## CSG140 Computer Graphics. Spring 2004. Quiz #1

### Prof. Futrelle

### This quiz is for Thursday 22 January - Closed book/notes

**Question 1.** For the figure below, transform the endpoints **a** and **b**
of the line segment to transform the line segment.
Each transform should be a 3x3 matrix (homogeneous coordinates).
The transforms you are to construct and apply are first: Construct a translation matrix
that moves the center of the line segment to the origin and then apply it to **a** and **b**.
Second, rotate each resulting point around the origin by -90° (minus 90 degrees).
Third, transform those resulting endpoints using the inverse of the original translation.
Draw the final state of the line segment, indicating each transformed
endpoint, **a'** and **b'**. Explain intuitively why you expect it to appear as you computed.

**Question 2.** Write out the 2x2 rotation matrix **R**(φ), for the general angle φ, and
another, **R**(-φ), for minus φ.
Form the product of **R**(φ) and **R**(-φ) and
show that it is the identity matrix.

**Question 3.** Two planes have [x,y] normal vectors **n**_{1} = [1,0]
and **n**_{2} = [1,1] (no z component).
Compute the dot product of the two using Cartesian coordinates and show that the result is
equal to the result obtained by using the formulation:
**n**_{1}·**n**_{2} = ||**n**_{1}|| ||**n**_{2}|| cosφ.