Rigid Motions & Symmetry

Three motions preserve size and shape:

• Translation: slide. (x, y) → (x + h, y + k)
• Reflection: flip over a line. Over the x-axis: (x, y) → (x, −y). Over the y-axis: (x, −y)... careful — over the y-axis it is (−x, y). Over y = x: (y, x).
• Rotation about the origin: 90° anticlockwise is (x, y) → (−y, x); 180° is (−x, −y); 270° anticlockwise is (y, −x).

All three produce a congruent image, which is exactly why congruence can be *defined* by them.

A figure has line symmetry if some reflection maps it onto itself, and rotational symmetry if some rotation under 360° does. A regular n-gon has n lines of symmetry and rotational symmetry every 360/n degrees.

Composing two reflections over parallel lines gives a translation; over intersecting lines it gives a rotation.

The plane isometries are translations, rotations, reflections and glide reflections. Each preserves distance, and hence angle measure, collinearity, betweenness and area.

Coordinate rules for the common cases: translation (x,y) → (x+h, y+k); reflection across the x-axis (x,−y), across the y-axis (−x,y), across y = x (y,x); rotation about the origin by 90° anticlockwise (−y,x), by 180° (−x,−y), by 270° anticlockwise (y,−x).

Isometries are classified by orientation: translations and rotations are direct (orientation-preserving), while reflections and glide reflections are opposite. Composing an even number of reflections yields a direct isometry, an odd number an opposite one — and every plane isometry is a composition of at most three reflections.

Composition of reflections across two parallel lines gives a translation by twice the distance between them; across two intersecting lines it gives a rotation about the intersection by twice the angle between them.

The symmetries of a figure form a group under composition; for a regular n-gon this is the dihedral group of order 2n, comprising n rotations and n reflections.