Week 2 - Linear and Affine Transformations, Homogeneous and 3D Coordinates

Question 1

What are linear transformations?

Answer 1

Transformations effected by a matrix product.

Question 2

Give 4 examples of linear transformations.

Answer 2

Rotation, scaling, shear, mirroring.

Question 3

What is the matrix transformation for rotation?

Answer 3

cosθ -sin θ
sin θ cos θ

Where θ is the angle of rotation around the origin.

Question 4

What is the matrix transformation for scaling?

Answer 4

λ1 0
0 λ2

Where λ1 and λ2 are the scaling factors.

Question 5

What is isotropic scaling?

Answer 5

Both x and y scaled by the same factor (equal entries on diagonal - λ1 = λ2).

Question 6

What is anisotropic scaling?

Answer 6

x and y scaled by the different factors (unequal entries on diagonal - λ1 =! λ2).

Question 7

What is the matrix transformation for shear?

Answer 7

Either:
1 λ
0 1

Or:
1 0
µ 1

(One coordinate left unchanged!)

Question 8

What is the formula for the general transformation?

Answer 8

x’ = Mx + t

Question 9

What are 3 properties of affine transformations?

Answer 9

Map straight lines into straight lines.

Map parallel lines into parallel lines.

Comprise all combinations of scaling, rotations, shears and translations.

Question 10

What is an affine transformation?

Answer 10

Linear transformation + translation.

Question 11

Why are homogeneous coordinates used?

Answer 11

To combine affine transformations without it getting too complicated.

Question 12

Give the homogeneous rotation matrix for 2D coordinates.

Answer 12

cosθ -sin θ 0
sin θ cos θ 0
0 0 1

Question 13

Give the homogeneous scaling matrix for 2D coordinates.

Answer 13

λx 0 0
0 λy 0
0 0 1

Question 14

Give the homogeneous shear matrix for 2D coordinates.

Answer 14

1 λ 0
0 1 0
0 0 1

or

1 0 0
µ 1 0
0 0 1

Question 15

Give the homogeneous translatation matrix for 2D coordinates.

Answer 15

1 0 tx
0 1 ty
0 0 1

Question 16

What are the 4 exceptions to “matrix multiplication is not commutative” in terms of translations?

Answer 16

Rotations in 2D commute with eachother (but not in 3D).

Translations commute with each other.

Scalings commute with each other.

Shears commute with each other.

Question 17

How do you rotate an object about a point?

Answer 17

Translate object back to the origin, T.

Rotate around the origin, R.

Translate back to original location, T^-1.

Question 18

How do you display the coordinates of a line segment?

Answer 18

Transform end points from local and world coordinates to display coordinates.

Draw line segment from joining these end points in display coordinates.

Question 19

How do you display the coordinates of a rectangle?

Answer 19

Transform corners from local and world coordinates to display coordinates.

Draw rectangle in display coordinates.

Question 20

How do you rotate by θ around the z-axis of a coordinate system (3D)?

Answer 20

cosθ -sin θ 0
sin θ cos θ 0
0 0 1

Question 21

How do you rotate by θ around the y-axis of a coordinate system (3D)?

Answer 21

cosθ 0 sin θ
0 1 0
-sin θ 0 cos θ

Question 22

How do you rotate by θ around the x-axis of a coordinate system (3D)?

Answer 22

1 0 0
0 cosθ -sin θ
0 sin θ cos θ

Question 23

What is the matrix for the inverse of the rotation matrix R?

Answer 23

R^T (the transpose).

Question 24

How do you rotate around an arbitrary axis?

Answer 24

Perform a first rotation to align v to one of the coordinate system’s axes (x, y or z).

Perform rotation around this axis.

Perform the inverse of step 1.