Quaternion

Quaternion §

A quaternion is a type of hypercomplex number that extends complex numbers. It is typically represented as:

$q=a+bi+cj+dk$

where:

• $a$, $b$, $c$, and $d$ are real numbers,
• $i$, $j$, and $k$ are the fundamental quaternion units.

Quaternions have properties that make them useful for representing rotations and orientations in three-dimensional space. They avoid the singularity and ambiguity problems of Euler angles and are more compact than rotation matrices.

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Operations with Quaternions §

Quaternions support addition, subtraction, multiplication, and division, but multiplication is not commutative:

$ij=k,ji=-k,jk=i,kj=-i,ki=j,ik=-j,i^2=j^2=k^2=ijk=-1$

Quaternion Conjugate §

The conjugate of a quaternion $q=a+bi+cj+dk$ is defined as:

$q^{\ast }=a-bi-cj-dk$

Norm of a Quaternion §

The norm of a quaternion is given by:

$\Vert q\Vert =\sqrt{a^2+b^2+c^2+d^2}$

Quaternion Inverse §

The inverse of a quaternion $q$ is defined as:

$q^{-1}=\frac{q^{\ast }}{\Vert q{\Vert }^2}$

Quaternion Matrix §

A quaternion matrix is a matrix where each element is a quaternion. These matrices can represent complex transformations in higher-dimensional spaces and are used in various fields, including theoretical physics, computer graphics, and robotics.

Applications §

Quaternion matrices are particularly useful for:

• Rotating points in three-dimensional space,
• Interpolating orientations (slerp),
• Simulating rigid body dynamics.

Their ability to represent rotations without suffering from gimbal lock makes them invaluable in 3D computer graphics and aerospace engineering.