CQ1

Sep. 10

Older 2D games are of great help in understanding the core fundamental concepts that we learn in the CIS4620/5620 Computer Animation class. Let's use this short clip from Capcom’s Megaman X4 as a reference for our first Conceptual Question of the semester:

In the clip above, we can see a rudimentary application of rotation and translation:

1. Megaman just ‘translates’ (moves) left/right and up/down,
2. The green beam rotates around the gun.

Note how the beam keeps rotating about the gun, even when the gun/Megaman is moving(‘translating’).

Now, consider the following snapshot from the clip and a hypothetical configuration:

Given:

• The global frame of reference of the world F0 represented by axes (x0, y0, z0).
• Megaman’s local frame of reference is FM , which is at the tip of the gun, is represented by axes (xM, yM, zM). The FM axes are aligned with F0 and just translated by the vector m.
• the gun’s local frame of reference FG is also at the tip of the gun and is represented by (xG, yG, zG) with axis xG pointing in the direction of the beam.
• the beam length is 5 units.

Question: What sequence of transformations can be used to compute the tip of the beam’s target point b in the world coordinate system?

Your answer should be in terms of a sequence of homogeneous transformation matrices, each representing either a pure rotation or pure translation. That is, homogeneous transformation matrices of the form $H_{rot}(axis, angle)$ or
$H_{trans}(d)$. When multiplied together, the sequence of transformation matrices should transform the tip of the beam $b^G$ expressed in the G coordinate system to its equivalent representation $b^0$ in the world coordinate system.

Answer

$$\begin{aligned} \vec{b}^0 &= F_G^0 \vec{b}^G \\ &= F_M^0 F_G^M \vec{b}^G \\ &= H_{\text{trans}}(\vec{m}) H_{\text{rot}}(Z_M, \theta) \vec{b}^G \end{aligned}$$