2023, no. 2

### MODIFICATION OF QUATERNION METHOD OF 3D-ROTATION DISCRIPTION BY MEANS OF ORTHOGONAL PROJECTIONS

#### Tsyplova I.E., Glukhova N.V., Golubkov A.V.

Quaternions are very important as a tool of scientific research in physics (quantum mechanics, physics of solid bodies, theory of relativity). In particular they help to describe rotations in spaces. The report is concerned with the problem of description of 3D space rotation of vectors around given axis. The known method of description of the rotations by means of quaternions is based on left-multiplication of the rotated vector by special quaternion q and right-multiplication of the result by its inverse $$(q^{-1})$$. We modified the method with the help of decomposition of rotated vector into its orthogonal projection and orthogonal component. As operator of rotation is linear, each part can be rotated separately, but the projection should not be modified, and orthogonal component can be rotated with the help of only left-multiplication and without division of the angle of rotation by 2. The only things one should do to find the quaternion-multiple for the rotation is to normalize any vector lying on the axis of rotation $$(p)$$ and use the formula $$q = \cos\phi + p \sin\phi$$ ( $$\phi$$ - is the angle of rotation). This approach simplifies calculations, which is evident from certain example solved by both methods. The result can be useful to researchers aimed to find the results of some 3D rotations even without special digital devices.

Keywords: 3D space rotation, vectors, orthogonal projection, orthogonal component, quaternions.

UDC: 512.64

PACS: 02.10

DOI: 10.17238/issn2226-8812.2023.2.47-53