10.1 Rigid Transformations: Translations, Reflections, Rotations & Symmetry

Foundations of Isometries and Rigid Transformations

In Euclidean plane geometry, a transformation is a bijective (one-to-one and onto) mapping $T: \mathbb{R}^2 \to \mathbb{R}^2$ that assigns every point $P(x, y)$ in the plane to a unique image point $P'(x', y')$. The original geometric figure is termed the pre-image, and the resulting figure following the transformation is termed the image.

A transformation is classified as an isometry (from the Greek isos, meaning "equal", and metron, meaning "measure"), or a rigid transformation (rigid motion), if and only if it preserves Euclidean distance between all pairs of points. Formally, for any two points $P$ and $Q$ and their corresponding images $P' = T(P)$ and $Q' = T(Q)$:

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

Invariant Properties Under Isometry

Because isometries strictly preserve Euclidean length, they preserve several foundational geometric attributes without alteration:

1. Distance (Length): The length of any line segment equals the length of its image segment ($AB = A'B'$).
2. Angle Measure: The measure of any interior or exterior angle is invariant ($m\angle ABC = m\angle A'B'C'$).
3. Collinearity: If three points $A$, $B$, and $C$ lie on a single straight line, their images $A'$, $B'$, and $C'$ lie on a single straight line.
4. Betweenness: If point $B$ lies between points $A$ and $C$ on a line segment, $B'$ lies between $A'$ and $C'$.
5. Parallelism: If line $\ell_1$ is parallel to line $\ell_2$, their images $\ell_1'$ and $\ell_2'$ are parallel.
6. Perpendicularity: If segment $\overline{AB} \perp \overline{CD}$, then $\overline{A'B'} \perp \overline{C'D'}$.
7. Area and Perimeter: Enclosed two-dimensional area and boundary perimeter are strictly invariant.

Because all corresponding side lengths and angle measures remain identical, an isometry guarantees that any pre-image polygon is strictly congruent to its image ($\Delta ABC \cong \Delta A'B'C'$). Rigid transformations serve as the modern transformational foundation for Euclidean congruence.

Direct vs. Opposite Isometries

Isometries are bifurcated into two topological categories based on orientation (the clockwise or counterclockwise cyclic ordering of vertices around a figure):

• Direct Isometries (Proper Motions): Preserve vertex orientation. If vertices $A \to B \to C$ traverse clockwise in the pre-image, $A' \to B' \to C'$ traverse clockwise in the image. Translations and rotations are direct isometries.
• Opposite Isometries (Improper Motions): Reverse vertex orientation. A clockwise pre-image yields a counterclockwise image. Reflections and glide reflections are opposite isometries.

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Translations: Vector Representation and Coordinate Rules

A translation is a direct isometry that displaces every point in the plane by a constant Euclidean distance in a specified direction. A translation is defined by a displacement vector $\vec{v} = \langle h, k \rangle$, where $h$ denotes horizontal shift along the $x$-axis and $k$ denotes vertical shift along the $y$-axis.

Algebraic Coordinate Mapping Rule

For a translation along vector $\vec{v} = \langle h, k \rangle$:

$$T_{\langle h, k \rangle}(x, y) = (x + h, y + k)$$

• If $h > 0$, the figure shifts right by $h$ units; if $h < 0$, it shifts left by $|h|$ units.
• If $k > 0$, the figure shifts upward by $k$ units; if $k < 0$, it shifts downward by $|k|$ units.

Fundamental Geometric Invariants of Translations

1. Parallel Vector Paths: For every point $P$, the directed segment $\overline{PP'}$ connecting pre-image to image is parallel to $\vec{v}$, with length equal to $|\vec{v}| = \sqrt{h^2 + k^2}$.
2. Parallelism of Segments: Any segment $\overline{AB}$ is strictly parallel to its translated image $\overline{A'B'}$ (unless the segment is collinear with the translation direction).
3. Fixed Points: A non-zero translation ($h \neq 0$ or $k \neq 0$) has zero fixed points; no point in the plane maps onto itself.

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Reflections: Lines of Reflection and Coordinate Mapping Rules

A reflection across a fixed line $m$ (the line of reflection or mirror line) is an opposite isometry that maps each point $P$ to a point $P'$ such that:

• If $P$ lies on line $m$, then $P' = P$ (points on the line of reflection are invariant fixed points).
• If $P$ does not lie on line $m$, then line $m$ is the perpendicular bisector of segment $\overline{PP'}$. This requires that line $m \perp \overline{PP'}$ and that the midpoint $M = \left(\frac{x_P + x_{P'}}{2}, \frac{y_P + y_{P'}}{2}\right)$ lies on line $m$.

Standard Reflection Coordinate Rules

1. Reflection Across the $x$-axis ($y = 0$):

   $$r_{x\text{-axis}}(x, y) = (x, -y)$$

   The horizontal coordinate remains unchanged; the vertical coordinate is negated.

2. Reflection Across the $y$-axis ($x = 0$):

   $$r_{y\text{-axis}}(x, y) = (-x, y)$$

   The vertical coordinate remains unchanged; the horizontal coordinate is negated.

3. Reflection Across the Line $y = x$:

   $$r_{y = x}(x, y) = (y, x)$$

   The coordinates exchange positions. This operation corresponds directly to computing the inverse of a mathematical relation.

4. Reflection Across the Line $y = -x$:

   $$r_{y = -x}(x, y) = (-y, -x)$$

   The coordinates exchange positions and both undergo sign negation.

5. Reflection Across Arbitrary Horizontal Line $y = c$:

   $$r_{y = c}(x, y) = (x, 2c - y)$$

   Because $c$ is the midpoint of $y$ and $y'$: $\frac{y + y'}{2} = c \implies y' = 2c - y$.

6. Reflection Across Arbitrary Vertical Line $x = c$:

   $$r_{x = c}(x, y) = (2c - x, y)$$

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Rotations: Centers, Angles, and Standard Counterclockwise Rules

A rotation $R_{(C, \theta)}$ is a direct isometry centered at a fixed point $C$ through a directed angle $\theta$. By mathematical convention, a positive angle $\theta > 0$ represents a counterclockwise (CCW) rotation, while a negative angle $\theta < 0$ represents a clockwise (CW) rotation.

Geometric Conditions for Rotation

For every point $P$ and its rotated image $P'$ about center $C$:

1. $CP = CP'$ (the distance from the center of rotation to the pre-image point equals the distance from the center to the image point).
2. $m\angle PCP' = |\theta|$.
3. The center of rotation $C$ is the unique fixed point of the transformation whenever $\theta$ is not an integer multiple of $360^\circ$.

Standard Counterclockwise Coordinate Rules About the Origin $(0, 0)$

1. $90^\circ$ Counterclockwise Rotation ($270^\circ$ Clockwise):

   $$R_{(0, 90^\circ)}(x, y) = (-y, x)$$

   Notice that the original quadrant $(x, y)$ transitions counterclockwise: Quadrant I $(+, +) \to$ Quadrant II $(-, +)$.

2. $180^\circ$ Rotation (Half-Turn / Point Reflection Through Origin):

   $$R_{(0, 180^\circ)}(x, y) = (-x, -y)$$

   Rotating $180^\circ$ counterclockwise produces the identical image as rotating $180^\circ$ clockwise. Both coordinates are negated.

3. $270^\circ$ Counterclockwise Rotation ($90^\circ$ Clockwise):

   $$R_{(0, 270^\circ)}(x, y) = (y, -x)$$

Rotation About an Arbitrary Center $(x_0, y_0)$

When the center of rotation is not the origin, apply a three-step algorithmic composition:

1. Translate the center $(x_0, y_0)$ to the origin: $(x - x_0, y - y_0)$.
2. Apply the standard origin rotation rule to the shifted coordinates.
3. Translate back to the original center: add $(x_0, y_0)$.

For a $90^\circ$ CCW rotation about $(x_0, y_0)$:

$$(x, y) \mapsto (x - x_0, y - y_0) \mapsto (-(y - y_0), x - x_0) \mapsto (x_0 - (y - y_0), y_0 + (x - x_0))$$

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Glide Reflections and Compositions of Rigid Motions

A glide reflection is the composite transformation formed by performing a reflection across a line $m$ followed by a translation along a vector $\vec{v}$ that is strictly parallel to line $m$:

$$G = T_{\vec{v}} \circ r_m \quad \text{where } \vec{v} \parallel m$$

Because the translation is parallel to the reflection line, the operations commute: $T_{\vec{v}} \circ r_m = r_m \circ T_{\vec{v}}$. A glide reflection is an opposite isometry that possesses no fixed points (unless $\vec{v} = \vec{0}$, which degenerates to a pure reflection).

Foundational Composition Theorems

According to the Cartan-Dieudonné Theorem, every rigid motion in the two-dimensional Euclidean plane can be generated by the composition of at most three reflections across lines:

1. Composition of Two Reflections Across Parallel Lines: Let lines $\ell_1$ and $\ell_2$ be parallel lines separated by perpendicular distance $d$. The composition $r_{\ell_2} \circ r_{\ell_1}$ is equivalent to a pure translation perpendicular to the lines by a distance of $2d$ in the direction from $\ell_1$ toward $\ell_2$:

   $$r_{\ell_2} \circ r_{\ell_1} = T_{\langle 2d, 0 \rangle} \quad (\text{for vertical lines separated by } d)$$

2. Composition of Two Reflections Across Intersecting Lines: Let lines $\ell_1$ and $\ell_2$ intersect at point $C$ with an acute/obtuse angle $\alpha$ between them. The composition $r_{\ell_2} \circ r_{\ell_1}$ is equivalent to a pure rotation centered at $C$ through directed angle $2\alpha$ in the direction from $\ell_1$ toward $\ell_2$:

   $$r_{\ell_2} \circ r_{\ell_1} = R_{(C, 2\alpha)}$$

3. Composition of Three Reflections Across Non-Concurrent Lines: Produces either a single reflection or a glide reflection.

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Line and Rotational Symmetry in Polygons

A symmetry of a geometric figure is an isometry that maps the figure onto itself.

Line (Reflectional) Symmetry

A figure has line symmetry (or reflectional symmetry) if there exists a line $m$ such that reflecting the figure across $m$ leaves the figure invariant ($r_m(F) = F$). Line $m$ is termed the axis of symmetry or line of reflection.

Rotational Symmetry

A figure has rotational symmetry if there exists a non-trivial counterclockwise angle $\theta$ ($0^\circ < \theta < 360^\circ$) about a central point $C$ such that rotating the figure leaves it invariant ($R_{(C, \theta)}(F) = F$).

• Order of Rotational Symmetry ($n$): The total number of distinct positions (including the full $360^\circ$ rotation) in which the figure matches its pre-image orientation. A figure possesses rotational symmetry if and only if $n \ge 2$.
• Fundamental Angle of Rotation: The smallest positive angle through which the figure must be rotated to coincide with itself: $$\theta_{\text{min}} = \frac{360^\circ}{n}$$
• Point Symmetry: A special case of rotational symmetry where the order is 2 and the angle of rotation is $180^\circ$. A figure with point symmetry looks identical right-side-up and upside-down.

Symmetries of Common Geometric Polygons

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Reference Summary of Coordinate Transformation Rules

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Worked Step-by-Step Examples

Worked Example 1: Multi-Step Composition of Rigid Transformations

Problem: Triangle $ABC$ has vertices $A(2, 5)$, $B(6, 1)$, and $C(4, -2)$. The triangle undergoes a two-step composite transformation:

1. First, reflection across the line $y = -x$.
2. Second, a $90^\circ$ counterclockwise rotation about the origin. Determine the coordinates of the final image vertices $A''$, $B''$, and $C''$, and determine whether the overall composition preserves or reverses orientation.

Solution: Step 1: Apply the reflection rule across $y = -x$, which maps $(x, y) \mapsto (-y, -x)$:

• $A(2, 5) \mapsto A'(-5, -2)$
• $B(6, 1) \mapsto B'(-1, -6)$
• $C(4, -2) \mapsto C'( -(-2), -4) = C'(2, -4)$

Step 2: Apply the $90^\circ$ counterclockwise rotation rule about the origin, which maps $(x', y') \mapsto (-y', x')$:

• $A'(-5, -2) \mapsto A''(-(-2), -5) = A''(2, -5)$
• $B'(-1, -6) \mapsto B''(-(-6), -1) = B''(6, -1)$
• $C'(2, -4) \mapsto C''(-(-4), 2) = C''(4, 2)$

Step 3: Analyze orientation and single-transformation equivalence: The first transformation is an opposite isometry (reverses orientation). The second transformation is a direct isometry (preserves orientation). The composition of an opposite isometry and a direct isometry is an opposite isometry (orientation is reversed). Comparing $(x, y)$ to $(x'', y'')$: $(2, 5) \mapsto (2, -5)$, $(6, 1) \mapsto (6, -1)$, and $(4, -2) \mapsto (4, 2)$. The mapping rule is $(x, y) \mapsto (x, -y)$, which represents a single reflection across the $x$-axis.

Worked Example 2: Symmetries of a Regular Octagon

Problem: A stop sign is modeled as a regular octagon. Determine:

1. The total number of lines of reflectional symmetry.
2. The order of rotational symmetry.
3. The minimum positive counterclockwise angle of rotation that maps the octagon onto itself.
4. Whether rotating the sign by $225^\circ$ maps the sign onto itself.

Solution: Step 1: For any regular $n$-gon, the number of lines of symmetry equals $n$. For a regular octagon ($n = 8$), there are exactly 8 lines of symmetry (4 passing through opposite pairs of vertices, and 4 passing through opposite midpoints of sides).

Step 2: The order of rotational symmetry for a regular $n$-gon is $n = 8$.

Step 3: The fundamental angle of rotation is:

$$\theta_{\text{min}} = \frac{360^\circ}{n} = \frac{360^\circ}{8} = 45^\circ$$

Step 4: Check if $225^\circ$ is an integer multiple of $45^\circ$:

$$\frac{225^\circ}{45^\circ} = 5$$

Because $225^\circ = 5 \times 45^\circ$, a rotation of $225^\circ$ represents 5 fundamental rotational increments and therefore maps the octagon exactly onto itself.

Worked Example 3: Composition of Two Reflections Across Parallel Lines

Problem: A point $P(3, 4)$ is reflected across the vertical line $x = 1$ to produce point $P'$, and $P'$ is subsequently reflected across the vertical line $x = 5$ to produce point $P''$. Find the coordinates of $P''$ and identify the single equivalent transformation.

Solution: Step 1: Reflect $P(3, 4)$ across line $x = 1$. Using $x' = 2c - x$ with $c = 1$:

$$x' = 2(1) - 3 = 2 - 3 = -1, \quad y' = 4 \implies P'(-1, 4)$$

Step 2: Reflect $P'(-1, 4)$ across line $x = 5$. Using $x'' = 2c - x'$ with $c = 5$:

$$x'' = 2(5) - (-1) = 10 + 1 = 11, \quad y'' = 4 \implies P''(11, 4)$$

Step 3: Analyze the single equivalent transformation: The pre-image $P(3, 4)$ mapped to $P''(11, 4)$. The vertical coordinate is unchanged, and the horizontal coordinate increased by $11 - 3 = 8$ units. The distance between the parallel lines $x = 1$ and $x = 5$ is $d = 5 - 1 = 4$ units. By the parallel reflection composition theorem, reflecting across two parallel lines separated by distance $d$ produces a translation of $2d = 2(4) = 8$ units in the direction from the first line toward the second (positive $x$-direction). The composite transformation is $T_{\langle 8, 0 \rangle}$.

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Diagnostic Misconceptions & Pedagogical Strategies

1. Assuming Parallelograms Have Reflectional Line Symmetry Along Diagonals: Students routinely assume that because the diagonal of a parallelogram divides it into two congruent triangles, the diagonal must be a line of symmetry. Pedagogical remedy: Have students fold a paper parallelogram along its diagonal. They immediately observe that the overlapping triangular flaps point in opposite directions and do not match up. Clarify that while the triangles are congruent, the reflection across the diagonal sends the opposite vertex outside the figure. A parallelogram possesses rotational symmetry of order 2 ($180^\circ$), not line symmetry (unless it is a rhombus or rectangle).
2. Confusion Between Clockwise and Counterclockwise Sign Conventions: In Cartesian trigonometry and transformational geometry, positive angles represent counterclockwise rotation, whereas students naturally associate positive rotation with clockwise clock hands. Emphasize that rotating from the positive $x$-axis toward the positive $y$-axis (Quadrant I to Quadrant II) defines a positive (counterclockwise) rotation.
3. Non-Commutative Nature of Compositions: Students frequently assume that transformations commute ($T_1 \circ T_2 = T_2 \circ T_1$). For example, translating 4 units right then reflecting across the $y$-axis yields $(-x - 4, y)$, whereas reflecting across the $y$-axis first then translating 4 units right yields $(-x + 4, y)$. Explicitly illustrate composite order using arrow diagrams and coordinate tracking.