Euler Angle To Quaternion Calculator

Euler to Quaternion Conversion:

\[ q = \begin{cases} q_w = \cos(\psi/2)\cos(\theta/2)\cos(\phi/2) + \sin(\psi/2)\sin(\theta/2)\sin(\phi/2) \\ q_x = \sin(\psi/2)\cos(\theta/2)\cos(\phi/2) - \cos(\psi/2)\sin(\theta/2)\sin(\phi/2) \\ q_y = \cos(\psi/2)\sin(\theta/2)\cos(\phi/2) + \sin(\psi/2)\cos(\theta/2)\sin(\phi/2) \\ q_z = \cos(\psi/2)\cos(\theta/2)\sin(\phi/2) - \sin(\psi/2)\sin(\theta/2)\cos(\phi/2) \end{cases} \]

ψ (psi) - Z rotation:

degrees

θ (theta) - Y rotation:

degrees

φ (phi) - X rotation:

degrees

Rotation Order:

Quaternion (w, x, y, z):

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1. What is Euler Angle to Quaternion Conversion?

The Euler Angle to Quaternion conversion transforms three sequential rotation angles (ψ, θ, φ) about the Z, Y, X axes into a single rotation represented as a quaternion. Quaternions provide a more efficient and numerically stable way to represent 3D rotations compared to Euler angles.

2. How Does the Calculator Work?

The calculator uses the following conversion formulas:

Where:

\( \psi \) — Rotation about Z axis (yaw)
\( \theta \) — Rotation about Y axis (pitch)
\( \phi \) — Rotation about X axis (roll)
All angles are in degrees

3. Importance of Quaternions

Details: Quaternions avoid gimbal lock and provide smoother interpolation between rotations compared to Euler angles. They are widely used in computer graphics, robotics, and aerospace applications.

4. Using the Calculator

Tips: Enter rotation angles in degrees. The default rotation order is ZYX (yaw, pitch, roll), but other common orders are available. The calculator outputs a unit quaternion (w, x, y, z) where w is the scalar part.

5. Frequently Asked Questions (FAQ)

Q1: Why use quaternions instead of Euler angles?
A: Quaternions avoid gimbal lock and provide more efficient rotation composition and interpolation.

Q2: What is the range of valid Euler angles?
A: While angles can be any real number, typical ranges are -180° to 180° or 0° to 360°.

Q3: How do I convert back to Euler angles?
A: The reverse conversion requires careful handling of edge cases and is more complex than the forward conversion.

Q4: What rotation conventions does this use?
A: The calculator uses right-handed, intrinsic rotations by default (ZYX order).

Q5: Are the quaternions normalized?
A: Yes, the output quaternion will always have unit length (norm = 1).