3D Coordinates, Transformations & Hierarchies

TAKEAWAYS:

• 3D coordinates use $(x,y,z)$ and homogeneous coordinates $(w=1)$ to represent points, vectors, and transformations uniformly.
• Transformations (translation, rotation, scaling) are matrices applied to objects or the camera/viewer; they follow the right-hand rule and matrix multiplication order.
• Hierarchical modeling breaks objects into parent-child relationships (e.g., a robot’s arm rotating around its shoulder) using local-to-world transformations.
• Composite transformations combine multiple operations (e.g., rotate then translate) into a single matrix for efficiency.
• Real-world use: Games (e.g., Pathao’s 3D delivery maps), CAD (e.g., NTC’s infrastructure modeling), and animation (e.g., YouTube’s 3D avatars).
• Exam focus: Derive transformation matrices, apply them to points, and explain hierarchical modeling with examples.

1. 3D Coordinate Systems

In 2D, we use $(x,y)$; in 3D, we add $z$ for depth. The right-hand rule defines axes:

• Thumb: $x$ (right)
• Index: $y$ (up)
• Middle: $z$ (out of screen)

Key terms:

• Point: A location $(x,y,z)$.
• Vector: A direction $(dx,dy,dz)$ (e.g., from point A to B).
• Normal vector: Perpendicular to a surface (used in lighting).

Worked Example: Convert the point $P(2,3,4)$ to homogeneous coordinates (used for transformations): $$P_{homogeneous}=\left[\begin{array}{c}2\\ 3\\ 4\\ 1\end{array}\right]$$ Why? Homogeneous coordinates unify points and vectors into a single matrix framework.

2. Basic 3D Transformations

Transformations move, rotate, or scale objects using 4×4 matrices. The general form for a point $P(x,y,z,1)$ is:

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cccc}{m}_{11}& m_{12}& m_{13}& t_x\\ m_{21}& m_{22}& m_{23}& t_y\\ m_{31}& m_{32}& m_{33}& t_z\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right]$$

A. Translation (Moving Objects)

Moves an object by $(t_x,t_y,t_z)$: $$T=\left[\begin{array}{cccc}1& 0& 0& t_x\\ 0& 1& 0& t_y\\ 0& 0& 1& t_z\\ 0& 0& 0& 1\end{array}\right]$$ Example: Move a cube from $(1,1,1)$ to $(3,2,0)$: $$\left[\begin{array}{c}3\\ 2\\ 0\\ 1\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 2\\ 0& 1& 0& 1\\ 0& 0& 1& -1\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]$$

B. Rotation (Around Axes)

Use the right-hand rule to determine rotation direction.

1. Rotation about X-axis (θ degrees): $$R_x(\theta )=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \theta & -\sin \theta & 0\\ 0& \sin \theta & \cos \theta & 0\\ 0& 0& 0& 1\end{array}\right]$$ Example: Rotate $P(0,1,0)$ by $90^{\circ }$ around $x$: $$\cos (90^{\circ })=0,\sin (90^{\circ })=1\Rightarrow P^{\prime }=(0,0,1)$$

2. Rotation about Y-axis: $$R_y(\theta )=\left[\begin{array}{cccc}\cos \theta & 0& \sin \theta & 0\\ 0& 1& 0& 0\\ -\sin \theta & 0& \cos \theta & 0\\ 0& 0& 0& 1\end{array}\right]$$

3. Rotation about Z-axis: $$R_z(\theta )=\left[\begin{array}{cccc}\cos \theta & -\sin \theta & 0& 0\\ \sin \theta & \cos \theta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

Worked Example (Composite Rotation): Rotate $P(1,0,0)$ by $90^{\circ }$ around $z$, then $90^{\circ }$ around $y$:

1. Apply $R_z(90^{\circ })$: $$P^{\prime }=(0,1,0)$$
2. Apply $R_y(90^{\circ })$: $$P^{\prime \prime }=(0,0,1)$$ Result: $P$ moves from $(1,0,0)$ to $(0,0,1)$.

C. Scaling (Resizing Objects)

Scales by factors $(s_x,s_y,s_z)$: $$S=\left[\begin{array}{cccc}{s}_x& 0& 0& 0\\ 0& s_y& 0& 0\\ 0& 0& s_z& 0\\ 0& 0& 0& 1\end{array}\right]$$ Example: Scale $P(2,3,4)$ by $(0.5,1,2)$: $$P^{\prime }=(1,3,8)$$

3. Matrix Multiplication Order

Transformations are applied right-to-left (matrix multiplication is not commutative): $$P^{\prime }=T\cdot R\cdot S\cdot P$$ Example: Translate then rotate vs. rotate then translate:

• Translate then rotate: $$P^{\prime }=R\cdot (T\cdot P)$$
• Rotate then translate: $$P^{\prime }=T\cdot (R\cdot P)$$ Result: Different final positions!

4. Hierarchical Modeling

Objects are modeled as trees where:

• Parent nodes define local transformations.
• Child nodes inherit transformations from parents.

Example: A robot arm with 3 joints:

1. Shoulder (root): Translates to $(0,0,0)$.
2. Elbow (child of shoulder): Rotates around $y$-axis by $45^{\circ }$.
3. Hand (child of elbow): Translates by $(0,0,2)$.

graph TD
    A["Shoulder"] --> B["Elbow"]
    B --> C["Hand"]
    A -->|"Local: T(0,0,0)"| D["World"]
    B -->|"Local: R_y(45°)"| D
    C -->|"Local: T(0,0,2)"| D

Global Transformation: $$T_{global}=T_{world}\cdot R_{elbow}\cdot T_{hand}$$

Real-World Tie-In:

• Pathao’s delivery maps: Uses hierarchical transformations to model moving vehicles and dynamic routes.
• 3D game characters: Skeletons (e.g., a knight’s armor) are parented to bones for realistic movement.

5. Composite Transformations

Combine multiple transformations into a single matrix for efficiency. For example, rotate then translate: $$T_{final}=T\cdot R$$ Example: Rotate $P(1,0,0)$ by $90^{\circ }$ around $z$, then translate by $(2,0,0)$:

1. Compute $R_z(90^{\circ })$: $$R_z=\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$
2. Compute $T(2,0,0)$: $$T=\left[\begin{array}{cccc}1& 0& 0& 2\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$
3. Multiply $T\cdot R$: $$T_{final}=\left[\begin{array}{cccc}0& -1& 0& 2\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$
4. Apply to $P$: $$P^{\prime }=(2,1,0)$$

6. Applications in Real World

A. E-Sewa & Government Services

• 3D modeling of infrastructure: NTC uses transformations to simulate power line routes in 3D space before construction.
• Virtual tours: E-Sewa’s citizen portals could use 3D transformations to show office layouts for appointments.

B. Gaming & Animation

• Pathao’s driver app: Uses hierarchical transformations to model moving vehicles and dynamic traffic routes.
• YouTube’s 3D avatars: Avatars are built using skeletal hierarchies (e.g., head rotates around neck).

C. CAD & Engineering

• Nepal’s NEPSE stock visualization: 3D charts use transformations to rotate and scale graphs for better analysis.
• Robotics: A robot arm’s movements (e.g., in industrial automation) rely on inverse kinematics and hierarchical transformations.

7. Common Pitfalls & Exam Tips

Exam Tip:

• Derive matrices: Show every step (e.g., trigonometric values for rotations).
• Visualize: Sketch the object before/after transformations.
• Hierarchy questions: Always show the tree and compute global transformations step-by-step.

8. Summary Table: Transformations

9. Worked Example: Daraz Delivery Route

Scenario: A Daraz delivery person starts at $(0,0,0)$, moves 5 units east ($+x$), then turns $90^{\circ }$ left (around $z$) and moves 3 units north ($+y$).

1. First translation (east): $$T_1=\left[\begin{array}{cccc}1& 0& 0& 5\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$ New position: $(5,0,0)$.

2. Rotation (90° left around $z$): $$R_z(90^{\circ })=\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$ Apply to $(5,0,0)$: $$P^{\prime }=(0,5,0)$$

3. Second translation (north): $$T_2=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 3\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$ Final position: $(0,8,0)$.

Visualization:

10. Real-World Picture: Robot Arm

graph TD
    A["Base"] --> B["Shoulder"]
    B --> C["Elbow"]
    C --> D["Wrist"]
    D --> E["Gripper"]
    A -->|"T(0,0,0)"| F["World"]
    B -->|"R_y(θ₁)"| F
    C -->|"T(0,0,L₁) + R_y(θ₂)"| F
    D -->|"T(0,0,L₂) + R_y(θ₃)"| F
    E -->|"T(0,0,L₃)"| F

11. Exam Tip: Matrix Derivation

For exams, always:

1. Write the general matrix form.
2. Substitute given values (e.g., $\theta =45^{\circ }\Rightarrow \cos \theta =\sin \theta =\frac{\sqrt{2}}{2}$).
3. Multiply step-by-step (show intermediate results).
4. Apply to a test point (e.g., $(1,0,0)$) to verify.

Example Question: "Derive the matrix to rotate a point $90^{\circ }$ around the $y$-axis and then translate it by $(2,-1,3)$." Solution:

1. $R_y(90^{\circ })$: $$\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$
2. $T(2,-1,3)$: $$\left[\begin{array}{cccc}1& 0& 0& 2\\ 0& 1& 0& -1\\ 0& 0& 1& 3\\ 0& 0& 0& 1\end{array}\right]$$
3. Composite $T\cdot R$: $$\left[\begin{array}{cccc}0& 0& 1& 2\\ 0& 1& 0& -1\\ -1& 0& 0& 3\\ 0& 0& 0& 1\end{array}\right]$$
4. Apply to $P(1,0,0)$: $$P^{\prime }=(2,-1,-1)$$