Transformations

…in 2D Computer Graphics

Transformations

Geometric transformations will map points in one space to points in another space:

\[(x',y') = f(x,y)\]

Transformations

The mapping function uses elementary operations, which include:

• Translation
• Rotation
• Scaling
• Shear
• Reflection
• Projection
• Warp

Transformations

Types of transformation preserve geometric properties of the object.

Rigid transformations

Euclidean transformations

Similarity transformations

Affine transformations

Projective transformations

Non-linear transformations

Object Representation

In graphics, we represent objects using points or vertices, which are connected to form polygons or faces.

Object Representation

Only the vertices are subjected to the transformations.

Object Representation

Question: How do we represent a vertex mathematically?

Tools for transformations

Transformations of an object are applied to each individual vertex of that object.

Tools for transformations

The mathematical entity used to perform a transformation to the vector of n vertex coordinates is:

A square matrix of size \(n\) x \(n\), where \(n\) is the dimension of the vertex vector.

Translation

Formally, we will represent vertex coordinates as a column vector:

\[\begin{bmatrix} x \\ y \end{bmatrix}\]

Translation

Translation is performed by adding a translation vector.

\[ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} t_x \\ t_y \end{bmatrix} \]

Translation

Example: Consider the vertex with coordinates \(~(-5, -2)\), that we wish to translate in the \(~x~\) direction 9 units, and in the \(~y~\) direction 5 units.

\[ \begin{bmatrix} 4 \\ 3 \end{bmatrix} = \begin{bmatrix} -5 \\ -2 \end{bmatrix} + \begin{bmatrix} 9 \\ 5 \end{bmatrix} \]

Rotation

• translation moves a single point
• rotation of a point is meaningless
• we need to perform rotations about an axis

Rotation

A mnemonic for trigonometry is SOH, CAH, TOA.

\[ sin(\theta) = \frac{O}{H},~ cos(\theta) = \frac{A}{H},~ tan(\theta) = \frac{O}{A} \]

Rotation

We can fit this mnemonic to our 2D plane:

\[ sin(\theta) = \frac{O}{H} = \frac{y}{r},~ cos(\theta) = \frac{A}{H} = \frac{x}{r},~ tan(\theta) = \frac{O}{A} = \frac{y}{x} \]

where \(r\) is the radius of a circle.

Matrix multiplication

Matrix multiplication is performed row by column:

\[ \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix} = \begin{bmatrix} a~ ~b \\ c~ ~d \end{bmatrix} \times \begin{bmatrix} x \\ y \end{bmatrix} \]

• Number of columns in the first operand must equal the number of rows in the second operand.

Rotation Matrix

Deriving the rotation matrix using trigonometric identities.

\[ r = \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \cos \beta \\ \sin \beta \end{bmatrix} \]

\[ r' = \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos (\alpha + \beta) \\ \sin (\alpha + \beta) \end{bmatrix} \]

Ptolomy’s identity

The sum of two angles:

\[\cos (\alpha + \beta) = \cos \alpha ~ \cos \beta - \sin \alpha ~ \sin \beta\] \[\sin (\alpha + \beta) = \sin \alpha ~ \cos \beta + \cos \alpha ~ \sin \beta\]

Rotation Matrix Derivation

using the identities:

\[ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \alpha ~ \cos \beta - \sin \alpha ~ \sin \beta \\ \sin \alpha ~ \cos \beta + \cos \alpha ~ \sin \beta \end{bmatrix} \]

Rotation Matrix Derivation

recall:

\[ \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \cos \beta \\ \sin \beta \end{bmatrix} \]

substitute \(x\) and \(y\):

\[ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \cos \alpha - y \sin \alpha \\ x \sin \alpha + y \cos \alpha \end{bmatrix} \]

Rotation Matrix Derivation

as a matrix multiplication:

\[ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \alpha &-\sin \alpha \\ \sin \alpha &~\cos \alpha \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \]