Game Mathematics

Official Week 8 Solutions

1.

Check the first property of linearity:

[Graphics:Images/solutions_gr_1.gif]

[Graphics:Images/solutions_gr_2.gif]

Therefore, [Graphics:Images/solutions_gr_3.gif], so the transformation is nonlinear.

2.

Check the first property of linearity:

[Graphics:Images/solutions_gr_4.gif]

Check the second property of linearity:

[Graphics:Images/solutions_gr_5.gif]

Constructing the induced matrix is a simple matter of applying the transformation to the columns of the [Graphics:Images/solutions_gr_6.gif] identity matrix. The result is shown below:

[Graphics:Images/solutions_gr_7.gif]

3.
The operation is not linear, since it involves translation, although it could be made linear under homogeneous coordinates.

4.

Define parent-child relationships between all the polygons composing the character such that the transformation of a parent will always be applied to the children (through matrix concatenation).

5.

[Graphics:Images/solutions_gr_8.gif]

6.

The transformation we seek is [Graphics:Images/solutions_gr_9.gif], since this effectively reflects the point off the x-axis. To get the induced matrix, we apply this transformation to the columns of the [Graphics:Images/solutions_gr_10.gif] identity matrix. The result is shown below:

[Graphics:Images/solutions_gr_11.gif]


Converted by Mathematica      December 13, 2001