2 - Linear and Affine Transformations

Question 1

linear transformations:

Answer 1

Transformations affected by a matrix product .
- rotation,
- scaling,
- shear
- mirroring

Question 2

Rotation around origin

Answer 2

Cos(θ) -Sin(θ)
Sin(θ) Cos(θ)

Leaves length of matrix unchanged

Question 3

Scaling

Answer 3

the scale matrix below is essentially a modified identity matrix

Isotropic = Equal entries on the diagonal ( makes something bigger or smaller without changing the aspect ratio)
Anisotropic = Different entries on the diagonal

| λ1 0 | |x|

0 λ2 | |y|

Question 4

Shear

Answer 4

One coordinate is left unchanged. (Pulling diagonally)

Question 5

Combining Linear Transformations

Answer 5

to combine example:
Rotation = R
Shear = S

combined result Z= (R⋅S)x
⋅ = Dot product

Question 6

Dot Product
|a1 | ⋅ | b1|
|a2 | ⋅ | b2|

Answer 6

a1b1 + a2b2

Question 7

Affine Transformations

Answer 7

that involve a traslation; they can either be standalone, or combined with a linear transformation.

Z = Mx + t (traslation)

Question 8

Properties of Affine Transformation

Answer 8

Affine transformations have three properties:
⦁ They map straight lines to straight lines
⦁ They map parallel lines to parallel lines
⦁ They compromise all combinations of scaling, rotations, shears and translations

Question 9

Combining Affine Transformations

Answer 9

homogenous coordinates

Question 10

Matrix Multiplication
|a b| x |e f|
|c d| |g h|

Answer 10

|ae+bg af+bh |
|ce+dg cf+dh |

To multiply two matrices, use the dot product of each row of the first matrix to the column of each second

Question 11

Cross Product