The reason Quaternions found applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, and orbital mechanics, is exactly their two distinct geometric interpretations that you are asking about. A. They can represent rotations in 3 and 4 space (and are called Rotation quaternions in this case) and B. They can represent an orientation (rotation relative to a reference position) in 3D space, they are called orientation quaternions or attitude quaternions. Algorithmically they require a less number of operations to implement rotations (when compared with the multiplying a vector with a rotation matrix that involves Euler angles). Any rotation of a 4D vector can be represented by the product of two Quaternions with that vector. Two excellent references describing these details from an application perspective are:

1. "Kuipers, Jack (2002), "Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality" (reprint edition), Princeton University Press. ISBN 0-691-10298-8" and

2. "Hanson, Andrew J. (2006), "Visualizing Quaternions", Elsevier: Morgan Kaufmann; San Francisco. ISBN 0-12-088400-3"

Two reminders:

1. Each of The algebras of Complex Numbers, Quaternions, Octonions, Sedenions etc (or 2n-ons) are generalizations algebras of the previous in the list and they contain all the previous ones as sub-algebras, while all are special cases of Cayley-Dixon Algerbas or Hyperlinear Algebras as some are calling them.

2. Although Quaternions are attributed to Hamilton (1843) it was C.F. Gauss who first discovered them earlier (1819) but published his work in 1900. Also, according to this http://www.rexresearch.com/maxwell.htm in 1864 J.C. Maxwell in his first paper (before the book) on Electromagnetism (James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field" ( Royal Society Transactions, Vol. CLV, 1865, p 459 );) formulated Electromagnetism in terms of 20 quaternion equations that also appeared in the 1873 edition of "A Treatise on Electricity & Magnetism" aka The BOOK.

Fully unambiguous interpretation of complex numbers and their 3D extension - quaternions, is only possible in terms of Geometric Algebra. For "complex" numbers the playground is a plane. "Complex" number is sum of two objects, scalar and oriented area on the plane. Orientation is defined by one of two possible options to move along are boundary. Sum means that you create a couple of geometrically not similar objects. Quaternions are the same couples though placed in 3D. Oriented plane area is spanned in basis planes of 3D. All that is explained in detail, for example, in http://arxiv.org/abs/1309.5002 or A. Soiguine, "Complex Conjugation - Realtive to What?," in Clifford Algebras with Numeric and Symbolic Computations, Boston, Birkhauser, 1996, pp. 285-294

Thank you Alex.

The reason Quaternions found applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, and orbital mechanics, is exactly their two distinct geometric interpretations that you are asking about. A. They can represent rotations in 3 and 4 space (and are called Rotation quaternions in this case) and B. They can represent an orientation (rotation relative to a reference position) in 3D space, they are called orientation quaternions or attitude quaternions. Algorithmically they require a less number of operations to implement rotations (when compared with the multiplying a vector with a rotation matrix that involves Euler angles). Any rotation of a 4D vector can be represented by the product of two Quaternions with that vector. Two excellent references describing these details from an application perspective are:

1. "Kuipers, Jack (2002), "Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality" (reprint edition), Princeton University Press. ISBN 0-691-10298-8" and

2. "Hanson, Andrew J. (2006), "Visualizing Quaternions", Elsevier: Morgan Kaufmann; San Francisco. ISBN 0-12-088400-3"

Two reminders:

1. Each of The algebras of Complex Numbers, Quaternions, Octonions, Sedenions etc (or 2n-ons) are generalizations algebras of the previous in the list and they contain all the previous ones as sub-algebras, while all are special cases of Cayley-Dixon Algerbas or Hyperlinear Algebras as some are calling them.

2. Although Quaternions are attributed to Hamilton (1843) it was C.F. Gauss who first discovered them earlier (1819) but published his work in 1900. Also, according to this http://www.rexresearch.com/maxwell.htm in 1864 J.C. Maxwell in his first paper (before the book) on Electromagnetism (James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field" ( Royal Society Transactions, Vol. CLV, 1865, p 459 );) formulated Electromagnetism in terms of 20 quaternion equations that also appeared in the 1873 edition of "A Treatise on Electricity & Magnetism" aka The BOOK.

Thank you Pr. Schmeikal

The new paper entitled "New fundamental parameters for attitude representation" helps define quaternion geometry in 3-dimensional space.  See the link.  - Russ

http://authors.elsevier.com/ sd/article/S0273117717303046

http://authors.elsevier.com/sd/article/S0273117717303046

In their paper "The geometric structure of unit dual quaternion with application in kinematic control" - 2012, Wang et al. investigated  the geometric structure, especially the Lie-group related properties, of unit dual quaternion.

A unit dual quaternion is a composition of a unit quaternion and a translational vector able to represent an arbitrary transformation (including rotation and translation) in the 3-D space globally without singularity.

Follow ttps://www.researchgate.net/profile/Xiangke_Wang/publication/256760046_The_geometric_structure_of_unit_dual_quaternion_with_application_in_kinematic_control/links/0deec5318885e2c6e3000000.pdf

Article The geometric structure of unit dual quaternion with applica...