mirror (or axis); a reflection across a line sends each point P to the point P' on the opposite side of m, the same perpendicular distance away, so that m is the perpendicular bisector of the segment PP'. Any point already sitting on the mirror does not move at all — the whole line m is held fixed. To find an image by hand, drop a perpendicular from P to m and continue an equal distance past it; that is P'.

A reflection preserves distance for the same kind of reason: reflecting both A and B across m flips a figure onto its mirror image without changing any lengths inside it. But here something genuinely new happens that neither the slide nor the turn ever does — the reflection reverses handedness. Trace the corners of a triangle 1, 2, 3 going counterclockwise; after a flip, the image reads 1, 2, 3 going clockwise. Your left hand has become a right hand. You cannot undo that by any amount of sliding and turning within the plane; a flip is a different species of motion.

Two families: direct and opposite

That observation about handedness splits all rigid motions cleanly into two families. A direct isometry preserves orientation — a counterclockwise loop stays counterclockwise — while an opposite isometry reverses it. Translations and rotations are direct; you can act them out with a real sliding tile, never lifting it off the table. A reflection is opposite; to perform it physically you must flip the tile over, which is why your hands feel it. This split into direct and opposite isometries is one of the deepest facts of the whole rung, and the next guide leans on it heavily.

RIGID MOTIONS OF THE PLANE

  direct  (keep handedness)   |  opposite (flip handedness)
  --------------------------- | ---------------------------
  translation  - slide        |  reflection  - flip
  rotation     - turn         |  glide reflection - flip+slide

  all four preserve every distance:  |A'B'| = |AB|

Here is the surprise the rest of the rung will earn: those four — translation, rotation, reflection, and the glide reflection still to come — are all there is. Every rigid motion of the plane, no matter how complicated it looks, is exactly one of these four. The upcoming guides build that classification carefully: first you will compose motions and meet the glide reflection, then you will discover that every isometry is secretly a product of at most three reflections, and from there the complete list falls out.

Reading a motion off two points

One last practical idea ties the rung together and shows just how rigid these motions are. Suppose you know where two points A and B land — A goes to A', B goes to B'. Astonishingly, that almost pins the entire motion down. Knowing the image of every point of a triangle would obviously fix it, but you do not even need the third corner: once you also decide whether handedness is kept or flipped, the images of just two points determine the rigid motion completely and uniquely. The plane is so stiff that two anchor points leave it nowhere to wiggle.

1. Decide the family first: does the figure read the same way around (a direct isometry — slide or turn) or mirror-reversed (an opposite one — a flip)?
2. If nothing turned and the slide is the same everywhere, it is a translation — read off the vector as A' minus A.
3. If it is direct but A and B swung around a still point, it is a rotation — its centre lies where the perpendicular bisectors of AA' and BB' cross.
4. If handedness flipped, it is a reflection (or, slide included, a glide reflection) — and the mirror is the line every point's segment-to-its-image crosses at a right angle.

Notice the centre-finding trick: a rotation's centre is equidistant from A and A' (both lie on a circle around it), so it sits on the perpendicular bisector of AA' — and likewise on the bisector of BB'. Two bisectors, one crossing point, and the centre is found. This little move, distance preservation forcing centres onto perpendicular bisectors, is the workhorse behind the classification you will prove next. You now hold the three primitive motions and the one idea — preserve every distance — from which the whole transformation rung is built.