COMP3145›Affine transformation in 3D

HCI & Computer GraphicsTopic 60 of 73

Affine transformation in 3D

7 minread

1,154words

Intermediatelevel

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.

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.

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

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.

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}

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}

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).

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

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.

Previous topic 59

Specifying views in 3D

Next topic 61

Projective transformations

Past Papers

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Click on Show Past Papers to see past papers.

COMP3145›Affine transformation in 3D

HCI & Computer GraphicsTopic 60 of 73

Affine transformation in 3D

7 minread

1,154words

Intermediatelevel

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.

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.

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

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.

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}

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}

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).

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

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.

Previous topic 59

Specifying views in 3D

Next topic 61

Projective transformations

Past Papers

Open this section to load past papers

Click on Show Past Papers to see past papers.