Quaternions are an extension of the complex number concept, playing a crucial role in 3D space rotation. To better understand the relationship between quaternions and 3D rotations, we begin with the fundamentals of complex numbers, as there are notable parallels between the properties of complex numbers and their role in 2D rotation.
Complex numbers consist of real and imaginary components, where the imaginary unit (i) is defined as ( \sqrt{-1} ), so any complex number (z) can be represented as ( z = a + bi ), with (a) as the real part and (b) as the imaginary part, where (i^2 = -1). Complex numbers can be viewed as points or vectors in the complex plane, where the real part corresponds to the x-axis and the imaginary part to the y-axis. Complex addition and subtraction are straightforward, performed by adding or subtracting the real and imaginary parts, respectively.
Multiplication of complex numbers, however, is slightly more complex, requiring expansion and simplification using (i^2 = -1). A significant feature of complex multiplication is its connection to matrix multiplication. Specifically, every complex number can be associated with a particular 2x2 real matrix. For example, the complex number ( z = a + bi ) can be represented by the matrix:
[ \begin{bmatrix} a & -b \\ b & a \end{bmatrix} ]
This matrix form allows us to use matrix multiplication to perform complex number multiplication. During this process, we observe that complex number multiplication effectively corresponds to a rotation and scaling transformation in 2D space. For instance, the unit complex number 1 corresponds to the identity matrix, leaving points in the complex plane unchanged, while the imaginary unit (i) corresponds to a 90-degree rotation matrix, meaning that multiplying a complex number by (i) rotates it 90 degrees clockwise.
After grasping these complex fundamentals, we move to quaternions. Quaternions are a further extension of complex numbers into four-dimensional space. Each quaternion consists of a real part and three imaginary parts, generally written as ( q = w + xi + yj + zk ), where (w) is the real part and (x), (y), and (z) are the imaginary parts. The units (i), (j), and (k) follow distinct multiplication rules in four-dimensional space, where quaternion multiplication is analogous to complex multiplication yet complies with four-dimensional rules.
Quaternions are highly beneficial in 3D rotation operations because they avoid gimbal lock, a common issue with Euler angle representation. This makes quaternions a stable and continuous method for rotation interpolation, essential in computer graphics for 3D animation and simulation. Through quaternions, we can use linear algebra and basic geometry principles to intuitively derive and comprehend 3D spatial rotation. Quaternions provide direct representation of the rotation axis and rotation angle, while interpolating quaternions enables smooth transitions between two rotations. This is extremely helpful in animation and real-time rendering for creating fluid movement. For example, in game development and flight simulation, quaternions are often used to achieve coherent flight paths, allowing engineers to implement smooth motion.