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Topic 6 of 15
06
2D Transformations
2D Transformations are operations used to change the position, size, or orientation of objects in a two-dimensional plane (X–Y plane).
📘 2D Transformations — Exam Notes
🔹 1. Introduction
2D Transformations are operations used to change the position, size, or orientation of objects in a two-dimensional plane (X–Y plane).
👉 These transformations are widely used in graphics applications like animation, games, and design.
🔹 2. Basic Idea
A point in 2D is represented as: [ P(x, y) ]
After transformation, it becomes: [ P'(x', y') ]
🔹 3. Types of 2D Transformations
🔸 3.1 Translation (Shifting)
✔️ Definition
Moves an object from one position to another without changing its shape or size.
✔️ Formula
[ x' = x + t_x,\quad y' = y + t_y ]
Where:
- (t_x), (t_y) = translation distances
✔️ Example
If (P(2,3)), (t_x=4), (t_y=1): [ P' = (6,4) ]
✔️ Matrix Form (Homogeneous Coordinates)
[
\begin{bmatrix}
x'
y'
1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & t_x
0 & 1 & t_y
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x
y
1
\end{bmatrix}
]
🔸 3.2 Scaling (Resizing)
✔️ Definition
Changes the size of an object.
✔️ Formula
[ x' = x \cdot s_x,\quad y' = y \cdot s_y ]
Where:
- (s_x), (s_y) = scaling factors
✔️ Types
- Uniform Scaling: (s_x = s_y)
- Non-uniform Scaling: (s_x \neq s_y)
✔️ Matrix Form
[
\begin{bmatrix}
s_x & 0 & 0
0 & s_y & 0
0 & 0 & 1
\end{bmatrix}
]
🔸 3.3 Rotation
✔️ Definition
Rotates an object around the origin (or any point).
✔️ Formula (about origin)
[ x' = x \cos\theta - y \sin\theta ] [ y' = x \sin\theta + y \cos\theta ]
✔️ Matrix Form
[
\begin{bmatrix}
\cos\theta & -\sin\theta & 0
\sin\theta & \cos\theta & 0
0 & 0 & 1
\end{bmatrix}
]
✔️ Important Points
- Positive angle → Anti-clockwise rotation
- Negative angle → Clockwise rotation
🔸 3.4 Reflection
✔️ Definition
Flips an object across a line (mirror effect).
✔️ Types & Matrices
-
About X-axis [ (x, y) \rightarrow (x, -y) ]
-
About Y-axis [ (x, y) \rightarrow (-x, y) ]
-
About Origin [ (x, y) \rightarrow (-x, -y) ]
🔸 3.5 Shearing
✔️ Definition
Distorts the shape of an object by shifting one coordinate.
✔️ Types
Horizontal Shear
[ x' = x + sh_x \cdot y,\quad y' = y ]
Vertical Shear
[ x' = x,\quad y' = y + sh_y \cdot x ]
✔️ Matrix Form (Horizontal)
[
\begin{bmatrix}
1 & sh_x & 0
0 & 1 & 0
0 & 0 & 1
\end{bmatrix}
]
🔹 4. Homogeneous Coordinates (Important)
✔️ Definition
A system that represents 2D points using 3 coordinates (x, y, 1) to simplify matrix operations.
✔️ Advantages
- All transformations can be represented using matrices
- Easy to combine multiple transformations
🔹 5. Composite Transformations
✔️ Definition
Combining two or more transformations into one.
✔️ Example
Rotate → then translate
👉 Multiply matrices in correct order: [ T = T_2 \cdot T_1 ]
✔️ Important Rule
⚠️ Order matters! Changing order changes result.
🔹 6. Rotation About an Arbitrary Point
✔️ Steps
- Translate point to origin
- Perform rotation
- Translate back
🔹 7. Diagram Descriptions
✔️ Translation
- Show object before and after shifting
✔️ Scaling
- Show same object larger/smaller
✔️ Rotation
- Show object rotating around origin
✔️ Reflection
- Show mirror image across axis
✔️ Shearing
- Show slanted shape
🔹 8. Applications of 2D Transformations
- Animation
- Game development
- Image processing
- CAD systems
- GUI design
🔹 9. Important Terms
- Transformation Matrix: Matrix used to perform operation
- Homogeneous Coordinates: Extended coordinate system
- Composite Transformation: Combination of transformations
📝 Likely Exam Questions
- Define 2D transformations.
- Explain translation with matrix representation.
- Describe scaling and its types.
- Explain rotation with formulas.
- What is reflection? Explain types.
- Define shearing with examples.
- What are homogeneous coordinates?
- Explain composite transformations.
- How do you rotate about an arbitrary point?
- Solve a numerical on transformation.
⚡ Quick Revision Summary
-
2D Transformations modify position, size, or shape
-
Main types:
- Translation
- Scaling
- Rotation
- Reflection
- Shearing
-
Use matrices + homogeneous coordinates
-
Order matters in composite transformations
-
Widely used in graphics & animation
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