Euler angles vs. quaternions · Gimbal lock demo · Smooth interpolation
Euler angles (yaw, pitch, roll) are intuitive but suffer from gimbal lock: when two rotation axes align (e.g. pitch = ±90°), one degree of freedom is lost and rotations become erratic. Quaternions — four-dimensional numbers q = w + xi + yj + zk discovered by Hamilton in 1843 — represent rotations without singularities. SLERP (Spherical Linear intERPolation) smoothly interpolates between two quaternion orientations along the shortest arc on the 4D hyper-sphere, producing constant-angular-velocity motion with no jitter.
William Rowan Hamilton carved the quaternion multiplication rules (i² = j² = k² = ijk = −1) into Brougham Bridge in Dublin on 16 October 1843. Quaternions are now essential in aerospace (spacecraft ADCS), robotics, and every 3D game engine — Unity, Unreal, and Godot all default to quaternion rotations internally. The Apollo guidance computer used a gimbal-based inertial measurement unit; near gimbal lock the astronauts had to reorient the spacecraft to avoid losing navigation.
This visualiser places two identical cubes side by side: the left one is rotated using Euler angles (yaw, pitch and roll applied as sequential rotation matrices), while the right one is driven by a unit quaternion q = w + xi + yj + zk. As you change the orientation, the quaternion side eases toward its target using SLERP — spherical linear interpolation along the shortest 4D arc — letting you watch how quaternions avoid the singularities that plague Euler angles.
Two Canvas 2D wireframe cubes rendered with a simple perspective projection and painter's-algorithm face sorting. The Euler cube is built from a yaw-pitch-roll matrix; the quaternion cube converts the equivalent unit quaternion to a 3x3 rotation matrix. Forcing pitch to 90 degrees collapses the Euler cube into gimbal lock while the quaternion side keeps all three axes independent.
Drag the Yaw, Pitch and Roll sliders (each -180 to 180 degrees) to set the target orientation. The Auto-rotate Speed slider (0-30) spins yaw and roll continuously. Tick "Force gimbal lock" to pin pitch at 90 degrees, toggle "SLERP interpolation" to compare smooth easing against instant snapping, and use "Show axes", Pause and Reset. The live w, x, y, z panel reports the current quaternion.
William Rowan Hamilton devised quaternions in a flash of insight on 16 October 1843 and carved the rule i squared = j squared = k squared = ijk = minus one into Dublin's Brougham Bridge. Today every major 3D engine — Unity, Unreal and Godot — stores rotations as quaternions internally.
A quaternion is a four-component number q = w + xi + yj + zk that encodes a 3D rotation as a point on a 4D unit hypersphere. A unit quaternion represents any orientation without the singularities or ambiguity of Euler angles, which is why it is the standard for graphics, robotics and aerospace attitude control.
Gimbal lock happens when two of the three Euler rotation axes line up, so one degree of freedom is lost. Ticking "Force gimbal lock" sets pitch to 90 degrees: on the left cube yaw and roll then rotate about the same visual axis, while the right quaternion cube keeps all three axes fully independent.
SLERP, or spherical linear interpolation, blends between the current and target quaternion along the shortest great-circle arc on the hypersphere at constant angular velocity. With it ticked the quaternion cube eases smoothly toward new orientations; unticked, it snaps instantly to the target each frame so you can feel the difference.
Yes. It uses the standard yaw-pitch-roll to quaternion conversion, normalises the result to a unit quaternion, and converts back via the canonical quaternion-to-matrix formula. The SLERP routine includes the usual sign-flip for the shortest path and a linear fallback when the two quaternions are nearly parallel, matching textbook implementations.
Euler angles are intuitive because yaw, pitch and roll map directly to everyday concepts like heading and tilt, making them ideal for user input and display. The trade-off is gimbal lock and awkward interpolation, so many systems take Euler input but convert to quaternions internally for the actual rotation maths.