What Are Quaternions?
Quaternions are a number system that extends complex numbers. Discovered by Sir William Rowan Hamilton in 1843, they are written as q = a + bi + cj + dk, where a, b, c, d are real numbers and i, j, k are the fundamental quaternion units. Quaternions are widely used in 3D computer graphics, robotics, and aerospace engineering for representing rotations.
Hamilton Product Rules
Fundamental Identities
The defining equations of quaternion algebra.
Product Rules
Products of the basis elements follow specific rules.
Anti-Commutativity
Reversing the order negates the result.
Quaternion Operations
Addition & Subtraction
Quaternion addition and subtraction are performed component-wise: (a1+b1i+c1j+d1k) + (a2+b2i+c2j+d2k) = (a1+a2) + (b1+b2)i + (c1+c2)j + (d1+d2)k.
Multiplication (Hamilton Product)
Quaternion multiplication is not commutative (q1 x q2 is generally not equal to q2 x q1). The product is computed by distributing and applying the multiplication rules for i, j, and k. The result components are:
- Real: a1a2 - b1b2 - c1c2 - d1d2
- i: a1b2 + b1a2 + c1d2 - d1c2
- j: a1c2 - b1d2 + c1a2 + d1b2
- k: a1d2 + b1c2 - c1b2 + d1a2
Conjugate, Norm & Inverse
The conjugate of q = a + bi + cj + dk is q* = a - bi - cj - dk. The norm is |q| = sqrt(a² + b² + c² + d²). The inverse is q⊃-1 = q* / |q|².
Applications
- 3D Rotations: Quaternions avoid gimbal lock and provide smooth interpolation (SLERP) for 3D rotations in games and simulations.
- Aerospace: Spacecraft attitude control systems use quaternions for orientation tracking.
- Robotics: Robot arm joint rotations are efficiently computed using quaternions.
- Computer Vision: Camera orientation and pose estimation rely on quaternion math.