Rotation about Y-axis ($R_y(\theta)$): $$ R_y(\theta) = \begin{pmatrix} \cos\theta & 0 & \sin\theta & 0 \\ 0 & 1 & 0 & 0 \\ -\sin\theta & 0 & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$

Rotation about Z-axis ($R_z(\theta)$): $$ R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 & 0 \\ \sin\theta & \cos\theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$

6. 2D vs. 3D Transformations

Feature	2D Transformations	3D Transformations
Dimensions	Operate on 2D points $(x, y)$.	Operate on 3D points $(x, y, z)$.
Homogeneous Coords	$(x, y, 1)$ for 2D points.	$(x, y, z, 1)$ for 3D points.
Matrix Size	$3 \times 3$ matrices.	$4 \times 4$ matrices.
Rotation Axes	Only about a point (effectively Z-axis in 2D plane).	About X, Y, Z axes, or arbitrary axis.
Complexity	Simpler, fewer degrees of freedom.	More complex, higher degrees of freedom.

7. Rotation about an Arbitrary Axis in 3D

To rotate an object about an arbitrary axis defined by two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ by angle $\theta$:
1. Translate $P_1$ to the origin: $T(-x_1, -y_1, -z_1)$.
2. Align the rotation axis with one of the coordinate axes (e.g., Z-axis) using two rotations ($R_x, R_y$).
3. Perform the desired rotation $\theta$ about the aligned axis ($R_z(\theta)$).
4. Undo the alignment rotations ($R_y^{-1}, R_x^{-1}$).
5. Undo the initial translation ($T(x_1, y_1, z_1)$).
The overall transformation matrix $M = T^{-1} \cdot R_x^{-1} \cdot R_y^{-1} \cdot R_z(\theta) \cdot R_y \cdot R_x \cdot T$.
The rotation axis vector $\vec{V} = P_2 - P_1 = (a, b, c)$.
Normalized vector $\vec{u} = (a', b', c')$, where $a' = a/|\vec{V}|$, etc.
Rotation angles to align with Z-axis:
- Rotate about X-axis by $\alpha_x$: $R_x(\alpha_x)$ to bring $\vec{V}$ into XZ-plane. $\cos \alpha_x = c'/d$, $\sin \alpha_x = b'/d$, where $d = \sqrt{b'^2+c'^2}$.
- Rotate about Y-axis by $\alpha_y$: $R_y(\alpha_y)$ to align with Z-axis. $\cos \alpha_y = d$, $\sin \alpha_y = a'$.

8. Composite Transformations in 3D

A sequence of transformations can be combined into a single composite transformation matrix by multiplying the individual transformation matrices in reverse order of application.
If transformations $M_1, M_2, \dots, M_n$ are applied sequentially to a point $P$, the final transformed point $P'$ is given by: $$ P' = (M_n \cdot \dots \cdot M_2 \cdot M_1) \cdot P $$
Matrix multiplication is not commutative, so the order of transformations is crucial.
Example: Scaling about an arbitrary point $P_f$ in 3D:
1. Translate $P_f$ to origin: $T(-x_f, -y_f, -z_f)$.
2. Scale: $S(s_x, s_y, s_z)$.
3. Translate back: $T(x_f, y_f, z_f)$.
Composite matrix $M = T(x_f, y_f, z_f) \cdot S(s_x, s_y, s_z) \cdot T(-x_f, -y_f, -z_f)$.

9. Parallel Projections

Projectors are parallel to each other and perpendicular or oblique