Minkowski geometric algebra of quaternion sets. (English) Zbl 1033.53004

Weiner, Ian; Gu, Weiqing

Int. J. Pure Appl. Math. 3, No. 4, 385-411 (2002).

As a natural extension of interval-arithmetic methods for monitoring propagation of errors or uncertainties in real-number computations, R. T. Farouki, H. P. Moon and B. Ravani [Geom. Dedicata 85, 283–315 (2001; Zbl 0987.51012)] introduced a geometric algebra of point sets in the complex plane which is based on the fundamental operations of forming Minkowski sums and Minkowski products, that is, for two sets one forms the sets of all sums or products, respectively, of elements, one from each operand.
In the paper under review the authors extend the definitions and methods of Farouki et al. from the complex numbers to the quaternions. Most rules remain valid but extra care has to be taken due to the non-commutativity of quaternion multiplication. Some geometric interpretations of Minkowski products are provided and simple examples relating to bodies of revolution with radial symmetry axes are given. In the last section of the paper the authors propose a generalisation which can be used to construct a large class of surfaces and solids, including all canal surfaces and ruled surfaces.

Reviewer: Günter F. Steinke (Christchurch)

MSC:

53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
20G20	Linear algebraic groups over the reals, the complexes, the quaternions
51B20	Minkowski geometries in nonlinear incidence geometry

Keywords:

Minkowski geometric algebra; quaternions; canal surfaces; ruled surfaces

Citations:

Zbl 0987.51012