9.5.2

508 • CHAPTER 8
TOOLS MENU • 509
Consider this example
 Take an object whose angle system is initially 0/0/0. First make a rotation
of 30° around heading, so that the X and Z axes of the object system are now
rotated compared to the X and Z axes of the world system. Call these new axes
X’ and Z’ (Y’ is identical to Y).
 A pitch of 20° now causes the object system to be rotated upward around
the object system’s X’ axis. Z’ becomes Z’’ and Y’ becomes Y’’ (now X’ and X’’
are identical).
 Lastly, we rotate around a bank angle of -45°. This rotation causes the rotated
object system to be tilted around the Z’’ axis. X’’ becomes X’’’ and Y’’ becomes
Y’’’ (this time Z’’ and Z’’’ are identical).
 The object has now achieved an angle system of 30/20/-45 by consecutive
rotations around H, P and B on a system in each case already rotated. Thus, HPB
rotates neither around object nor around world axes. That bank is identical to a
rotation around an object axis is purely coincidental. There are several different
Euler systems, each one with a particular rotation order.
While this all seems rather impractical, the Euler system has a major advantage: Rotations of objects
are decoupled from one another as much as possible, which is not the case with rotations around
object axes. Heading does not affect bank, bank does not affect pitch. Imagine if the X position of an
object always affected the Y and Z positions ...
A further example claries the decoupling problem
This example should to clarify why CINEMA 4D uses the Euler system. You’ll nd this system easy to
use once you are familiar with it.
 Suppose for a moment that CINEMA 4D doesn’t use Euler angles. Imagine a
point on the X-axis in position 100/0/0.
 Rotate the point 90° around the Y-axis. It then lies exactly on the Z-axis at
0/0/100. Now keep rotating, this time for 30° around the X-axis. The point now
lies in the ZY plane at 0/87/50.
 So far, so good. Now, however, you reverse the rotation order. The point at
100/0/0 is still at the position 100/0/0 after a rotation of 30° around the X-axis.
Subsequently, you rotate again around the Y-axis for 90°. The point is now at
0/0/100, a completely different position.
 So, due to the mathematical properties of rotations, the sequence of rotations
around the object axes is not commutative (i.e. rotation A plus rotation B does
not equal rotation B plus rotation A). This leads to unexpected results with
animation.