Plane isometries: move a figure, preserve its shape

A copy of an ornamental motif must go somewhere else. Sliding, rotating and reflecting are different operations, although its dimensions can stay the same. How do we describe the intended operation unambiguously?

A plane isometry preserves distance between every pair of points. Reflection, rotation and translation produce figures congruent to the original, while changing position and possibly orientation. The order of several motions generally matters. We use triangle congruence, coordinates and vectors.

What stays the same

A transformation assigns an image F(X) to each point X. The motions considered here map the entire plane bijectively onto itself: each image has one preimage. The isometry condition is F(X)F(Y)=XY for every pair X,Y. Preserving one length or one area is insufficient.

It follows that segment lengths, betweenness, angle sizes and perpendicularity are preserved. A triangle’s three image points have the same three side lengths, so SSS gives the same corresponding angles. Polygon perimeters and areas are preserved too. Individual point coordinates may change. AMSI, congruence through motions.

Reflection in a line

Under reflection in line l, XX′ is perpendicular to l and its midpoint lies on l. Points on the axis remain fixed. Reflection in the y-axis is (x;y)→(−x;y); reflection in x is (x;y)→(x;−y).

Any figure can be reflected; it need not already have a symmetry axis. An axis of symmetry of a figure is a line whose reflection maps that figure onto itself. Our scalene triangle has none. Its mirror image reverses the orientation of the labelled vertex order; translations and rotations alone cannot superimpose it on the original in the plane.

Reflection in y

Central symmetry

Rotation +90°

Translation (3;0)

Dashes show the same original A(1;1),B(4;1),C(2;3) in every panel; orange is its image. Axes and unit scale stay fixed; every triangle has sides 3,√8,√5 and area 3. A and A′ are marked; under translation A′ coincides with the original B. Reflection reverses orientation, while the other three motions preserve it. The middle two motions are centered at O(0;0). Each formula maps every point, rather than only vertex A.

Central symmetry is a half-turn

Relative to center O, each X maps to X′ with O the midpoint of XX′. O itself stays fixed. For O(0;0), (x;y)→(−x;−y); for O(a;b), (x;y)→(2a−x;2b−y).

This is rotation by 180°, rather than reflection in a line. It preserves a triangle’s orientation. A figure has central symmetry only when this motion maps it onto itself: a parallelogram has a center at its diagonal intersection, while our triangle has none.

Rotation: center, angle and direction

Specify center O and directed angle α. Distance OX stays fixed while ray OX turns by α. In our coordinates positive rotation is counterclockwise. A 90° rotation about the origin is (x;y)→(−y;x). A clockwise 90° rotation would be (x;y)→(y;−x).

For A(1;1),B(4;1),C(2;3), positive 90° rotation gives A′(−1;1),B′(−1;4),C′(−3;2). Side lengths stay unchanged. This actively rotates the figure with fixed axes, rather than changing the coordinate system. Rotation by 360° restores every point. AMSI, constructing motions.

Translation: one change for everyone

For vector t=(p;q), map every point by (x;y)→(x+p;y+q). With t=(3;0), our triangle becomes A′(4;1),B′(7;1),C′(5;3). Moving A alone is insufficient: apply the same change to all points.

Why is distance preserved? Differences of image coordinates contain (x₂+p)−(x₁+p)=x₂−x₁, and similarly for y. Added constants cancel in the distance formula. Translation by the zero vector is the identity.

Two operations: test their order on a point

Let R be 90° rotation about the origin and T translation by (3;0). For P(2;1):

First R, then T: (2;1)→(−1;2)→(2;2).
First T, then R: (2;1)→(5;1)→(−1;5).

The results differ. In T∘R, the right-hand operation R acts first: T(R(P)). Composing two isometries preserves distance again: the second preserves the distance after the first, which preserved the original distance. VCAA–AMSI, transformations and composition.

With formulasOptional · safe to skip

Any rotation and the way back

For center O(a;b), subtract center coordinates, rotate, then add the center back:

$x'=a+(x-a)\cos\alpha-(y-b)\sin\alpha,$

$y'=b+(x-a)\sin\alpha+(y-b)\cos\alpha.$

These formulas accept degrees or radians provided the sine and cosine mode matches. Inverse rotation has angle −α; inverse translation uses vector −t; reflecting in the same line twice gives the identity. Some compositions look different from the four operations shown: reflection combined with translation along its axis can produce a glide reflection. We do not need the complete classification here.

To superimpose a drawing element, record the motion’s parameters and order as well as its name: axis; center and directed angle; or vector. Then check matching vertices and the distances between them.