## Important information

- the test will be Friday October 21 in class.

## Practice questions

### 1. transforms

#### 2D

Give an example in 2D that shows that scaling and rotations are not always commutative.

#### 3D

Give a transformation matrix that maps the black triangle to the
gray one. You may leave your answer as a product of matrices, with
rotations given symbolically as e.g. *R*_{z}(45) meaning rotate
CCW 45 degrees about the *z*
axis.

### 2. Camera Coordinates

Explain how to compute the view matrix give the *eye*
position, the *up* vector, and the point *center* being
looked at.

### 3. Projection matrices

#### Example projection/depth transform matrix

Consider the matrix

P =

1 0 0 0

0 1 0 0

0 0 3 -2

0 0 1 0

Describe the effect of the matrix, both geometrically and in the context of a rendering pipeline.

Describe the effect of applying this transformation twice.

#### Perspective projection matrix

Give a matrix that implements the following perspective transformation (D is the distance to the near clipping plane). Explain your answer.

```
[x,y,z,w]ᵀ → [Dx/z, Dy/z, D]
```

### 4. Clipping

Give robust inequlities to clip a homogeneous point `[x,y,z,w]ᵀ`

against the standard cube `[-1,1]×[-1,1]×[-1,1]`

. Are there any special
cases? If so, explain.