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.

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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

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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).

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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.

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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.

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C. Homography

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

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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

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6. Summary Table

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

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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.