8.2 Rigid Motions: Translations, Reflections, Rotations & Symmetry

8.2 Rigid Motions: Translations, Reflections, Rotations & Symmetry

1. Isometries and Invariant Geometric Properties

In transformational geometry, an isometry (or rigid motion) is a distance-preserving bijective mapping $T: \mathbb{R}^2 \to \mathbb{R}^2$. For any two points $P$ and $Q$:

$$d(T(P), T(Q)) = d(P, Q)$$

Because distance is invariant under an isometry, several key geometric properties are preserved:

1. Segment Length: Segments map to congruent segments ($\overline{AB} \cong \overline{A'B'}$).
2. Angle Measure: Angles map to congruent angles ($\angle ABC \cong \angle A'B'C'$).
3. Collinearity and Betweenness: Points on a line map to points on a line, preserving betweenness. Parallel lines map to parallel lines.
4. Area and Perimeter: Polygons preserve perimeter and two-dimensional area.

Direct vs. Opposite Isometries

Isometries are classified by their effect on orientation (handedness):

• Direct Isometries (Proper): Preserve the clockwise or counterclockwise cyclic ordering of vertices. Examples: translations and rotations.
• Opposite Isometries (Improper): Reverse cyclic ordering, turning clockwise orientations counterclockwise. Examples: reflections and glide reflections.

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2. Coordinate Mapping Rules for Fundamental Rigid Motions

In the Cartesian plane, rigid motions follow explicit coordinate mapping rules.

Translations

A translation shifts every point by displacement vector $\vec{v} = \langle a, b \rangle$:

$$T_{\langle a, b \rangle}(x, y) = (x + a, y + b)$$

Translations have no fixed points unless $\vec{v} = \langle 0, 0 \rangle$. Segments connecting pre-image points to image points are parallel and equal in length to $\vec{v}$.

Reflections

A reflection flips points across a line of reflection $m$. Points on $m$ are fixed ($T(P) = P$). For $P \notin m$, line $m$ is the perpendicular bisector of $\overline{PP'}$.

• Over $x$-axis ($y = 0$): $r_{x\text{-axis}}(x, y) = (x, -y)$
• Over $y$-axis ($x = 0$): $r_{y\text{-axis}}(x, y) = (-x, y)$
• Over $y = x$: $r_{y=x}(x, y) = (y, x)$
• Over $y = -x$: $r_{y=-x}(x, y) = (-y, -x)$
• Over vertical line $x = h$: $r_{x=h}(x, y) = (2h - x, y)$
• Over horizontal line $y = k$: $r_{y=k}(x, y) = (x, 2k - y)$

Rotations

A rotation turns points by angle $\theta$ about a center of rotation. By convention, positive angles represent counterclockwise (CCW) rotation.

• $90^\circ$ CCW Rotation ($270^\circ$ CW): $R_{90^\circ}(x, y) = (-y, x)$
• $180^\circ$ Rotation: $R_{180^\circ}(x, y) = (-x, -y)$
• $270^\circ$ CCW Rotation ($90^\circ$ CW): $R_{270^\circ}(x, y) = (y, -x)$
• General Angle $\theta$ CCW: $R_\theta(x, y) = (x \cos\theta - y \sin\theta, \; x \sin\theta + y \cos\theta)$

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3. Compositions of Rigid Motions & Structural Theorems

Compositions of transformations are denoted $(T_2 \circ T_1)(P) = T_2(T_1(P))$, applied right to left.

Reflections Across Parallel Lines: The Translation Theorem

Reflecting across two parallel lines $L_1$ and $L_2$ separated by distance $d$ produces a pure translation:

$$r_{L_2} \circ r_{L_1} = T_{\vec{v}}$$

The translation vector $\vec{v}$ is perpendicular to the lines, directed from $L_1$ toward $L_2$, with magnitude equal to twice the distance between the lines: $\|\vec{v}\| = 2d$.

Reflections Across Intersecting Lines: The Rotation Theorem

Reflecting across two lines $L_1$ and $L_2$ intersecting at point $C$ at angle $\theta$ produces a pure rotation:

$$r_{L_2} \circ r_{L_1} = R_{C, 2\theta}$$

The rotation is centered at intersection point $C$ through angle $2\theta$, directed from $L_1$ toward $L_2$.

Glide Reflections

A glide reflection combines a reflection across line $m$ with a non-zero translation vector $\vec{v}$ strictly parallel to $m$:

$$G = T_{\vec{v}} \circ r_m = r_m \circ T_{\vec{v}}$$

Because the vector is parallel to the line, translation and reflection commute. A glide reflection reverses orientation and has no fixed points.

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4. Line Symmetry and Rotational Symmetry

A figure exhibits symmetry if a non-trivial isometry maps the figure onto itself.

Line (Reflectional) Symmetry

A figure has line symmetry if reflection across line $m$ leaves the figure invariant: $r_m(\mathcal{F}) = \mathcal{F}$.

• Isosceles triangle: $1$ line of symmetry.
• Equilateral triangle: $3$ lines of symmetry.
• Rectangle (non-square): $2$ lines of symmetry (connecting opposite midpoints).
• Square: $4$ lines of symmetry (2 medians, 2 diagonals).
• Any regular $n$-gon has $n$ lines of symmetry.

Rotational Symmetry

A figure has rotational symmetry if a rotation about a center by angle $\alpha$ ($0^\circ < \alpha < 360^\circ$) maps the figure onto itself.

• Order ($n$): The number of times the figure matches itself in a full $360^\circ$ rotation.
• Magnitude: The smallest positive angle of rotational symmetry: $$\text{Magnitude} = \frac{360^\circ}{n}$$

For a regular octagon ($n = 8$), the order is $8$ and magnitude is $\frac{360^\circ}{8} = 45^\circ$. A figure with order 2 ($180^\circ$) rotational symmetry possesses point symmetry about its center.