1. Definition

An affine transformation in 3D is a linear mapping that preserves points, straight lines, and planes, and allows translation, scaling, rotation, reflection, and shear of 3D objects.

• Affine transformations maintain parallelism of lines but not necessarily distances or angles.
• Represented conveniently using 4×4 matrices with homogeneous coordinates.

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2. Homogeneous Coordinates in 3D

A 3D point $(x, y, z)$ is represented in homogeneous coordinates as:

$$P = \begin{bmatrix} x \ y \ z \ 1 \end{bmatrix}$$

• Using 4×4 matrices allows translation to be expressed as matrix multiplication, along with rotation, scaling, and shear.

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3. Types of 3D Affine Transformations

A. Translation

Moves a point by a vector $(tx, ty, tz)$:

$$T = \begin{bmatrix} 1 & 0 & 0 & tx \ 0 & 1 & 0 & ty \ 0 & 0 & 1 & tz \ 0 & 0 & 0 & 1 \end{bmatrix}, \quad P' = T \cdot P$$

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B. Scaling

Scales coordinates along x, y, z axes:

$$S = \begin{bmatrix} Sx & 0 & 0 & 0 \ 0 & Sy & 0 & 0 \ 0 & 0 & Sz & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}, \quad P' = S \cdot P$$

• $Sx, Sy, Sz > 1$ enlarges, $0 < S < 1$ shrinks.

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C. Rotation

Rotation about x, y, or z axes:

About X-axis (angle $\theta$):

$$R_x = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & \cos\theta & -\sin\theta & 0 \ 0 & \sin\theta & \cos\theta & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$$

About Y-axis (angle $\theta$):

$$R_y = \begin{bmatrix} \cos\theta & 0 & \sin\theta & 0 \ 0 & 1 & 0 & 0 \ -\sin\theta & 0 & \cos\theta & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$$

About Z-axis (angle $\theta$):

$$R_z = \begin{bmatrix} \cos\theta & -\sin\theta & 0 & 0 \ \sin\theta & \cos\theta & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$$

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D. Shear

Shifts one coordinate proportionally to another:

$$Sh = \begin{bmatrix} 1 & Sh_{xy} & Sh_{xz} & 0 \ Sh_{yx} & 1 & Sh_{yz} & 0 \ Sh_{zx} & Sh_{zy} & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}$$

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4. Concatenation of 3D Affine Transformations

• Multiple transformations can be combined into a single matrix:

$$M_{final} = T \cdot R \cdot S \cdot Sh$$

• Apply once to a point/vector:

$$P' = M_{final} \cdot P$$

• Order matters (matrix multiplication is not commutative).

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5. Summary Table

Transformation	Matrix Form	Effect
Translation	4×4 matrix with last column $(tx, ty, tz, 1)$	Move object in space
Scaling	Diagonal 4×4 matrix	Resize object along axes
Rotation	4×4 matrix $(R_x, R_y, R_z)$	Rotate object about axes
Shear	Off-diagonal elements	Skew object along axes

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Key Points:

• Affine transformations in 3D preserve straight lines and parallelism.
• Using homogeneous coordinates, translation, rotation, scaling, and shear can all be represented as matrix multiplication.
• Essential for 3D modeling, animation, and graphics pipelines.