Why use quaternions for 3D rotations?

Quaternions avoid gimbal lock, interpolate smoothly between orientations (slerp), and are more numerically stable than Euler angles or repeated matrix multiplications. A unit quaternion corresponds to a 3D rotation and can be applied efficiently to vectors.

What does this quaternion calculator do?

This calculator lets you add, subtract and multiply quaternions, compute their conjugate, norm, inverse and normalized form. It can also build a rotation quaternion from an axis–angle pair, rotate 3D vectors, and visualise the rotation in an interactive 3D canvas.

How do I build a rotation quaternion from axis and angle?

Given a unit axis vector u = (ux, uy, uz) and an angle θ in radians, the corresponding unit quaternion is q = (cos(θ/2), ux sin(θ/2), uy sin(θ/2), uz sin(θ/2)). The calculator implements this formula directly from your axis and angle inputs.

Do quaternions commute under multiplication?

No. Quaternion multiplication is associative but not commutative: in general q1 · q2 ≠ q2 · q1. This is consistent with the non-commutative nature of composing 3D rotations.

Do I need a unit quaternion for rotations?

To represent a pure rotation, the quaternion must be unit-length (norm equal to 1). The calculator provides a normalize function and an option to auto-normalize before applying a rotation, which is standard practice in graphics, robotics and simulation engines.

Quaternion Calculator

Q: What is a quaternion?

A quaternion is a four-dimensional extension of complex numbers, usually written as q = w + xi + yj + zk with real coefficients w, x, y, z. Quaternions form a non-commutative division algebra and are widely used to represent rotations in 3D space.

3D rotations & algebra

Work with quaternions for 3D rotations and algebraic operations. Define quaternions, add/subtract, multiply, conjugate, normalize, invert, build rotation quaternions from axis–angle and rotate 3D vectors with an interactive 3D visualization of the original and rotated vector.

quaternion arithmetic 3D rotation axis–angle & unit quaternions graphics & robotics

Quaternion inputs & algebra

Quaternion q₁

Form: q = w + x i + y j + z k

Typical rotation quaternions are unit length: \(\|q\| = 1\).

Quaternion q₂ (optional)

For addition, subtraction or chained rotations.

Use q₂ for composing rotations (q = q₂·q₁, etc.).

Operation

Quaternion multiplication is not commutative: q₁·q₂ ≠ q₂·q₁ in general.

Presets

Axis–angle → q₁

Uses q = (cos(θ/2), u·sin(θ/2)).

3D rotation with quaternion & interactive visualisation

Unit quaternion required for pure rotation

Rotation quaternion q_rot

You can reuse q₁ above, or type directly.

Auto-normalize q_rot before rotating

Vector to rotate v

3D vector v = (v_x, v_y, v_z).

v_x

v_y

v_z

Rotated vector & rotation matrix

Original vector v

Rotated vector v′

Simple orthographic-like projection. Rotate around different axes to see the path from v to v′.

Rotation matrix R (3×3) equivalent to q_rot

v′ = R · v

For a unit quaternion q = (w, x, y, z), the matrix R is orthogonal with determinant 1, so it preserves lengths and angles.

What is a quaternion?

A quaternion is a hyper-complex number of the form \[ q = w + x \mathbf{i} + y \mathbf{j} + z \mathbf{k}, \] where \(w, x, y, z \in \mathbb{R}\) and \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) satisfy \(\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{ijk} = -1\). Quaternions extend complex numbers and form a non-commutative division algebra.

Basic quaternion operations

Addition: component-wise — \(q_1 + q_2 = (w_1 + w_2,\; x_1 + x_2,\; y_1 + y_2,\; z_1 + z_2)\).
Conjugate: \(\overline{q} = (w,\; -x,\; -y,\; -z)\).
Norm: \(\|q\| = \sqrt{w^2 + x^2 + y^2 + z^2}\).
Inverse: for \(q \neq 0\), \(q^{-1} = \overline{q} / \|q\|^2\).

Multiplication is more involved but encodes rotation when \(\|q\| = 1\).

Quaternions and 3D rotations

A unit quaternion (\(\|q\| = 1\)) can represent a 3D rotation. Given an axis–angle pair (a unit axis \( \mathbf{u} = (u_x,u_y,u_z)\) and angle \(\theta\)):

\[ q = \left(\cos\frac{\theta}{2},\; u_x \sin\frac{\theta}{2},\; u_y \sin\frac{\theta}{2},\; u_z \sin\frac{\theta}{2}\right). \]

To rotate a vector \(\mathbf{v} = (v_x,v_y,v_z)\), embed it as a pure quaternion \(p = (0, v_x, v_y, v_z)\) and compute:

\[ p' = q \, p \, q^{-1}. \] The rotated vector is the imaginary part of \(p'\).

Rotation matrix from quaternion

For a unit quaternion \(q = (w,x,y,z)\), the corresponding \(3\times3\) rotation matrix is

\[ R = \begin{bmatrix} 1 - 2(y^2 + z^2) & 2(xy - wz) & 2(xz + wy) \\ 2(xy + wz) & 1 - 2(x^2 + z^2) & 2(yz - wx) \\ 2(xz - wy) & 2(yz + wx) & 1 - 2(x^2 + y^2) \end{bmatrix}. \]

The calculator uses this matrix internally to show the linear map \( \mathbf{v}' = R\mathbf{v} \) alongside the direct quaternion formula.

Typical applications

Game development & graphics: camera orientation, character bones, camera rigs.
Robotics & drones: attitude representation, sensor fusion, control loops.
Simulation & physics: rigid body orientation and numerical integration of angular velocity.
Interpolation: smooth orientation interpolation using slerp between two unit quaternions.

Quaternion Calculator

Quaternion inputs & algebra

Quaternion algebra results

3D rotation with quaternion & interactive visualisation

Rotated vector & rotation matrix

Rotation matrix R (3×3) equivalent to q_rot

What is a quaternion?

Quaternions and 3D rotations

Rotation matrix from quaternion

Typical applications

Related algebra & 3D math tools

Quaternion Calculator

Quaternion inputs & algebra

Quaternion algebra results

3D rotation with quaternion & interactive visualisation

Rotated vector & rotation matrix

Rotation matrix R (3×3) equivalent to qrot

What is a quaternion?

Quaternions and 3D rotations

Rotation matrix from quaternion

Typical applications

Related algebra & 3D math tools

Rotation matrix R (3×3) equivalent to q_rot