Transformation tricks that make geometry move

TRRD (T=Translation/slide, R=Rotation/turn, R=Reflection/flip, D=Dilation/resize): Translation, Rotation, Reflection, Dilationion (scale)

TTranslation — slide

RRotation — turn

RReflection — flip

DDilation — scale

Reflect over x-axis: negate y → (x,-y). Over y-axis: negate x → (-x,y).

Reflection Coordinate Rules — Over x-axis: (x,y) → (x,-y). Over y-axis: (x,y) → (-x,y). Over y=x: swap coordinates → (y,x). Over y=-x: swap and negate → (-y,-x).

90° counterclockwise: (x,y) → (-y,x). 90° clockwise: (x,y) → (y,-x).

Rotation Rules — 90° CCW: (x,y) → (-y,x). 90° CW: (x,y) → (y,-x). 180°: (x,y) → (-x,-y). 270° CCW = 90° CW.

Dilation from origin: multiply both coordinates by scale factor k

Dilation Rules — k > 1: enlargement. 0 < k < 1: reduction. k < 0: reduction + 180° rotation. Area scales by k² (scale factor squared).

Composition of transformations: order matters — apply right to left

Composition of Transformations — T₂∘T₁ means apply T₁ first, then T₂. Changing the order usually gives a different result. Two reflections over intersecting lines = a rotation. Two reflections over parallel lines = a translation.

Isometry: transformation that preserves distance and shape. Translation, rotation, reflection are isometries. Dilation is NOT.

Isometries — Isometry (rigid motion): preserves distances between all points → preserves size and shape. Translations, rotations, and reflections are all isometries — the image is congruent to the pre-image. Dilation: NOT an isometry — changes size (unless scale factor = 1). Dilated image is similar but not congruent.

Glide reflection: translate then reflect over a line parallel to the translation direction

Glide Reflection — Glide reflection = translation + reflection over a line parallel to the translation direction. The order doesn't matter here — same result either way. Footprints in snow: alternating left and right footprints form a glide reflection pattern. Classified as an isometry (preserves distance).

Symmetry: line symmetry (reflection maps figure to itself). Rotational symmetry (rotation maps to itself).

Types of Symmetry — Line (reflective) symmetry: one or more lines where reflection produces the same figure. Regular polygon with n sides has n lines of symmetry. Rotational symmetry: rotation of less than 360° maps figure to itself. Order of rotation: number of times it maps to itself in one full rotation. Regular hexagon: 6-fold rotational symmetry.

Tessellation: tiles fill a plane with no gaps or overlaps. Regular polygons that tessellate: equilateral triangle, square, regular hexagon.

Tessellations — For a regular polygon to tessellate: interior angle must divide evenly into 360°. Equilateral triangle: 60° → 6 fit around a point ✓. Square: 90° → 4 fit ✓. Regular hexagon: 120° → 3 fit ✓. Regular pentagon: 108° → 360/108 = 3.33... ✗. Only these three regular polygons tessellate alone.

Scale factor > 1: enlargement. Scale factor between 0 and 1: reduction. Scale factor = 1: identity (no change).

Dilation Scale Factors — Center of dilation: the fixed point from which the figure is scaled. Scale factor k > 1: image larger than pre-image. 0 < k < 1: image smaller. k = 1: identical image (identity transformation). k < 0: reduction AND 180° rotation. Image points: multiply each coordinate by k (if center is origin).

Rotation center: each point travels the same angle on a circular arc centered at the rotation center

Properties of Rotations — Every point moves through the same angle. The distance from each point to the center of rotation stays constant — points travel on circular arcs. The rotation center is the only point that stays fixed. Rotating 90° CCW four times = 360° = back to start. Composition of two reflections over intersecting lines = rotation.

Two reflections over parallel lines = translation. Distance = twice the distance between the lines.

Composition of Reflections — Reflect over two parallel lines: result is a translation. Translation distance = 2× the distance between the lines. Direction: perpendicular to both lines. Reflect over two intersecting lines: result is a rotation. Rotation angle = 2× the angle between the lines. These relationships connect the four transformation types.