Transformation Matrices

Last updated Aug 15, 2023 Edit Source

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Textbook: Cambridge - Chapter 16

• Transformations in the 2D plane are ways of systematically changing all points in the plane ($x,\\ y$) to a new point in the plane ($x’,\\ y’$)
• Linear transformations have the rule $(x,\\ y) → (ax + by,\\ cx + dy)$

• Each ordered pair can be written as a $2\\ ×\\ 1$ matrix also known as a column vector
  • $(x,\\ y) = [{x \atop y}]$
  • Now we can rewrite the linear transformation rule: $(x, y) → (ax + by,\\ cx + dy)$ using matrix multiplication:
    • $[{x’ \atop y’}] = A[{x \atop y}] = [{a \atop c}{b \atop d}][{x \atop y}] = [{ax\\ +\\ by \atop cx\\ +\\ dy}]$
  • Matrix transformations will involve multiplying ($x,\\ y$) by a $2\\ ×\\ 2$ matrix
    • The resultant will still be a $2\\ ×\\ 1$ matrix allowing it to be plotted

• The unit square has vertices $(0, 0), (1, 0), (0, 1)$ and $(1, 1)$
• The effect of a linear transformation can be seen by looking at the image of the unit square while applying the transformation of a matrix
  • Check out → Desmos Unit Square
• In the matrix of a linear transformation:
  • The first column is the image of $(1, 0)$, written as a column vector $[{1 \atop 0}]$
  • The second column is the image of $(0, 1)$, written as a column vector $[{0 \atop 1}]$

• Since we can transform shapes on a 2D plain using matrices, there are several important transformations that are geometric in nature
• Important note: shear transformations are not required in the course, however they are an interesting and useful transformation
  • Read about shears here: Cambridge - Shears
• Matrix multiplication can be thought as applying one linear transformation and then another
  • This is why the order in matrix multiplication matters
  • The transformations happen from right to left
    • e.g. $[{1 \atop 0}{0 \atop -1}][{3 \atop 0}{0 \atop 1}]$ applies the transformation of the matrix $[{3 \atop 0}{0 \atop 1}]$ first and then the transformation of $[{1 \atop 0}{0 \atop -1}]$ afterwards
    • Think of it as a transformation of the plane as seen in the 3Blue1Brown video in matrices

• Reflection over a line maps each point in the plane to its mirror image on the other side of the line
  • Reflection over the $x$-axis and $y$-axis
    • A reflection in the $x$-axis is defined by $(x,\\ y) → (x,\\ −y)$
      • This can be represented in matrix multiplication: $[{x’ \atop y’}] = [{1 \atop 0}{\\ 0 \atop -1}][{x \atop y}]$
    • A reflection in the y-axis is defined by $(x,\\ y) → (−x,\\ y)$
      • This can be represented in matrix multiplication: $[{x’ \atop y’}] = [{-1 \atop 0}{\\ 0 \atop 1}][{x \atop y}]$
  • Reflection in the line $y = x$ and $y = -x$
    • A reflection in the line $y = x$ is defined by $(x,\\ y) → (y,\\ x)$
      • This can be represented in matrix multiplication: $[{x’ \atop y’}] = [{0 \atop 1}{\\ 1 \atop 0}][{x \atop y}]$ 250
    • A reflection in the line $y = -x$ is defined by $(x,\\ y) → (-y,\\ -x)$
      • This can be represented in matrix multiplication: $[{x’ \atop y’}] = [{0 \atop -1}{\\ -1 \atop 0}][{x \atop y}]$ 250
  • Reflections in the line $\textcolor{#00ddff}{y = mx}$
    • Textbook: Cambridge - Reflections in the line y = mx
      • The textbook explains in depth how to find the reflection matrix
    • Let’s suppose that the angle between the positive direction of the $x$-axis and the line $y = mx$ is $\theta$, then $\tan\theta = m$
      • $\therefore\\ y = mx = x\tan\theta$
      • The reflection matrix is $[{x’ \atop y’}] = [{\cos(2\theta) \atop \sin(2\theta)}{\\ \\ \\ \sin(2\theta) \atop -\cos(2\theta)}][{x \atop y}]$
    • Review Trigonometry and Double Angle Formulas
  • Reflections in the line $\textcolor{#00ddff}{y = mx + c}$
    • Steps
      1. Vertically translate line of reflection to intersect with origin
      2. Apply reflection matrix
      3. Vertically translate line of reflection to original position
    • Formula
      • $[{x’ \atop y’}] = [{\cos(2\theta) \atop \sin(2\theta)}{\\ \\ \\ \sin(2\theta) \atop -\cos(2\theta)}]([{x \atop y}] - [{0 \atop c}{0 \atop c}{… \atop …}{0 \atop c}]) + [{0 \atop c}{0 \atop c}{… \atop …}{0 \atop c}]$

• Dilation is the process of stretching a shape on the graph from an axis by multiplying points by a value
  • Dilation from the $y$-axis and $x$-axis
    • A dilation from the $y$-axis is defined by $(x,\\ y) → (cx,\\ y)$ where $c > 0$
      • The $x$-coordinate is scaled by a factor of $c$, but the $y$-coordinate is unchanged
      • This can be represented in matrix multiplication: $[{x’ \atop y’}] = [{c \atop 0}{0 \atop 1}][{x \atop y}]$
    • A dilation from the $x$-axis is defined by $(x,\\ y) → (x,\\ cy)$ where $c > 0$
      • The $y$-coordinate is scaled by a factor of $c$, but the $x$-coordinate is unchanged
      • This can be represented in matrix multiplication: $[{x’ \atop y’}] = [{1 \atop 0}{0 \atop c}][{x \atop y}]$
    • A dilation along both the $x$-axis and $y$-axis is defined by $(x, y) → (cx, dy)$ where $c$ and $d > 0$
      • The $x$-coordinate is scaled by a factor of $c$, the $y$-coordinate is scaled by a factor of $d$
      • This can be represented in matrix multiplication: $[{x’ \atop y’}] = [{c \atop 0}{0 \atop d}][{x \atop y}]$
    • If $c$ or $d$ were negative it would also involve a reflection and therefore, a pure dilation requires $c$ to be greater than $0$

• Projections are the process of projecting a point onto either the $x$-axis or $y$-axis
  • A shape is flattened into a 2D line across either access
• The transformation is defined as $(x, y) → (x, 0)$ or $(x, y) → (0, y)$
  • $(x, y) → (x, 0)$ projects onto the $x$-axis
    • This can be represented in matrix multiplication: $[{x’ \atop y’}] = [{1 \atop 0}{0 \atop 0}][{x \atop y}]$
  • $(x, y) → (0, y)$ projects onto the $y$-axis
    • This can be represented in matrix multiplication: $[{x’ \atop y’}] = [{0 \atop 0}{0 \atop 1}][{x \atop y}]$

• A translation moves all points on the plane in the same direction and by the same distance
  • A translation is defined as $(x, y) → (x + a,\\ y + b)$
    • $a$ shifts the graph in the $x$-direction
    • $b$ shifts the graph in the $y$-direction
  • A translation is expressed with vector addition
    • $[{x’ \atop y’}] = [{x \atop y}] + [{a \atop b}]$
    • Translations cannot be represented with matrix multiplication
• Translations are not linear transformations

• A rotation moves all points around a fixed point without changing the distance from that point
  • Textbook: Cambridge - Rotations
  • Rotation around the origin
    • This can be represented in matrix multiplication: $[{x’ \atop y’}] = [{\cos\theta \atop \sin\theta}{-\sin\theta \atop \\ \\ \\ \cos\theta}][{x \atop y}]$
      • $\theta$ is the angle from the positive direction of the $x$-axis
      • Rotation in an anti-clockwise direction
  • Rotation around point $\textcolor{#00ddff}{(p, q)}$
    • Steps:
      1. Translate centre of rotation to the origin
      2. Apply rotation matrix
      3. Translate centre of rotation back to original position
    • Formula:
      • $[{x’ \atop y’}] = [{\cos\theta \atop \sin\theta}{-\sin\theta \atop \\ \\ \\ \cos\theta}]([{x \atop y}] - [{p \atop q}{p \atop q}{… \atop …}{p \atop q}]) + [{p \atop q}{p \atop q}{… \atop …}{p \atop q}]$

• We know that geometric transformations can be represented by a matrix, however, how do we apply multiple transformations to an object all at once?
• Lets say we want to apply transformation $A$, $B$ and $C$ (in this order) to object $[{x \atop y}]$
  • Transformations can be applied through matrix multiplication:
    • $C\cdot B\cdot A\cdot [{x \atop y}]$
    • Notice how the order of multiplication is reversed from the order of transformations
    • The order of multiplication matters, just like the order that the transformations are applied
• $AB$ is the matrix of transformation $B$ followed by $A$
• $BA$ is the matrix of transformation $A$ followed by $B$

• Straight lines always remain as a straight line after undergoing a linear transformation
• Linear transformations can be approached like a simultaneous equation
  • We can represent $x$ and $y$ (the original equation) and $x’$ and $y’$ (the transformed equation) as a matrix
• We want to represent transformed matrix $[{x’ \atop y’}]$ in terms of matrix $[{x \atop y}]$
  • e.g. applying a reflection over the $y$-axis:
    • $[{1 \atop 0}{\\ \\ 0 \atop -1}][{x \atop y}] = [{x’ \atop y’}] → [{x \atop y}] = [{1 \atop 0}{\\ \\ 0 \atop -1}]^{-1}[{x’ \atop y’}] → [{x \atop y}] = [{-1 \atop \\ \\ \\ 0}{0 \atop 1}][{x’ \atop y’}] → [{-x’ \atop y’}]$
      • If this transformation was applied to $y = 3x + 7$, the output equation would be $y = -3x + 7$
• When a shape with area $A$ is transformed by a matrix with determinant $k$, the area of the image is $|k|A$

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