COMP3145›Projective transformations

HCI & Computer GraphicsTopic 61 of 73

Projective transformations

4 minread

636words

Beginnerlevel

1. Definition

A projective transformation (also called a perspective transformation) is a type of geometric transformation that maps points in one plane or space to another while preserving straight lines, but not necessarily parallelism or ratios of distances.

Used to simulate perspective in 2D or 3D graphics, i.e., how objects appear smaller as they move farther from the viewer.
Extends affine transformations by allowing projection onto a plane.

2. Key Characteristics

Property	Affine Transformation	Projective Transformation
Line preservation	Yes	Yes
Parallelism preservation	Yes	No
Ratio of distances	Preserved	Not preserved
Can represent perspective	No	Yes

3. Homogeneous Coordinates Representation

In 2D, a point $(x, y)$ is represented in homogeneous coordinates as:

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

A 2D projective transformation is represented by a 3×3 matrix:

P' = \begin{bmatrix} x' \\ y' \\ w' \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}

The transformed Cartesian coordinates are obtained by:

x_{cart} = \frac{x'}{w'}, \quad y_{cart} = \frac{y'}{w'}

In 3D, projective transformations use a 4×4 matrix similarly, mapping 3D points to 2D projection plane (screen).

4. Examples of Projective Transformations

A. Perspective Projection

Projects 3D points onto a 2D plane as seen from a viewpoint.
Objects farther away appear smaller, simulating human vision.

Perspective matrix in 3D (simplified):

P = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 1/d \ 0 & 0 & 0 & 1 \end{bmatrix}

$d$ = distance from viewer to projection plane.

B. Vanishing Points

Parallel lines in 3D may converge at a vanishing point in the projected 2D view.
Essential for realistic rendering in 3D graphics and CAD.

C. Homography

Mapping between two planes (e.g., image transformations in computer vision).
All straight lines remain straight, but shapes can appear distorted.

5. Applications of Projective Transformations

3D rendering and perspective in games and simulations
Computer vision (camera calibration, image rectification)
Architectural and CAD drawings
Augmented reality overlays

6. Summary Table

Feature	Affine	Projective
Lines	Preserved	Preserved
Parallel lines	Preserved	Not necessarily
Distance ratios	Preserved	Not preserved
Perspective simulation	No	Yes

Key Points:

Projective transformations are more general than affine transformations.
They allow realistic rendering of 3D scenes on 2D screens.
Often implemented using homogeneous coordinates and 4×4 matrices in graphics pipelines.

Previous topic 60

Affine transformation in 3D

Next topic 62

Ray tracing

Past Papers

Open this section to load past papers

Click on Show Past Papers to see past papers.

COMP3145›Projective transformations

HCI & Computer GraphicsTopic 61 of 73

Projective transformations

4 minread

636words

Beginnerlevel

1. Definition

Used to simulate perspective in 2D or 3D graphics, i.e., how objects appear smaller as they move farther from the viewer.
Extends affine transformations by allowing projection onto a plane.

2. Key Characteristics

Property	Affine Transformation	Projective Transformation
Line preservation	Yes	Yes
Parallelism preservation	Yes	No
Ratio of distances	Preserved	Not preserved
Can represent perspective	No	Yes

3. Homogeneous Coordinates Representation

In 2D, a point $(x, y)$ is represented in homogeneous coordinates as:

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

A 2D projective transformation is represented by a 3×3 matrix:

P' = \begin{bmatrix} x' \\ y' \\ w' \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}

The transformed Cartesian coordinates are obtained by:

x_{cart} = \frac{x'}{w'}, \quad y_{cart} = \frac{y'}{w'}

In 3D, projective transformations use a 4×4 matrix similarly, mapping 3D points to 2D projection plane (screen).

4. Examples of Projective Transformations

A. Perspective Projection

Projects 3D points onto a 2D plane as seen from a viewpoint.
Objects farther away appear smaller, simulating human vision.

Perspective matrix in 3D (simplified):

P = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 1/d \ 0 & 0 & 0 & 1 \end{bmatrix}

$d$ = distance from viewer to projection plane.

B. Vanishing Points

Parallel lines in 3D may converge at a vanishing point in the projected 2D view.
Essential for realistic rendering in 3D graphics and CAD.

C. Homography

Mapping between two planes (e.g., image transformations in computer vision).
All straight lines remain straight, but shapes can appear distorted.

5. Applications of Projective Transformations

3D rendering and perspective in games and simulations
Computer vision (camera calibration, image rectification)
Architectural and CAD drawings
Augmented reality overlays

6. Summary Table

Feature	Affine	Projective
Lines	Preserved	Preserved
Parallel lines	Preserved	Not necessarily
Distance ratios	Preserved	Not preserved
Perspective simulation	No	Yes

Key Points:

Projective transformations are more general than affine transformations.
They allow realistic rendering of 3D scenes on 2D screens.
Often implemented using homogeneous coordinates and 4×4 matrices in graphics pipelines.

Previous topic 60

Affine transformation in 3D

Next topic 62

Ray tracing

Past Papers

Open this section to load past papers

Click on Show Past Papers to see past papers.