LiveGraphics3D Example:
Euler Angles

Main Convention
 There are several ways to describe rotations in three dimensions. For example any rotation can be defined by its axis and angle. This information can also be put into a mathematical object called quaternion. (This is how most computer programs and in fact LiveGraphics3D deal with rotations.) However in physics three so-called Euler angles (or Euler's angles or Eulerian angles) are often preferred (mainly for quantum mechnical reasons) to define a rotation. The idea of Euler angles is to split the complete rotation of a cartesian coordinate system into three simpler rotations about the axes of this system. This picture trys to visualize the main convention (according to H. Goldstein, Classical Mechanics, 2nd ed., p. 145) which uses a first rotation (blue) about the z-axis, then a second rotation (green) about the rotated x-axis (denoted as x') and finally a third rotation (red) about the new z-axis (z'').

x-Convention
 This graphic demonstrates another convention for the Euler angles. In this x-convention the three axes of rotation are the z-, the (not rotated!) x- and again the z-axis. The same angles are used in this example but in reversed order, such that the complete rotation is identical to the rotation of the example above. These examples are probably the most colorful explanations of Euler angles ever created. However, they are primarily intended to let LiveGraphics3D draw some greek symbols. (If you cannot see any greek symbols, the Java virtual machine of your web browser does probably not support Unicode characters.)

Martin Kraus, April 15, 2021