Quaternion Functions: Multiplication, Inverse, Conjugate, and Norm Operations

Quaternions are mathematical tools that extend complex numbers, widely used in 3D graphics, computer vision, and robotics to represent rotations. A quaternion consists of one real component and three imaginary components, typically expressed as q = w + xi + yj + zk. ### Quaternion Multiplication Quaternion multiplication is non-commutative, with operations based on the fundamental properties i² = j² = k² = ijk = -1. When multiplying two quaternions, each component must be expanded and like terms combined. In code implementation, this involves computing the Hamilton product: q1·q2 = (w1w2 - x1x2 - y1y2 - z1z2) + (w1x2 + x1w2 + y1z2 - z1y2)i + (w1y2 - x1z2 + y1w2 + z1x2)j + (w1z2 + x1y2 - y1x2 + z1w2)k ### Quaternion Conjugate The quaternion conjugate is defined by negating the imaginary components: q* = w - xi - yj - zk. The conjugate operation is frequently used in calculating the quaternion inverse. In programming, this can be implemented by simply flipping the signs of the i, j, and k components while keeping the real part unchanged. ### Quaternion Inverse The quaternion inverse is defined as the conjugate divided by the squared norm: q⁻¹ = q* / ||q||². The inverse exists only when the quaternion's norm is non-zero. For implementation, first compute the norm squared (w² + x² + y² + z²), then multiply the conjugate by the reciprocal of this value. For unit quaternions, the inverse equals the conjugate, optimizing rotation calculations. ### Quaternion Norm The quaternion norm resembles vector magnitude, calculated as ||q|| = √(w² + x² + y² + z²). Normalized quaternions (unit quaternions) have a norm of 1 and are commonly used to represent rotations. Code implementation typically involves computing the square root of the sum of squared components, often optimized using fast inverse square root algorithms in performance-critical applications. ### Vector Rotation Example When rotating a 3D vector using quaternions, the vector is first represented as a pure quaternion (real part 0), then transformed through quaternion multiplication. The implementation steps include: Constructing the rotation quaternion q = [cos(θ/2), sin(θ/2)·n], where n is the rotation axis (unit vector) and θ is the rotation angle. Converting the target vector v to quaternion form v' = [0, v]. Computing the rotated vector v'' = q·v'·q⁻¹, where q⁻¹ is the conjugate of q (since for unit quaternions, the inverse equals the conjugate). Extracting the imaginary components of v'' as the coordinates of the rotated vector. This method is more efficient than rotation matrices and avoids gimbal lock problems, making it widely adopted in 3D graphics processing. The code implementation typically involves three quaternion multiplications and careful component management to ensure numerical stability.