Question

Distinguishing Types of Transformations
Compare a dilation to the other transformations: translation, reflection, rotation.

Answer 1

Short answer
- A dilation rescales distances from a fixed center by a scale factor k; translations, reflections and rotations do not change sizes (they are isometries).
- Translation, reflection and rotation preserve all distances and areas; a dilation multiplies lengths by |k| and areas by k^2 (in the plane).
- All four are "rigid or similarity" plane transformations: translations/reflections/rotations = isometries (distance-preserving); dilations = similarity transformations (angle-preserving, distance changed by a constant factor).

Detailed comparison (useful properties)
- Definition
- Translation: every point P → P + v (move by vector v).
- Rotation: rotate about a point by an angle θ.
- Reflection: mirror across a line (in the plane).
- Dilation: with center O and scale factor k, map P → O + k(P − O).

- Size and shape
- Translation/rotation/reflection: preserve lengths and areas (congruence).
- Dilation: multiplies all lengths by |k| and areas by k^2; shapes stay similar (same angles), not congruent unless |k| = 1.

- Angles and parallelism
- All four preserve angles between curves/lines.
- Dilation and translation preserve parallelism (a line not through the center becomes a parallel line). Rotations do not in general map a line to a parallel line (unless 0° or 180°). Reflections map lines to lines (angle depends).

- Orientation
- Translation and rotation preserve orientation.
- Reflection reverses orientation.
- Dilation with positive k preserves orientation; with negative k the effect includes a half-turn component (sign of k changes direction along rays).

- Fixed points
- Translation: no fixed points (unless zero vector).
- Rotation: the center is fixed (other points move).
- Reflection: every point on the mirror line is fixed.
- Dilation: the center O is fixed; other points move (unless k = 1).

- Algebra / matrices (plane)
- Translation: affine translation (not a linear map unless you work in homogeneous coordinates).
- Rotation: orthogonal matrix with determinant +1.
- Reflection: orthogonal matrix with determinant −1.
- Dilation: scalar matrix kI; determinant = k^2 (area scaling).

- Composition facts
- Any composition of translations/rotations/reflections is an isometry.
- Composing a dilation with an isometry gives a similarity (size changed but shape preserved).
- Two dilations about the same center compose to another dilation with product of scale factors.

- Special/singular cases
- k = 1: dilation = identity.
- k = −1: dilation about O sends every point to its point symmetric about O (a half-turn/180° rotation about O).
- k = 0 collapses everything to the center.

Quick examples
- Translate triangle by vector (3, 2): same size and orientation, just moved.
- Rotate the triangle 60° about a point: same size and orientation, turned.
- Reflect across a line: same size, mirror image (orientation reversed).
- Dilate by k = 2 about its centroid: triangle twice as large, same angles, centroid fixed.

Summary
- Dilation = similarity (changes scale, preserves shape/angles).
- Translation/reflection/rotation = isometries (preserve distances and area).
- Use dilations when you want to change size but keep shape; use the other three when size must remain unchanged.