3.1 The Basic Transformations (2D)

• Translation: Dragging it with your finger (Move).
• scaling: Pinching to zoom (Resize).
• Rotation: Twisting your fingers (Spin).

1. Translation (Moving)

Adding a value to move the point.

• $x' = x + t_x$
• $y' = y + t_y$

2. Scaling (Resizing)

Multiplying to stretch or shrink.

• $x' = x \cdot s_x$
• $y' = y \cdot s_y$

3. Rotation (Spinning)

Rotating about the Origin $(0,0)$.

• $x' = x \cos\theta - y \sin\theta$
• $y' = x \sin\theta + y \cos\theta$

mindmap
  root((2D Moves))
    Translation
      Move
      Add (T)
    Scaling
      Resize
      Multiply (S)
    Rotation
      Spin
      Trig (Cos/Sin)
            

3.2 Reflection & Shearing

Reflection (Mirroring)

Flipping an object. It changes the "handedness".

Shearing (Slanting)

• X-Shear: Pushes horizontal lines. $x' = x + sh_x \cdot y$, $y' = y$.
• Y-Shear: Pushes vertical lines. $x' = x$, $y' = y + sh_y \cdot x$.

mindmap
  root((Mirror & Slant))
    Reflection
      Flip Signs
      Change Handedness
    Shear
      Push Top/Side
      Deform Shape
            

3.3 3D Transformations

Same concepts as 2D, but with a Z-axis. Points become $(x, y, z)$.

1. 3D Translation

$$x' = x + t_x, \quad y' = y + t_y, \quad z' = z + t_z$$

2. 3D Scaling

$$x' = x \cdot s_x, \quad y' = y \cdot s_y, \quad z' = z \cdot s_z$$

3. 3D Rotation (The Tricky Part)

Rotation happens around an axis.

• About Z-Axis (Like 2D):
  $x' = x\cos\theta - y\sin\theta$
  $y' = x\sin\theta + y\cos\theta$
  $z' = z$
• About X-Axis:
  $x' = x$
  $y' = y\cos\theta - z\sin\theta$
  $z' = y\sin\theta + z\cos\theta$
• About Y-Axis:
  $x' = z\sin\theta + x\cos\theta$
  $y' = y$
  $z' = z\cos\theta - x\sin\theta$

mindmap
  root((3D Space))
    Concept
      Add Z-Axis
      Depth
    3D Rotation
      About Z (Standard)
      About X (Tilt)
      About Y (Pan)
            

3.4 Composite & Affine Transformations

Homogeneous Coordinates

Add a dummy coordinate to enable matrix multiplication: $(x, y) \to (x, y, 1)$ or $(x,y,z,1)$.

Composite Rule

Matrix multiplication is NOT commutative ($A \cdot B \neq B \cdot A$). Multiply Right-to-Left.

Common Recipes

A. Pivot Point Rotation

1. Translate Pivot to Origin.
2. Rotate.
3. Inverse Translate back.

B. Fixed Point Scaling

Same Recipe: $T^{-1} \cdot S \cdot T$.

Rigid vs. Non-Rigid Body

• Rigid: Shape/Size constant (Translate, Rotate, Reflect).
• Non-Rigid: Shape distorts (Scale, Shear).

mindmap
  root((Composite Logic))
    Rules
      Right-to-Left
      Homogeneous (1)
    Pivot Point
      1. Move to Origin
      2. Rotate
      3. Move Back
    Scaling
      Same steps
      Fixed Point
            

3.5 Advanced Concepts

Reflection About Arbitrary Line ($y = mx + c$)

1. Translate line to pass through Origin.
2. Rotate line to align with X-axis.
3. Reflect about X-axis.
4. Inverse Rotate.
5. Inverse Translate.

Rapid-Fire Formulas (Exam Tricks)

Quick tips:

• Inverse Translation: Negate $t_x, t_y$.
• Inverse Rotation: Negate $\theta$.
• Reflect about $y=k$: $y' = -y + 2k$.
• Reflect about $x=k$: $x' = -x + 2k$.

mindmap
  root((Advanced Magic))
    Arbitrary Reflection
      5-Step Algo
      Move -> Rot -> Ref -> InvRot -> InvMove
    Shortcuts
      Inverse = Negate
      y=k line Reflection