Geometry

Transformation

Transformation refers to the movement of a shape or figure in a coordinate plane without changing its size or shape (except in dilation). There are four main types of transformations: Reflection, Rotation, Translation, and Dilation.

Reflection

A reflection (Flipping over a line) flips a figure over a given axis or line. The formulas depend on the axis of reflection. The formulae are given below:

- Over the x-axis: \( P(x, y) \xrightarrow{\text{Re: x - axis}} P'(x, -y) \) 

- Over the y-axis: \( P(x, y) \xrightarrow{\text{Re: y - axis}} P'(-x, y) \) 

- Over the line \( (y = x) \): \( P(x, y) \xrightarrow{\text{Re: y = x }} P'(y, x) \) 

- Over the line \( (y = -x) \): \( P(x, y) \xrightarrow{\text{Re: y = -x }} P'(-y, -x) \)

- Over the line \( (x = a) \): \( P(x, y) \xrightarrow{\text{Re: x = a }} P'(2a – x, y )\)

- Over the line \( (y = b) \): \( P(x, y) \xrightarrow{\text{Re: y = b }} P'(x, 2b - y )\)

Rotation

A rotation (Turning around a point) rotates a figure counterclockwise about the origin (0, 0) by a given angle: 

- 90° counterclockwise: \( P(x, y) \xrightarrow{\text{Ro: 0, +90}} P'(-y, x) \) 

- 180° counterclockwise: \( P(x, y) \xrightarrow{\text{Ro: 0, +180}} P'(-x, -y) \) 

- 270° counterclockwise: \( P(x, y) \xrightarrow{\text{Ro: 0, +270}}  P'(y, -x) \) 

- For clockwise rotation, reverse the sign of the angle. 

- Note: The rotation for [Ro: 0, 360] will remain the same like \( P(x, y) \xrightarrow{Ro: 0, \pm 360^0}  P’(x, y) \) 

Translation

A translation (Sliding without rotating) moves a figure without rotating or flipping it. Let \(T\bigl(\begin{smallmatrix} a \\ b \end{smallmatrix}\bigr)\) be the translation vector, then

- Formula: \( P(x, y) \xrightarrow{T\bigl(\begin{smallmatrix} a \\ b \end{smallmatrix}\bigr)} P’(x + a, y + b)\) 

  - Where \( a \) is the horizontal shift 

  - \( b \) is the vertical shift 

Dilation

A dilation (Resizing the shape) changes the size of a figure but keeps its shape the same. It is defined by a scale factor \( k \): 

- Formula: 

a. Center (0, 0):  \( P(x, y) \xrightarrow{[(0,0), k]} P’(kx, ky) \) 

b. Center (a, b): \(P(x, y) \xrightarrow{[(0,0), k]} P’(k(x-a)+a, k(y-b)+b)\)

  - If \( k > 1 \), the figure enlarges 

  - If \( 0 < k < 1 \), the figure shrinks 

  - If \( k = 1 \), the figure remains the same