What are quaternions 1

Quaternions were first described in 1853 by W. R. Hamilton, today they are used in computer graphics.

Definition: A quaternion is a quadruple = (,,,) with,,, . The set of quaternions is given with designated.

In an addition and a multiplication are defined in a certain way. Addition and multiplication in can be traced back to the addition or multiplication of real numbers.

One possibility for this is to understand quaternions as hyper-complex numbers (with 3 imaginary parts). The term then corresponds to a quaternion (,,,)

 + i + j + k

Here are i, j and k three different types of imaginary numbers. The addition and multiplication of quaternions are now obtained by applying the calculation rules for terms of real numbers, whereby for the multiplication of i, j and k the following link table is used as a basis:

 

  ·    i    j    k  
i-1k-j
j-k-1i
kj-i-1

 

As can be seen from the link table, the multiplication is not commutative, because it is e.g. i·j = k, but j·i = -k.

Theorem: The set of quaternions (, +, ·) Forms a sloping body.

Quaternions can also be understood as 2 × 2 matrices of complex numbers or 4 × 4 matrices of real numbers. The links matrix addition and matrix multiplication result in the same inclined body.

The complex 2 × 2 matrix corresponds to a quaternion (,,,)

 + i    + i
- + i – i

or the real 4 × 4 matrix

 - -
--
 --

Definition: The quaternion conjugated to a quaternion = (,,,)

  =  (, -, -, -)

The amount of is defined as

||  =  ·  =  2 + 2 + 2 + 2

Claim: The quaternion is multiplicatively inverse to a quaternion

 

-1  =   

Proof: It is

·-1  =    =    =  1

The immediate consequence is that for a quaternion with the amount 1:-1  =  .

Computation of rotations with quaternions

A rotation around the axis by an angle α is represented by the quaternion = (cos (α / 2), sin (α / 2), 0, 0). One point = (0, 0, 0) is given by the quaternion = (0, 0, 0, 0) represents. The rotated point ° results as

°  =   · ·

The rotation about any normalized vector (,,) by an angle α is given by the quaternion

= (cos (α / 2), sin (α / 2), sin (α / 2), sin (α / 2))

represents.

In fact, every quaternion of magnitude 1 represents a rotation. The set of rotations represented in this way forms the surface of a unit sphere in the 4. You can interpolate between two rotations by interpolating between two points on the surface of the sphere.

literature

[TL 94]K.D. Tönnies, H.U. Lemke: 3D computer graphic representations. Oldenbourg (1994)

 

Further: