Activity2.6.4
In this activity, we will use homogeneous coordinates and matrix transformations to move our Jessie into a variety of poses.

Since we regard our character as living in \(\real^3\text{,}\) we will consider matrix transformations defined by matrices
\begin{equation*} \left[\begin{array}{rrr} a \amp b \amp c \\ d \amp e \amp f \\ 0 \amp 0 \amp 1 \\ \end{array}\right] \text{.} \end{equation*}Verify that such a matrix transformation transforms points in the plane \(z=1\) into other points in this plane; that is, verify that
\begin{equation*} \left[\begin{array}{rrr} a \amp b \amp c \\ d \amp e \amp f \\ 0 \amp 0 \amp 1 \\ \end{array}\right] \threevec{x}{y}{1} = \threevec{x'}{y'}{1} \text{.} \end{equation*}Express the coordinates of the resulting point \(x'\) and \(y'\) in terms of the coordinates of the original point \(x\) and \(y\text{.}\)
ANIMATE

Find the matrix transformation that translates Jessie to a new position in the plane, as shown in Figure 13

As originally drawn, Jessie is waving with one of her hands. In one of the movie's scenes, we would like her to wave with her other hand, as shown in Figure 14. Find the matrix transformation that moves her into this pose.

Later, Jessie performs a cartwheel by moving through the sequence of poses shown in Figure 15. Find the matrix transformations that create these poses.

Next, we would like to find the transformations that zoom in on Jessie's face, as shown in Figure 16. To do this, you should think about composing matrix transformations. This can be accomplished in the diagram by using the Compose button, which makes the current pose, displayed on the right, the new beginning pose, displayed on the left. What is the matrix transformation that moves Jessie from the original pose, shown in the upper left, to the final pose, shown in the lower right?

We would also like to create Jessie's shadow, shown in the sequence of poses in Figure 17. Find the sequence of matrix transformations that achieves this. In particular, find the matrix transformation that take Jessie from her original pose to her shadow in the lower right.
Write a final scene to the movie and describe how to construct a sequence of matrix transformations that create your scene.