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Topic 9 of 15
09
3D Transformations
3D Transformations are operations used to change the position, size, orientation, or shape of objects in three-dimensional space.
📘 3D Transformations — Exam Notes
🔹 1. Introduction
3D Transformations are operations used to change the position, size, orientation, or shape of objects in three-dimensional space.
👉 These transformations work on objects defined in (x, y, z) coordinates.
🔹 2. Basic Representation
A point in 3D space: [ P(x, y, z) ]
Using homogeneous coordinates: [ P(x, y, z, 1) ]
👉 This allows us to use matrix multiplication for all transformations.
🔹 3. Types of 3D Transformations
🔸 3.1 Translation (Moving Object)
✔️ Definition
Shifts an object from one position to another.
✔️ Formula
[ x' = x + t_x,\quad y' = y + t_y,\quad z' = z + t_z ]
✔️ Matrix Form
[
\begin{bmatrix}
1 & 0 & 0 & t_x
0 & 1 & 0 & t_y
0 & 0 & 1 & t_z
0 & 0 & 0 & 1
\end{bmatrix}
]
🔸 3.2 Scaling (Resizing Object)
✔️ Definition
Changes the size of an object.
✔️ Formula
[ x' = x \cdot s_x,\quad y' = y \cdot s_y,\quad z' = z \cdot s_z ]
✔️ Matrix Form
[
\begin{bmatrix}
s_x & 0 & 0 & 0
0 & s_y & 0 & 0
0 & 0 & s_z & 0
0 & 0 & 0 & 1
\end{bmatrix}
]
✔️ Types
- Uniform Scaling: (s_x = s_y = s_z)
- Non-uniform Scaling: Different scaling factors
🔸 3.3 Rotation (Very Important)
✔️ Definition
Rotates an object around an axis.
✔️ Rotation About X-axis
[
\begin{aligned}
x' &= x
y' &= y \cos\theta - z \sin\theta
z' &= y \sin\theta + z \cos\theta
\end{aligned}
]
✔️ Rotation About Y-axis
[
\begin{aligned}
x' &= x \cos\theta + z \sin\theta
y' &= y
z' &= -x \sin\theta + z \cos\theta
\end{aligned}
]
✔️ Rotation About Z-axis
[
\begin{aligned}
x' &= x \cos\theta - y \sin\theta
y' &= x \sin\theta + y \cos\theta
z' &= z
\end{aligned}
]
✔️ Matrix Form (Z-axis Example)
[
\begin{bmatrix}
\cos\theta & -\sin\theta & 0 & 0
\sin\theta & \cos\theta & 0 & 0
0 & 0 & 1 & 0
0 & 0 & 0 & 1
\end{bmatrix}
]
🔸 3.4 Reflection
✔️ Definition
Produces a mirror image of an object.
✔️ Examples
-
Reflection about XY-plane: [ (x, y, z) \rightarrow (x, y, -z) ]
-
Reflection about YZ-plane: [ (x, y, z) \rightarrow (-x, y, z) ]
-
Reflection about XZ-plane: [ (x, y, z) \rightarrow (x, -y, z) ]
🔸 3.5 Shearing
✔️ Definition
Distorts the shape by shifting coordinates.
✔️ Example (X-direction shear)
[ x' = x + sh_x \cdot y + sh_z \cdot z ]
🔹 4. Homogeneous Coordinates (Important)
✔️ Definition
Representation of 3D points as: [ (x, y, z, 1) ]
✔️ Advantages
- Simplifies transformations
- Allows combination of multiple operations
🔹 5. Composite Transformations
✔️ Definition
Combination of multiple transformations.
✔️ Example
Scale → Rotate → Translate
👉 Represented as: [ T = T_3 \cdot T_2 \cdot T_1 ]
✔️ Important Rule
⚠️ Order matters Changing order gives different results.
🔹 6. Rotation About an Arbitrary Axis
✔️ Steps
- Translate object so axis passes through origin
- Align axis with coordinate axis
- Perform rotation
- Reverse alignment
- Translate back
🔹 7. Scaling About a Fixed Point
✔️ Steps
- Translate object to origin
- Apply scaling
- Translate back
🔹 8. Diagram Descriptions
✔️ 3D Translation
- Show object moved from one position to another
✔️ Rotation
- Show object rotating around X, Y, or Z axis
✔️ Scaling
- Show object becoming larger/smaller
✔️ Reflection
- Show mirror image across a plane
🔹 9. Applications of 3D Transformations
- Animation
- Game development
- Virtual reality
- CAD systems
- Robotics
🔹 10. Important Terms
- Transformation Matrix: Matrix used for transformation
- Homogeneous Coordinates: Extended coordinate system
- Composite Transformation: Combination of operations
- Axis of Rotation: Line about which rotation occurs
📝 Likely Exam Questions
- Define 3D transformations.
- Explain translation with matrix form.
- Describe scaling and its types.
- Explain rotation about X, Y, Z axes.
- What is reflection in 3D?
- Define shearing in 3D.
- What are homogeneous coordinates?
- Explain composite transformations.
- How do you rotate about an arbitrary axis?
- Solve a numerical problem on 3D transformation.
⚡ Quick Revision Summary
-
3D transformations modify objects in (x, y, z) space
-
Types:
- Translation
- Scaling
- Rotation
- Reflection
- Shearing
-
Use 4×4 matrices + homogeneous coordinates
-
Order is important in composite transformations
-
Widely used in 3D graphics, animation, and games
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