1. Affine Transformations – Definition

Definition: An affine transformation is a geometric transformation that preserves:

• Points, straight lines, and planes
• Parallelism of lines (parallel lines remain parallel)

Affine transformations are widely used in 2D and 3D computer graphics for moving, rotating, resizing, or skewing objects.

Mathematically, in 2D, an affine transformation can be represented as:

$$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} + \begin{bmatrix} tx \ ty \end{bmatrix}$$

Or in homogeneous coordinates (common in graphics):

$$\begin{bmatrix} x' \ y' \ 1 \end{bmatrix} = \begin{bmatrix} a & b & tx \ c & d & ty \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ 1 \end{bmatrix}$$

Where $(tx, ty)$ are translation components, and the 2×2 matrix defines rotation, scaling, or shear.

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2. Types of Affine Transformations

A. Translation

Definition:

• Moving an object from one position to another without changing its shape, size, or orientation.

Formula in 2D:

$$x' = x + tx, \quad y' = y + ty$$

Matrix Form (homogeneous coordinates):

$$T = \begin{bmatrix} 1 & 0 & tx \ 0 & 1 & ty \ 0 & 0 & 1 \end{bmatrix}$$

Example:

• Moving a triangle 5 units right and 3 units up.

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B. Rotation

Definition:

• Rotating an object around the origin or a pivot point by an angle θ.

Formula in 2D (around origin):

$$x' = x \cos \theta - y \sin \theta, \quad y' = x \sin \theta + y \cos \theta$$

Matrix Form:

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

Note: To rotate around a point $(px, py)$, first translate to origin, rotate, then translate back.

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C. Scaling

Definition:

• Changing the size of an object in the x and/or y direction.

Formula in 2D:

$$x' = sx \cdot x, \quad y' = sy \cdot y$$

Matrix Form:

$$S = \begin{bmatrix} sx & 0 & 0 \ 0 & sy & 0 \ 0 & 0 & 1 \end{bmatrix}$$

Notes:

• Uniform scaling: $sx = sy$ → preserves shape
• Non-uniform scaling: $sx \neq sy$ → may stretch/compress

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D. Shear

Definition:

• Shifting one coordinate proportionally to another, effectively slanting the shape.

Formulas in 2D:

• X-shear: $x' = x + sh_x \cdot y, \quad y' = y$
• Y-shear: $x' = x, \quad y' = y + sh_y \cdot x$

Matrix Forms:

$$\text{X-shear: } H_x = \begin{bmatrix} 1 & sh_x & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}, \quad \text{Y-shear: } H_y = \begin{bmatrix} 1 & 0 & 0 \ sh_y & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$$

Example:

• Slanting a rectangle so that its top edge moves more than the bottom edge.

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3. Key Properties of Affine Transformations

• Straight lines remain straight
• Parallel lines remain parallel
• Ratios of distances along a line are preserved
• Shapes may change size or orientation but basic collinearity is preserved

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4. Summary Table

Transformation	Effect	Matrix Form (2D Homogeneous)
Translation	Move object	$\begin{bmatrix}1 & 0 & tx \ 0 & 1 & ty \ 0 & 0 & 1\end{bmatrix}$
Rotation	Rotate object	$\begin{bmatrix}\cosθ & -\sinθ & 0 \ \sinθ & \cosθ & 0 \ 0 & 0 & 1\end{bmatrix}$
Scaling	Resize object	$\begin{bmatrix}sx & 0 & 0 \ 0 & sy & 0 \ 0 & 0 & 1\end{bmatrix}$
Shear	Slant object	X-shear: $\begin{bmatrix}1 & sh_x & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix}$Y-shear: $\begin{bmatrix}1 & 0 & 0 \ sh_y & 1 & 0 \ 0 & 0 & 1\end{bmatrix}$