Calculate Euler Angles and Quaternions

Convert Euler angles and quaternions, and visualize the corresponding three-dimensional rotations. This calculator works in both directions, generating a graphic showing how the three axes rotate in space. The default setting is to calculate a quaternion from the Euler angles. Here, please enter the three angles in degrees and select the calculation order. For the reverse calculation, enter the four values ​​of the quaternion. The Euler angles for the sequence xyz will be calculated. The mathematical background is complex; some explanation can be found below the calculator and 3D visualizer.
The x-axis is drawn in blue, the y-axis in red, and the z-axis in yellow. If the axis is dashed, it points away from the viewer. The shorter the axis, the more it points towards the viewer or exactly opposite. Axes perpendicular to the viewing direction are the longest. Axes in the viewing direction are not visible.

The following is a very simplified explanation of the mathematical context:
Euler angles describe orientation in space. Angle α rotates around the x-axis, β around the y-axis, and γ around the z-axis. A change in α causes roll, β pitch, and γ yaw. These terms are used in aviation. If you imagine an airplane, roll means the wings move up or down. Pitch makes the nose and tail move up or down. Yaw makes the nose and tail turn left or right.
Quaternions are four-dimensional numbers discovered by William Rowan Hamilton in 1843. They generalize the two-dimensional complex numbers to four dimensions. However, multiplication is not commutative with quaternions. Therefore, the order in which the angles are applied is important for their calculation. Quaternions are primarily used to describe rotations in 3D space. Quaternions avoid problems like gimbal lock, which can occur with Euler angles.
Gimbal lock occurs when two of the three axes of an Euler angle system are aligned. In this case, one degree of rotational freedom is lost, and certain directions can no longer be controlled independently. Quaternions avoid this problem because they represent orientation in space without an axis sequence and do not have singular positions like these.