Question

Question 2: 2D Homogeneous Matrices [30 Marks] For each of the following homogenous matrices, write the decomposition into simple 2D transformations (translation, rotate, scale and shear). [6 Marks each] For example, the matrix M 10 0 0 1 Can be written as b) M2 0 0 d) M1 1 0 0.5 0.866 0 e) M 0.866 0.5 0

Answer 1

Homogeneous matrices are representation of homogeneous coordinates , that is three values are used to represent a 2D point.

This matrices are used to represent translation as a 3x3 multiplication operation.

Below are the representation of various transformations performed on a 2D point. From the given matrix, the transformations can be decomposed by the elements present at specific positions in the matrix.

• For translation, look for tx and ty in the translation matrix.
• For scaling, look for Sx and Sy in the Scaling matrix.
• For rotation, look for cos and sin values.
• For shear, look for a and b.

In the given matrix , look at the positions above mentioned to decompose the transformation operations..

Below are the reference matrices for different different transformations, which we have to use to decompose any given matrix.

This is carried out below for various given matrices.

o,y)

Sheah 20 o 0 01 0 o,0 fine Can G2 0,0

1 ne lete -1 1 0 0 0 1

Ris idi 5) M= 0.5 の866 o 0866 0S 0 0 0 h 0.60

Compare the transformation matrix and the given matrix and you will get your values.