Rotations by Euler angles

Any rotation can be described by three Euler angles. In this example the original blue orthonormal vector triplet sigma1, sigma2, sigma3 is turned

1) by an angle psi around the sigma 3 axis (result: sigma 3 unchanged, red duplet)
2) by an angle theta around the line of nodes n, here the sigma1 axis, (result: green triplet)
3) by an angle phi around the e3 axis

finally resulting in the new yellow triplet e1, e2, e3.

In geometric algebra this is expressed by the rotor combination

R = exp(1/2 i sigma3 psi) exp(1/2 i n theta) exp(1/2 i e3 phi).

with i = unit pseudoscalar of Euclidean 3-space.

You can experiment with the graph by dragging sigma2 (changes the scale), and the dark red, green and golden points. (Sometimes the vectors will jump to other positions. To my understanding this is caused by the definition of the end points through intersections of lines and circles. Sometimes Cinderella chooses the other intersection. In this case just press the refresh button of your bowser to return to the original positions.)

Please enable Java for an interactive construction (with Cinderella).

[ GA with Cinderella ]

Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).