Activity 2.6.4

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

  1. 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 \(y\text{.}\)


  2. Find the matrix transformation that translates our character to a new position in the plane, as shown in Figure 2.6.13

    Figure 2.6.13 Translating to a new position.
  3. As originally drawn, our character is waving with one of their hands. In one of the movie's scenes, we would like her to wave with their other hand, as shown in Figure 2.6.14. Find the matrix transformation that moves them into this pose.

    Figure 2.6.14 Waving with the other hand.
  4. Later, our chracter performs a cartwheel by moving through the sequence of poses shown in Figure 2.6.15. Find the matrix transformations that create these poses.

    Figure 2.6.15 Performing a cartwheel.
  5. Next, we would like to find the transformations that zoom in on our character's face, as shown in Figure 2.6.16. 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 the character from the original pose, shown in the upper left, to the final pose, shown in the lower right?

    Figure 2.6.16 Zooming in on our characters' face.
  6. We would also like to create our character's shadow, shown in the sequence of poses in Figure 2.6.17. Find the sequence of matrix transformations that achieves this. In particular, find the matrix transformation that take our character from their original pose to their shadow in the lower right.

    Figure 2.6.17 Casting a shadow.
  7. Write a final scene to the movie and describe how to construct a sequence of matrix transformations that create your scene.