Geometry Unit 5: Transformations

Foundations of Geometric Transformations and Rigid Motions

• A transformation is a mathematical function that alters the position, size, or shape of a geometric figure or graph on a coordinate plane.
• The transformation establishes a direct mapping from an original figure, termed the preimage (the initial or "before" configuration), to a resulting figure, termed the image (the final configuration following the transformation).
• Symbolic mapping notation designates a preimage point using a standard uppercase letter (such as $A$) and its corresponding image point using prime notation (such as $A'$). Subsequent successive transformations on the same point are designated with multiple primes (such as $A''$ and $A'''$).
• An isometry, also defined as a rigid motion, is a specific category of transformation that preserves both the size and shape of a geometric figure.
• Rigid motions preserve four fundamental geometric properties:
  • Distance: The lengths of all line segments between corresponding points remain invariant; side lengths are congruent before and after the transformation.
  • Angle Measures: The degrees of all interior and exterior angles remain invariant; corresponding angles are congruent.
  • Collinearity: Points that lie along a common straight line in the preimage remain on a straight line in the image.
  • Betweenness: If a point lies between two other points along a segment in the preimage, its image maintains the exact same relative intermediate position.

Translations and Vector Operations in the Coordinate Plane

• A translation is a rigid motion (isometry) that maps every point of a figure or graph across the exact same distance and in the exact same direction, shifting the entire figure without inducing rotation or reflection.
• A vector is a geometric and algebraic quantity characterized by both magnitude (length or size) and direction. A vector initiates at an initial point and terminates at a terminal (stopping) point.
• In coordinate geometry, translations are formally expressed using vectors written in component form as $\langle a, b \rangle$, where $a$ denotes the directed horizontal change along the $x$-axis and $b$ denotes the directed vertical change along the $y$-axis.
• Functional vector notation denotes the mapping of an arbitrary point $A$ under translation vector $v$ as $T_v(A) = A'$.
• Distinction between points and vectors:
  • An ordered pair written with round parentheses, such as $(4, -2)$, represents a specific, fixed position or location in the coordinate plane.
  • An ordered pair written with angle brackets, such as $\langle 4, -2 \rangle$, defines a displacement vector that indicates an operational shift of $4$ units horizontally and $-2$ units vertically from any given initial point.
  • A purely horizontal vector exhibits zero vertical displacement and is formally expressed in component form as $\langle a, 0 \rangle$.
• Coordinate Plane Translation Rules:
  • Horizontal translation by $a$ units to the right (positive horizontal direction): $(x, y) \rightarrow (x + a, y)$
  • Horizontal translation by $a$ units to the left (negative horizontal direction): $(x, y) \rightarrow (x - a, y)$
  • Vertical translation by $b$ units up (positive vertical direction): $(x, y) \rightarrow (x, y + b)$
  • Vertical translation by $b$ units down (negative vertical direction): $(x, y) \rightarrow (x, y - b)$
• Translation Coordinate Mapping Exercises:
  • Translating a polygon with vertices $A(0, 6)$, $B(4, 8)$, $C(4, 4)$, and $D(0, 2)$ by the displacement vector $\langle 4, -2 \rangle$ produces the image vertices:
  • $A'(0 + 4, 6 - 2) = A'(4, 4)$
  • $B'(4 + 4, 8 - 2) = B'(8, 6)$
  • $C'(4 + 4, 4 - 2) = C'(8, 2)$
  • $D'(0 + 4, 2 - 2) = D'(4, 0)$
  • Mapping preimage coordinates $A(-4, 4)$, $B(-1, 2)$, $C(-2, 0)$, and $D(-5, 1)$ using the vector $\langle 6, -3 \rangle$ generates the image coordinates:
  • $A'(-4 + 6, 4 - 3) = A'(2, 1)$
  • $B'(-1 + 6, 2 - 3) = B'(5, -1)$
  • $C'(-2 + 6, 0 - 3) = C'(4, -3)$
  • $D'(-5 + 6, 1 - 3) = D'(1, -2)$
  • Mapping preimage coordinates $A(-4, 4)$, $B(-1, 2)$, $C(-2, 0)$, and $D(-5, 1)$ using the vector $\langle 6, -4 \rangle$ generates the image coordinates:
  • $A'(-4 + 6, 4 - 4) = A'(2, 0)$
  • $B'(-1 + 6, 2 - 4) = B'(5, -2)$
  • $C'(-2 + 6, 0 - 4) = C'(4, -4)$
  • $D'(-5 + 6, 1 - 4) = D'(1, -3)$
  • Determining an unknown translation vector from preimage vertex $A(-4, 4)$ to image vertex $A'(-6, 9)$: the horizontal shift is $-6 - (-4) = -2$ and the vertical shift is $9 - 4 = 5$, yielding the translation vector $\langle -2, 5 \rangle$.
• Verbal Translation Directives and Corresponding Algebraic Mappings:
  • Shift $4$ units right and $5$ units down: $(x, y) \rightarrow (x + 4, y - 5)$
  • Shift $4$ units right and $3$ units down: $(x, y) \rightarrow (x + 4, y - 3)$
  • Shift $4$ units left and $2$ units down: $(x, y) \rightarrow (x - 4, y - 2)$
  • Shift $3$ units left and $3$ units down: $(x, y) \rightarrow (x - 3, y - 3)$
  • Shift $2$ units up: $(x, y) \rightarrow (x, y + 2)$
  • Shift $3$ units right and $5$ units up: $(x, y) \rightarrow (x + 3, y + 5)$
  • Shift $2$ units left and $1$ unit up: $(x, y) \rightarrow (x - 2, y + 1)$
  • Shift $5$ units left: $(x, y) \rightarrow (x - 5, y)$
  • Applying the rule $(x, y) \rightarrow (x, y + 5)$ to points $H(-3, -2)$ and $A(0, -1)$ produces $H'(-3, 3)$ and $A'(0, 4)$.
  • Applying the rule $(x, y) \rightarrow (x + 4, y - 2)$ translates every coordinate by vector $\langle 4, -2 \rangle$.
  • Applying the translation vector $\langle -3, 1 \rangle$ to a preimage shifts every vertex $3$ units left and $1$ unit up.

Rotations and Angular Coordinate Rules

• A rotation is a rigid motion that turns a geometric figure through a specified angle about a fixed center point $P$.
• Properties of rotational transformations:
  • Every point of the preimage and its corresponding image point maintain the exact same radial distance from the center of rotation $P$.
  • All angles with vertex $P$ formed by connecting a preimage point, the center point $P$, and the corresponding image point are congruent to the angle of rotation.
  • Distance, segment lengths, and angle measures are strictly preserved, confirming rotation as an isometry.
• Directional Conventions:
  • When no direction is explicitly stated, standard mathematical convention assumes a counterclockwise (ccw) rotation.
  • Rotations may proceed either clockwise (cw) or counterclockwise (ccw).
  • Circular angle equivalences about the origin:
  • A $90^\circ$ clockwise rotation is equivalent to a $270^\circ$ counterclockwise rotation.
  • A $180^\circ$ clockwise rotation is equivalent to a $180^\circ$ counterclockwise rotation.
  • A $270^\circ$ clockwise rotation is equivalent to a $90^\circ$ counterclockwise rotation.
  • A $360^\circ$ rotation in either direction returns every point to its original coordinates (the identity transformation).
• Standard Coordinate Rules for Counterclockwise Rotations About the Origin $(0, 0)$:
  • $90^\circ$ Counterclockwise Rotation: $(x, y) \rightarrow (-y, x)$
  • $180^\circ$ Counterclockwise Rotation: $(x, y) \rightarrow (-x, -y)$
  • $270^\circ$ Counterclockwise Rotation: $(x, y) \rightarrow (y, -x)$
  • $360^\circ$ Counterclockwise Rotation: $(x, y) \rightarrow (x, y)$
• Verification via tracing paper: Aligning the corner marked with the origin over $(0, 0)$, copying a triangle's vertices, and executing quarter turns confirms the mapped coordinates ($A \rightarrow A'$, $B \rightarrow B'$, $C \rightarrow C'$) conform to these algebraic sign and coordinate-swap rules.

Reflections and Coordinate Line Inversions

• A reflection is a transformation across a line called the line of reflection.
• Geometric Definition: The line of reflection functions as the perpendicular bisector of every line segment connecting an original preimage point to its corresponding image point.
• Coordinate Rules for Reflections Across Standard Axes and Diagonal Lines:
  • Reflection across the horizontal $x$-axis ($y = 0$): $(x, y) \rightarrow (x, -y)$
  • Reflection across the vertical $y$-axis ($x = 0$): $(x, y) \rightarrow (-x, y)$
  • Reflection across the identity line $y = x$: $(x, y) \rightarrow (y, x)$
  • Reflection across the opposite diagonal line $y = -x$: $(x, y) \rightarrow (-y, -x)$
• Reflection Across Arbitrary Horizontal and Vertical Lines:
  • For reflections across an arbitrary vertical line $x = h$, the horizontal distance from the preimage $x$-coordinate to $h$ is mirrored on the opposite side: $(x, y) \rightarrow (2h - x, y)$.
  • Reflection across $x = 1$: $(x, y) \rightarrow (2(1) - x, y) = (2 - x, y)$
  • Reflection across $x = 5$: $(x, y) \rightarrow (2(5) - x, y) = (10 - x, y)$
  • Reflection across $x = -1$: $(x, y) \rightarrow (2(-1) - x, y) = (-2 - x, y)$
  • For reflections across an arbitrary horizontal line $y = k$, the vertical distance from the preimage $y$-coordinate to $k$ is mirrored on the opposite side: $(x, y) \rightarrow (x, 2k - y)$.
  • Reflection across $y = 1$: $(x, y) \rightarrow (x, 2(1) - y) = (x, 2 - y)$
  • Reflection across $y = -1$: $(x, y) \rightarrow (x, 2(-1) - y) = (x, -2 - y)$
• Coordinate Application on Quadrilateral $PQRS$ with vertices $P(0, 7)$, $Q(4, 6)$, $R(2, 3)$, and $S(-1, 3)$:
  • Reflection across the horizontal line $y = 1$:
  • $P(0, 7) \rightarrow P'(0, 2(1) - 7) = P'(0, -5)$
  • $Q(4, 6) \rightarrow Q'(4, 2(1) - 6) = Q'(4, -4)$
  • $R(2, 3) \rightarrow R'(2, 2(1) - 3) = R'(2, -1)$
  • $S(-1, 3) \rightarrow S'(-1, 2(1) - 3) = S'(-1, -1)$
  • Reflection across the diagonal line $y = -x$:
  • Applying $(x, y) \rightarrow (-y, -x)$ maps $P(0, 7) \rightarrow P'(-7, 0)$, $Q(4, 6) \rightarrow Q'(-6, -4)$, $R(2, 3) \rightarrow R'(-3, -2)$, and $S(-1, 3) \rightarrow S'(-3, 1)$.

Reflectional and Rotational Symmetry

• A geometric figure possesses symmetry if there exists a rigid transformation that maps the figure directly onto itself.
• Reflectional Symmetry (Line Symmetry):
  • A line of symmetry is a line across which a figure can be reflected such that the preimage coincides identically with its image.
  • A line of symmetry divides a two-dimensional figure into two congruent, mirror-image halves.
  • Regular polygons with $n$ sides possess exactly $n$ lines of symmetry:
  • An equilateral triangle ($3$-gon) has $3$ lines of symmetry.
  • A square ($4$-gon) has $4$ lines of symmetry.
  • A regular pentagon ($5$-gon) has $5$ lines of symmetry.
  • A regular hexagon ($6$-gon) has $6$ lines of symmetry.
  • A regular dodecagon ($12$-gon) has $12$ lines of symmetry.
  • In numerical symbology, digits that possess reflectional line symmetry include $0$, $3$, and $8$.
• Rotational Symmetry:
  • A figure possesses rotational symmetry if it can be rotated about its center point by an angle of $180^\circ$ or less such that it maps directly onto itself.
  • The angle of rotational symmetry is defined as the smallest positive angle that rotates the figure onto itself.
  • For any regular polygon with $n$ sides, the fundamental angle of rotational symmetry is calculated using the formula:     $\text{Angle of Rotational Symmetry} = \frac{360^\circ}{n}$
  • Successive angles of rotation up to and including $360^\circ$ are integer multiples of the fundamental angle:
  • Equilateral triangle ($n = 3$): $\frac{360^\circ}{3} = 120^\circ$. Angles of rotation onto itself: $120^\circ$, $240^\circ$, and $360^\circ$.
  • Regular pentagon ($n = 5$): $\frac{360^\circ}{5} = 72^\circ$. Angles of rotation onto itself: $72^\circ$, $144^\circ$, $216^\circ$, $288^\circ$, and $360^\circ$.
  • Regular hexagon ($n = 6$): $\frac{360^\circ}{6} = 60^\circ$. Angles of rotation onto itself: $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, $300^\circ$, and $360^\circ$.
  • Regular dodecagon ($n = 12$): $\frac{360^\circ}{12} = 30^\circ$. Fundamental angle of rotational symmetry is $30^\circ$.
  • Real-world items exhibiting rotational symmetry include sliced circular foods such as pizzas, whole pies, circular donuts, and citrus/orange cross-sections.
• Reconstructing Figures from Symmetry Axes:
  • If the coordinate $x$-axis and $y$-axis are designated as lines of symmetry for an image, a shape defined in one quadrant must be reflected across the $x$-axis, across the $y$-axis, and across the origin (combined reflection across both axes) to generate the complete four-quadrant symmetric figure.

Dilations, Stretches, and Compressions

• A dilation is a non-rigid transformation that changes the size of a geometric figure by a proportional factor in all directions, centered at a specific fixed point.
• Scale Factor ($k$):
  • The scale factor $k$ is defined as the ratio of the image length to the corresponding preimage length:     $k = \frac{\text{Image Length}}{\text{Preimage Length}}$
  • When centered at the coordinate origin $(0, 0)$, a dilation applies the algebraic coordinate rule:     $(x, y) \rightarrow (kx, ky)$
• Classifications of Dilations Based on the Value of $k$:
  • Enlargement: Occurs when $k > 1$. The image is proportionally larger than the preimage.
  • Reduction: Occurs when $0 < k < 1$. The image is proportionally smaller than the preimage.
  • Identity / Congruence: When $k = 1$, the dimensions remain completely unchanged.
• Scale Factor Classification Examples:
  • $k = \frac{1}{3} \implies$ Reduction ($0 < \frac{1}{3} < 1$)
  • $k = \frac{5}{4} \implies$ Enlargement ($\frac{5}{4} = 1.25 > 1$)
  • $k = 0.93 \implies$ Reduction ($0 < 0.93 < 1$)
  • $k = 2 \implies$ Enlargement ($2 > 1$)
  • $k = 1.5 \implies$ Enlargement ($1.5 > 1$)
  • $k = \frac{1}{2} \implies$ Reduction ($0 < \frac{1}{2} < 1$)
  • $k = \frac{1}{4} \implies$ Reduction ($0 < \frac{1}{4} < 1$)
• Calculating Unknown Scale Factors from Preimage and Image Vertex Coordinates:
  • Preimage $A(3, 4)$ mapped to Image $R(9, 12)$: $k = \frac{9}{3} = \frac{12}{4} = 3$ (Enlargement)
  • Preimage $A(-9, 12)$ mapped to Image $R(6, 8)$: $k = \frac{6}{-9} = \frac{8}{12} = -\frac{2}{3}$; with positive magnitude, $k = \frac{2}{3}$ (Reduction)
  • Preimage $A(-2, -3)$ mapped to Image $R(-10, -15)$: $k = \frac{-10}{-2} = \frac{-15}{-3} = 5$ (Enlargement)
• Computing Image Coordinates of Dilated Segments and Figures:
  • Line segment with endpoints $A(4, 4)$ and $B(8, 12)$ dilated by scale factor $k = \frac{3}{4}$:
  • $A'(4 \times \frac{3}{4}, 4 \times \frac{3}{4}) = A'(3, 3)$
  • $B'(8 \times \frac{3}{4}, 12 \times \frac{3}{4}) = B'(6, 9)$
  • Line segment with endpoints $A(0, 0)$ and $B(-3, 2)$ dilated by scale factor $k = 5$:
  • $A'(0 \times 5, 0 \times 5) = A'(0, 0)$
  • $B'(-3 \times 5, 2 \times 5) = B'(-15, 10)$
  • Quadrilateral $ABCD$ with vertices $A(-4, -2)$, $B(-6, -4)$, $C(-4, -6)$, and $D(-2, -4)$ dilated by scale factor $k = \frac{1}{2}$ centered at the origin:
  • $A'(-4 \times \frac{1}{2}, -2 \times \frac{1}{2}) = A'(-2, -1)$
  • $B'(-6 \times \frac{1}{2}, -4 \times \frac{1}{2}) = B'(-3, -2)$
  • $C'(-4 \times \frac{1}{2}, -6 \times \frac{1}{2}) = C'(-2, -3)$
  • $D'(-2 \times \frac{1}{2}, -4 \times \frac{1}{2}) = D'(-1, -2)$
  • Polygon with vertices $A(-5, 5)$, $B(-5, 10)$, and $C(10, 0)$ dilated by scale factor $k = \frac{3}{5}$:
  • $A'(-5 \times \frac{3}{5}, 5 \times \frac{3}{5}) = A'(-3, 3)$
  • $B'(-5 \times \frac{3}{5}, 10 \times \frac{3}{5}) = B'(-3, 6)$
  • $C'(10 \times \frac{3}{5}, 0 \times \frac{3}{5}) = C'(6, 0)$
• Stretches and Compressions (Distortion Transformations):
  • Unlike dilations, which scale both coordinate axes equally and preserve geometric similarity, stretches and compressions scale axes disproportionately, altering the shape of the figure.
  • Horizontal Stretch: Expands the horizontal coordinate by a factor $a > 1$ while keeping the vertical coordinate constant: $(x, y) \rightarrow (ax, y)$ (for example, $(x, y) \rightarrow (3x, y)$).
  • Horizontal Compression: Contracts the horizontal coordinate by a factor $0 < a < 1$: $(x, y) \rightarrow (ax, y)$.
  • Vertical Stretch: Expands the vertical coordinate by a factor $b > 1$ while keeping the horizontal coordinate constant: $(x, y) \rightarrow (x, by)$ (for example, $(x, y) \rightarrow (x, 3y)$).
  • Vertical Compression: Contracts the vertical coordinate by a factor $0 < b < 1$: $(x, y) \rightarrow (x, by)$ (for example, $(x, y) \rightarrow (x, 0.3y)$).
• Rigid Versus Non-Rigid Classification Summary:
  • Translation: $(x, y) \rightarrow (x + 3, y)$ (Rigid transformation; preserves size and shape)
  • Dilation: $(x, y) \rightarrow (3x, 3y)$ (Non-rigid transformation; preserves shape but changes size)
  • Horizontal Stretch: $(x, y) \rightarrow (3x, y)$ (Non-rigid transformation; alters shape and size)
  • Vertical Stretch: $(x, y) \rightarrow (x, 3y)$ (Non-rigid transformation; alters shape and size)
  • Vertical Compression: $(x, y) \rightarrow (x, 0.3y)$ (Non-rigid transformation; alters shape and size)
  • $270^\circ$ Rotation: $(x, y) \rightarrow (y, -x)$ (Rigid transformation; preserves size and shape)

Multi-Step Composite Transformations and Coordinate Paths

• A composite transformation is a sequence of two or more successive geometric transformations executed such that the output image of the preceding transformation serves as the input preimage for the succeeding transformation.
• Multi-Step Transformation Paths (Treasure Hunt Scenarios):
  • Route A:
  • Step 0: Start at the origin $(0, 0)$.
  • Step 1: Slide right $3$ steps: $(0 + 3, 0) = (3, 0)$.
  • Step 2: Move up $2$ steps: $(3, 0 + 2) = (3, 2)$.
  • Step 3: Rotate $90^\circ$ counterclockwise ($(x, y) \rightarrow (-y, x)$, centered at origin): $(-2, 3)$.
  • Step 4: Slide left $1$ step: $(-2 - 1, 3) = (-3, 3)$.
  • Step 5: Reflect across the $y$-axis ($(x, y) \rightarrow (-x, y)$): $(-(-3), 3) = (3, 3)$.
  • Final Position for Route A: $(3, 3)$.
  • Route B:
  • Step 0: Start at the origin $(0, 0)$.
  • Step 1: Step up $2$ steps: $(0, 2)$.
  • Step 2: Slide left $3$ steps: $(-3, 2)$.
  • Step 3: Reflect across the line $y = -x$ ($(x, y) \rightarrow (-y, -x)$): $(-2, -(-3)) = (-2, 3)$.
  • Step 4: Rotate $180^\circ$ ($(x, y) \rightarrow (-x, -y)$): $(2, -3)$.
  • Step 5: Move right $7$ steps: $(2 + 7, -3) = (9, -3)$.
  • Final Position for Route B: $(9, -3)$.
  • Route C:
  • Step 0: Start at the origin $(0, 0)$.
  • Step 1: Slide down $4$ steps: $(0, -4)$.
  • Step 2: Rotate $90^\circ$ counterclockwise: $(-(-4), 0) = (4, 0)$.
  • Step 3: Move down $4$ steps: $(4, 0 - 4) = (4, -4)$.
  • Step 4: Reflect across the $y$-axis: $(-4, -4)$.
  • Step 5: Reflect across the line $y = -x$: $(-(-4), -(-4)) = (4, 4)$.
  • Final Position for Route C: $(4, 4)$.
  • Route D:
  • Step 0: Start at the origin $(0, 0)$.
  • Step 1: Move left $4$ steps: $(-4, 0)$.
  • Step 2: Rotate $360^\circ$: $(-4, 0)$.
  • Step 3: Move up $2$ steps: $(-4, 2)$.
  • Step 4: Move right $7$ steps: $(-4 + 7, 2) = (3, 2)$.
  • Step 5: Reflect across the line $y = -x$: $(-2, -3)$.
  • Final Position for Route D: $(-2, -3)$.
  • Spicy Treasure Hunt ($10$-Step Composite Sequence):
  • Step 0: Initial starting position at $(0, 0)$.
  • Step 1: Slide right $3$ steps: $(3, 0)$.
  • Step 2: Move up $2$ steps: $(3, 2)$.
  • Step 3: Rotate $90^\circ$ counterclockwise: $(-2, 3)$.
  • Step 4: Slide left $1$ step: $(-3, 3)$.
  • Step 5: Reflect across the $y$-axis: $(3, 3)$.
  • Step 6: Rotate $270^\circ$ counterclockwise ($(x, y) \rightarrow (y, -x)$): $(3, -3)$.
  • Step 7: Move up $4$ steps: $(3, -3 + 4) = (3, 1)$.
  • Step 8: Reflect across the line $y = x$ ($(x, y) \rightarrow (y, x)$): $(1, 3)$.
  • Step 9: Rotate $180^\circ$ ($(x, y) \rightarrow (-x, -y)$): $(-1, -3)$.
  • Step 10: Move down $2$ steps: $(-1, -3 - 2) = (-1, -5)$.
  • Final Terminus for Spicy Sequence: $(-1, -5)$.
• Three-Stage Polygon Composite Verification on Triangle $TAG$ with vertices $T(-4, 1)$, $A(-5, 0)$, and $G(-1, -2)$:
  • Part A: Translation using the displacement rule $\langle 4, 1 \rangle$ yielding $T'A'G'$:
  • $T'(-4 + 4, 1 + 1) = T'(0, 2)$
  • $A'(-5 + 4, 0 + 1) = A'(-1, 1)$
  • $G'(-1 + 4, -2 + 1) = G'(3, -1)$
  • Part B: Reflection of $T'A'G'$ across the $y$-axis ($(x, y) \rightarrow (-x, y)$) yielding $T''A''G''$:
  • $T''(0, 2)$
  • $A''(1, 1)$
  • $G''(-3, -1)$
  • Part C: Clockwise rotation of $90^\circ$ of $T''A''G''$ about the origin ($(x, y) \rightarrow (y, -x)$) yielding $T'''A'''G'''$:
  • $T'''(2, 0)$
  • $A'''(1, -1)$
  • $G'''(-1, 3)$

Practical Spatial Modeling: Dorm Room Design Project

• Spatial Modeling Parameters:
  • A college dormitory room is modeled as a closed coordinate grid measuring $10 \times 10$ units, incorporating an entry doorway location.
  • Furniture pieces are modeled as two-dimensional geometric polygons defined by four coordinate vertices ($A$, $B$, $C$, and $D$) matching specific spatial footprint dimensions:
  • Bed (Cama): Rectangular footprint of $4 \times 6$ units.
  • Desk (Escritorio): Rectangular footprint of $2 \times 4$ units.
  • Chair (Silla): Square footprint of $2 \times 2$ units.
  • Dresser (Cómoda): Rectangular footprint of $3 \times 2$ units.
  • Bookcase #1 (Librería #1): Rectangular footprint.
  • Bookcase #2 (Librería #2): Rectangular footprint.
• Layout Transformations (Semester Reorganization Workflow):
  • Initial semester arrangement: Each piece of furniture is placed at initial coordinate boundaries defined by vertex coordinates $A(x_A, y_A)$, $B(x_B, y_B)$, $C(x_C, y_C)$, and $D(x_D, y_D)$.
  • Second semester reorganization: To refresh room flow without lifting heavy components, furniture is repositioned across the floor plane.
  • Every spatial rearrangement is formally documented using geometric transformations:
  • Translations: Sliding a piece along a displacement vector $\langle a, b \rangle$.
  • Rotations: Turning a piece by an angle of $90^\circ$, $180^\circ$, or $270^\circ$ around a pivot vertex to adjust alignment with room walls.
  • Reflections: Flipping the orientation of an item across a reference axis to mirror layout configurations.
  • For each furniture unit, mathematical bookkeeping records the Preimage vertices ($A, B, C, D$) and the corresponding new Image vertices ($A', B', C', D'$).